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1508.00629v2.A_Critical_Analysis_of_the_Feasibility_of_Pure_Strain_Actuated_Giant_Magnetostrictive_Nanoscale_Memories.pdf

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

1610.04598v2.Nambu_mechanics_for_stochastic_magnetization_dynamics.pdf

arXiv:1610.04598v2 [cond-mat.mes-hall] 19 Jan 2017Nambu mechanics for stochastic magnetization dynamics Pascal Thibaudeaua,∗, Thomas Nusslea,b, Stam Nicolisb aCEA DAM/Le Ripault, BP 16, F-37260, Monts, FRANCE bCNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350), F´ ed´ eration de Recherche ”Denis Poisson” (FR2964), D´ epartement de Physi que, Universit´ e de Tours, Parc de Grandmont, F-37200, Tours, FRANCE Abstract The Landau-Lifshitz-Gilbert (LLG) equation describes the dynamic s of a damped magnetization vector that can be understood as a generalization o f Larmor spin precession. The LLG equation cannot be deduced from the Hamilton ian frame- work, by introducing a coupling to a usual bath, but requires the int roduction of additional constraints. It is shown that these constraints can be formulated ele- gantly and consistently in the framework of dissipative Nambu mecha nics. This has many consequences for both the variational principle and for t opological as- pects of hidden symmetries that control conserved quantities. W e particularly study how the damping terms of dissipative Nambu mechanics affect t he con- sistent interaction of magnetic systems with stochastic reservoir s and derive a master equation for the magnetization. The proposals are suppor ted by numer- ical studies using symplectic integrators that preserve the topolo gical structure of Nambu equations. These results are compared to computations performed by direct sampling of the stochastic equations and by using closure a ssumptions for the moment equations, deduced from the master equation. Keywords: Magnetization dynamics, Fokker-Planck equation, magnetic ordering ∗Corresponding author Email addresses: pascal.thibaudeau@cea.fr (Pascal Thibaudeau), thomas.nussle@cea.fr (Thomas Nussle), stam.nicolis@lmpt.univ-tours.fr (Stam Nicolis) Preprint submitted to Elsevier September 18, 20181. Introduction In micromagnetism, the transverse Landau-Lifshitz-Gilbert (LLG ) equation (1 +α2)∂si ∂t=ǫijkωj(s)sk+α(ωi(s)sjsj−ωj(s)sjsi) (1) describes the dynamics of a magnetization vector s≡M/MswithMsthe sat- uration magnetization. This equation can be seen as a generalization of Larmor spin precession, for a collection of elementary classical magnets ev olving in an effective pulsation ω=−1 ¯hδH δs=γBand within a magnetic medium, charac- terized by a damping constant αand a gyromagnetic ratio γ[1].His here identified as a scalar functional of the magnetization vector and ca n be consis- tently generalized to include spatial derivatives of the magnetizatio n vector [2] as well. Spin-transfer torques, that are, nowadays, of particula r practical rele- vance [3, 4] can be, also, taken into account in this formalism. In the following, we shall work in units where ¯ h= 1, to simplify notation. It is well known that this equation cannot be derived from a Hamiltonia n variational principle, with the damping effects described by coupling t he magne- tization to a bath, by deforming the Poisson bracket of Hamiltonian m echanics, even though the Landau–Lifshitz equation itself is Hamiltonian. The r eason is that the damping cannot be described by a “scalar” potential, but b y a “vector” potential. This has been made manifest [5] first by an analysis of the quantum ve rsion of the Landau-Lifshitz equation for damped spin motion including arb itrary spin length, magnetic anisotropy and many interacting quantum spin s. In par- ticular, this analysis has revealed that the damped spin equation of m otion is an example of metriplectic dynamical system [6], an approach which t ries to unite symplectic, nondissipative and metric, dissipative dynamics into one com- mon mathematical framework. This dissipative system has been see n afterwards nothing but a natural combination of semimetric dynamics for the dis sipative part and Poisson dynamics for the conservative ones [7]. As a conse quence, this provided a canonical description for any constrained dissipative sy stems through 2an extension of the concept of Dirac brackets developed originally f or conserva- tive constrained Hamiltonian dynamics. Then, this has culminated rec ently by observing the underlying geometrical nature of these brackets a s certain n-ary generalizations of Lie algebras, commonly encountered in conserva tive Hamilto- nian dynamics [8]. However, despite the evident progresses obtaine d, no clear direction emerges for the case of dissipative n-ary generalizations, and even no variational principle have been formulated, to date, that incorp orates such properties. What we shall show in this paper is that it is, however, possible to de- scribe the Landau–Lifshitz–Gilbert equation by using the variationa l principle of Nambu mechanics and to describe the damping effects as the resu lt of in- troducing dissipation by suitably deforming the Nambu–instead of th e Poisson– bracket. In this way we shall find, as a bonus, that it is possible to de duce the relation between longitudinal and transverse damping of the ma gnetization, when writing the appropriate master equation for the probability de nsity. To achieve this in a Hamiltonian formalism requires additional assumptions , whose provenance can, thus, be understood as the result of the prope rties of Nambu mechanics. We focus here on the essential points; a fuller account will be pro- vided in future work. Neglecting damping effects, if one sets H1≡ −ω·sandH2≡s·s/2, eq.(1) can be recast in the form ∂si ∂t={si,H1,H2}, (2) where for any functions A,B,Cofs, {A,B,C} ≡ǫijk∂A ∂si∂B ∂sj∂C ∂sk(3) is the Nambu-Poisson (NP) bracket, or Nambu bracket, or Nambu t riple bracket, a skew-symmetric object, obeying both the Leibniz rule and the Fun damental Identity [9, 10]. One can see immediately that both H1andH2are constants of motion, because of the anti-symmetric property of the bracket. This provides the generalization of Hamiltonian mechanics to phase spaces of arbitrar y dimension; 3in particular it does not need to be even. This is a way of taking into acc ount constraints and provides a natural framework for describing the magnetization dynamics, since the magnetization vector has, in general, three co mponents. The constraints–and the symmetries–can be made manifest, by no ting that it is possible to express vectors and vector fields in, at least, two wa ys, that can be understood as special cases of Hodge decomposition. For the three–dimensional case that is of interest here, this mean s that a vector field V(s) can be expressed in the “Helmholtz representation” [11] in the following way Vi≡ǫijk∂Ak ∂sj+∂Φ ∂si(4) whereAis a vector potential and Φ a scalar potential. On the other hand, this same vector field V(s) can be decomposed according to the “Monge representation” [12] Vi≡∂C1 ∂si+C2∂C3 ∂si(5) which defines the “Clebsch-Monge potentials”, Ci. If one identifies as the Clebsch–Monge potentials, C2≡H1,C3≡H2and C1≡D, Vi=∂D ∂si+H1∂H2 ∂si, (6) and the vector field V(s)≡˙s, then one immediately finds that eq. (2) takes the form ∂s ∂t={s,H1,H2}+∇sD (7) that identifies the contribution of the dissipation in this context, as the expected generalization from usual Hamiltonian mechanics. In the absence of the Gilbert term, dissipation is absent. More generally, the evolution equation for any function, F(s) can be written as [13] ∂F ∂t={F,H1,H2}+∂D ∂si∂F ∂si(8) for a dissipation function D(s). 4The equivalence between the Helmholtz and the Monge representat ion im- plies the existence of freedom of redefinition for the potentials, CiandDand Aiand Φ. This freedom expresses the symmetry under symplectic tra nsforma- tions, that can be interpreted as diffeomorphism transformations , that leave the volume invariant. These have consequences for the equations of m otion. For instance, the dissipation described by the Gilbert term in the Lan dau– Lifshitz–Gilbert equation (1) ∂D ∂si≡α(˜ωi(s)sjsj−˜ωj(s)sjsi) (9) cannot be derived from a scalar potential, since the RHS of this expr ession is not curl–free, so the function Don the LHS is not single valued; but it does conserve the norm of the magnetization, i.e. H2. Because of the Gilbert expression, bothωandηare rescaled such as ˜ω≡ω/(1 +α2) andη→η/(1 +α2). So there are two questions: (a) Whether it can lead to stochastic e ffects, that can be described in terms of deterministic chaos and/or (b) Whethe r its effects can be described by a bath of “vector potential” excitations. The fi rst case was described, in outline in ref. [14], where the role of an external to rque was shown to be instrumental; the second will be discussed in detail in the following sections. While, in both cases, a stochastic description, in terms of a probability density on the space of states is the main tool, it is much easier to pre sent for the case of a bath, than for the case of deterministic chaos, which is much more subtle. Therefore, we shall now couple our magnetic moment to a bath of flu ctuating degrees of freedom, that will be described by a stochastic proces s. 2. Nambu dynamics in a macroscopic bath To this end, one couples linearly the deterministic system such as (8) , to a stochastic process, i.e. a noise vector, random in time, labelled ηi(t), whose law of probability is given. This leads to a system of stochastic differen tial equations, that can be written in the Langevin form ∂si ∂t={si,H1,H2}+∂D ∂si+eij(s)ηj(t) (10) 5whereeij(s) can be interpreted as the vielbein on the manifold, defined by the dynamical variables, s. It should be noted that it is the vector nature of the dynamical variables that implies that the vielbein, must, also, carry in dices. We may note that the additional noise term can be used to “renorma lize” the precession frequency and, thus, mix, non-trivially, with the Gilb ert term. This means that, in the presence of either, the other cannot be ex cluded. When this vielbein is the identity matrix, eij(s) =δij, the stochastic cou- pling to the noise is additive, whereas it is multiplicative otherwise. In th at case, if the norm of the spin vector has to remain constant in time, t hen the gradient of H2must be orthogonal to the gradient of Dandeij(s)si= 0∀j. However, it is important to realize that, while the Gilbert dissipation te rm is not a gradient, the noise term, described by the vielbein is not so co nstrained. For additive noise, indeed, it is a gradient, while for the case of multiplic ative noise studied by Brown and successors there can be an interesting interference between the two terms, that is worth studying in more detail, within N ambu mechanics, to understand, better, what are the coordinate art ifacts and what are the intrinsic features thereof. Because {s(t)}, defined by the eq.(10), becomes a stochastic process, we can define an instantaneous conditional probability distribution Pη(s,t), that depends, on the noise configuration and, also, on the magnetizatio ns0at the initial time and which satisfies a continuity equation in configuration sp ace ∂Pη(s,t) ∂t+∂( ˙siPη(s,t))) ∂si= 0. (11) An equation for /an}b∇acketle{tPη/an}b∇acket∇i}htcan be formed, which becomes an average over all the possible realizations of the noise, namely ∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂t+∂/an}b∇acketle{t˙siPη/an}b∇acket∇i}ht ∂si= 0, (12) once the distribution law of {η(t)}is provided. It is important to stress here that this implies that the backreaction of the spin degrees of freed om on the bath can be neglected–which is by no means obvious. One way to chec k this is by showing that no “runaway solutions” appear. This, however, do es not ex- 6haust all possibilities, that can be found by working with the Langevin equation directly. For non–trivial vielbeine, however, this is quite involved, so it is useful to have an approximate solution in hand. To be specific, we consider a noise, described by the Ornstein-Uhlen beck process [15] of intensity ∆ and autocorrelation time τ, /an}b∇acketle{tηi(t)/an}b∇acket∇i}ht= 0 /an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht=∆ τδije−|t−t′| τ where the higher point correlation functions are deduced from Wick ’s theorem and which can be shown to become a white noise process, when τ→0. We assume that the solution to eq.(12) converges, in the sense of ave rage over-the- noise, to an equilibrium distribution, that is normalizable and, whose co rrelation functions, also, exist. While this is, of course, not at all obvious to p rove, evi- dence can be found by numerical studies, using stochastic integra tion methods that preserve the symplectic structure of the Landau–Lifshitz e quation, even under perturbations (cf. [16] for earlier work). 2.1. Additive noise Walton [17] was one of the first to consider the introduction of an ad ditive noise into an LLG equation and remarked that it may lead to a Fokker- Planck equation, without entering into details. To see this more thoroughly and to illustrate our strategy, we consider the case of additive noise, i.e. w heneij= δijin our framework. By including eq.(10) in (12) and in the limit of white noise, expressions like /an}b∇acketle{tηiPη/an}b∇acket∇i}htmust be defined and can be evaluated by either an expansion of the Shapiro-Loginov formulae of differentiation [18] an d taking the limit ofτ→0, or, directly, by applying the Furutsu-Novikov-Donsker theore m [19, 20, 21]. This leads to /an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂si. (13) 7where ˜∆≡∆/(1 +α2). Using the dampened current vector Ji≡ {si,H1,H2}+ ∂D ∂si, the (averaged) probability density /an}b∇acketle{tPη/an}b∇acket∇i}htsatisfies the following equation ∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂t+∂ ∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂2/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂si∂si= 0 (14) where˜˜∆≡∆/(1 +α2)2and which is of the Fokker-Planck form [22]. This last partial differential equation can be solved directly by several nume rical methods, including a finite-element computer code or can lead to ordinary differ ential equations for the moments of s. For example, for the average of the magnetization, one obtains th e evolution equation d/an}b∇acketle{tsi/an}b∇acket∇i}ht dt=−/integraldisplay dssi∂/an}b∇acketle{tPη(s,t)/an}b∇acket∇i}ht ∂t=/an}b∇acketle{tJi/an}b∇acket∇i}ht. (15) For the case of Landau-Lifshitz-Gilbert in a uniform precession field B, we obtain the following equations, for the first and second moments, d dt/an}b∇acketle{tsi/an}b∇acket∇i}ht=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α[˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht] (16) d dt/an}b∇acketle{tsisj/an}b∇acket∇i}ht= ˜ωl(ǫilk/an}b∇acketle{tsksj/an}b∇acket∇i}ht+ǫjlk/an}b∇acketle{tsksi/an}b∇acket∇i}ht) +α[˜ωi/an}b∇acketle{tslslsj/an}b∇acket∇i}ht + ˜ωj/an}b∇acketle{tslslsi/an}b∇acket∇i}ht−2˜ωl/an}b∇acketle{tslsisj/an}b∇acket∇i}ht] + 2˜˜∆δij (17) where ˜ω≡γB/(1 +α2). In order to close consistently these equations, one can truncate the hierarchy of moments; either on the second /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 or third cumulants /an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e. /an}b∇acketle{tsisj/an}b∇acket∇i}ht=/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht, (18) /an}b∇acketle{tsisjsk/an}b∇acket∇i}ht=/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht+/an}b∇acketle{tsisk/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht+/an}b∇acketle{tsjsk/an}b∇acket∇i}ht/an}b∇acketle{tsi/an}b∇acket∇i}ht −2/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht. (19) Because the closure of the hierarchy is related to an expansion in po wers of ∆, for practical purposes, the validity of eqs.(16,17) is limited to low v alues of the coupling to the bath (that describes the fluctuations). For example, if one sets /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, eq.(16) produces an average spin motion independent of value that ∆ may take. This is in contradiction with the numerical expe riments 8performed by the stochastic integration and noise average of eq.( 10) quoted in reference [23] and by experiments. This means that it is mandatory to keep at least eqs.(16) and (17) together in the numerical evaluation of t he thermal behavior of the dynamics of the average thermal magnetization /an}b∇acketle{ts/an}b∇acket∇i}ht. This was previously observed [24, 25] and circumvented by alternate secon d-order closure relationships, but is not supported by direct numerical experiment s. This can be illustrated by the following figure (1). For this given set of Figure 1: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, connected to an additive noise. The upper graphs (a) plot som e of the first–order moments of the averaged magnetization vector over 102realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text). Parameters of the simulations : {∆ = 0.13 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{ts1(0)s1(0)/an}bracketri}ht= 1. parameters, the agreement between the stochastic average an d the effective model is fairly decent. As expected, for a single noise realization, th e norm of the spin vector in an additive stochastic noise cannot be conserv ed during the dynamics, but, by the average-over-the-noise accumulation process, this is 9observed for very low values of ∆ and very short times. However, t his agreement with the effective equations is lost, when the temperature increase s, because of the perturbative nature of the equations (16-17). Agreement c an, however, be restored by imposing this constraint in the effective equations, for a given order in perturbation of ∆, by appropriate modifications of the hierarchic al closing relationships /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Bij(∆) or/an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Cijk(∆). It is of some interest to study the effects of the choice of initial con ditions. In particular, how the relaxation to equilibrium is affected by choosing a c omponent of the initial magnetization along the precession axis in the effective m odel, e.g. s(0) = (1/√ 2,0,−1/√ 2) and by taking all the initial correlations, /an}b∇acketle{tsi(0)sj(0)/an}b∇acket∇i}ht= 1 20−1 2 0 0 0 −1 201 2 (20) The results are shown in figure (2). Both in figures (1) and (2), it is observed that the average norm of the spin vector increases over time. This can be understood with the above arguments. In general, according to eq.(10) and because Jis a transverse vector, (1 +α2)sidsi dt=eij(s)siηj(t). (21) This equation describes how the LHS depends on the noise realization ; so the average over the noise can be found by computing the averages of the RHS. The simplest case is that of the additive vielbein, eij(s) =δij. Assuming that the average-over-the noise procedure and the time derivative commu te, we have d dt/angbracketleftbig s2/angbracketrightbig =2/an}b∇acketle{tsiηi/an}b∇acket∇i}ht 1 +α2. (22) For any Gaussian stochastic process, the Furutsu-Novikov-Don sker theorem states that /an}b∇acketle{tsi(t)ηi(t)/an}b∇acket∇i}ht=/integraldisplay+∞ −∞dt′/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht/angbracketleftbiggδsi(t) δηj(t′)/angbracketrightbigg . (23) In the most general situation, the functional derivativesδsi(t) δηj(t′)can be calculated [26], and eq.(23) admits simplifications in the white noise limit. In this limit, 10-2-1012 0 1 2 3 4 5 t (ns)-2-1012 sxsy sz(a) (b) Figure 2: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con- nected to an additive noise. The upper graphs (a) plot some of the first–order moments of the averaged magnetization vector over 103realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text). Parameters of the simulations : {∆ = 0.0655 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆ t= 10−4ns,s(0) =/an}bracketle{ts(0)/an}bracketri}ht= (1/√ 2,0,−1/√ 2),/an}bracketle{tsisj/an}bracketri}ht(0) = 0 except for (11)=1/2, (13)=(31)=-1/2, (33)=1/2 }. 11the integration is straightforward and we have /angbracketleftbig s2(t)/angbracketrightbig =s2(0) + 6˜˜∆t, (24) which is a conventional diffusion regime. It is also worth noticing that w hen computing the trace of (17), the only term which remains is indeed d dt/an}b∇acketle{tsisi/an}b∇acket∇i}ht= 6˜˜∆ (25) which allows our effective model to reproduce exactly the diffusion re gime. Fig- ure (3) compares the time evolution of the average of the square n orm spin vector. Numerical stochastic integration of eq.(10) is tested by in creasing the 0 1 2 3 4 5 t (ns)11,522,53 <|s|2>mean over 103 runs mean over 104 runs diffusion regime Figure 3: Mean square norm of the spin in the additive white no ise case for the following conditions: integration step of 10−4ns; ∆ = 0 .0655 rad.GHz; s(0) = (0 ,1,0);α= 0.1; ω= (0,0,18) rad.GHz compared to the expected diffusion regime (see te xt). size of the noise sampling and reveals a convergence to the predicte d linear diffusion regime. 122.2. Multiplicative noise Brown [27] was one of the first to propose a non–trivial vielbein, tha t takes the form eij(s) =ǫijksk/(1 +α2) for the LLG equation. We notice, first of all, that it is present, even if α= 0, i.e. in the absence of the Gilbert term. Also, that, since the determinant of this matrix [ e] is zero, this vielbein is not invertible. Because of its natural transverse character, this vie lbein preserves the norm of the spin for any realization of the noise, once a dissipation fu nctionD is chosen, that has this property. In the white-noise limit, the aver age over-the- noise continuity equation (12) cannot be transformed strictly to a Fokker-Planck form. This time /an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂ ∂sj(eji/an}b∇acketle{tPη/an}b∇acket∇i}ht), (26) which is a generalization of the additive situation shown in eq.(13). The conti- nuity equation thus becomes ∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂t+∂ ∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂ ∂si/parenleftbigg eij∂ ∂sk(ekj/an}b∇acketle{tPη/an}b∇acket∇i}ht)/parenrightbigg = 0. (27) What deserves closer attention is, whether, in fact, this equation is invariant under diffeomeorphisms of the manifold [28] defined by the vielbein, o r whether it breaks it to a subgroup thereof. This will be presented in future w ork. In the context of magnetic thermal fluctuations, this continuity equatio n was encoun- tered several times in the literature [22, 29], but obtaining it from fir st principles is more cumbersome than our latter derivation, a remark already qu oted [18]. Moreover, our derivation presents the advantage of being easily g eneralizable to non-Markovian noise distributions [23, 30, 31], by simply keeping th e partial derivative equation on the noise with the continuity equation, and so lving them together. Consequently, the evolution equation for the average magnetizat ion is now supplemented by a term provided by a non constant vielbein and one h as d/an}b∇acketle{tsi/an}b∇acket∇i}ht dt=/an}b∇acketle{tJi/an}b∇acket∇i}ht+˜˜∆/angbracketleftbigg∂eil ∂skekl/angbracketrightbigg . (28) With the vielbein proposed by Brown and assuming a constant extern al field, 13one gets d/an}b∇acketle{tsi/an}b∇acket∇i}ht dt=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α(˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht) −2∆ (1 +α2)2/an}b∇acketle{tsi/an}b∇acket∇i}ht. (29) This equation highlights both a transverse part, coming from the av erage over the probability current Jand a longitudinal part, coming from the average over the extra vielbein term. By imposing, further, the second-or der cumulant approximation /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e. “small” fluctuations to keep the distribution of sgaussian, a single equation can be obtained, in which a longitudinal rela xation timeτL≡(1 +α2)2/2∆ may be identified. This is illustrated by the content of figure (4). In that case, the ap proxima- Figure 4: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con- nected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments of the averaged magnetization vector over 102realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param- eters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{tsx(0)sx(0)/an}bracketri}ht= 1. 14tion/an}b∇acketle{t/an}b∇acketle{tsisjsj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 has been retained in order to keep two sets of equations, three for the average magnetization components and nine on the averag e second-order moments, that have been solved simultaneously using an eight-orde r Runge- Kutta algorithm with variable time-steps. This is the same numerical im ple- mentation that has been followed for the studies of the additive nois e, solving eqs.(16) and (17) simultaneously. We have observed numerically tha t, as ex- pected, the average second-order moments are symmetrical by an exchange of their component indices, both for the multiplicative and the additive n oise. In- terestingly, by keeping identical the number of random events tak en to evaluate the average of the stochastic magnetization dynamics between th e additive and multiplicative noise, we observe a greater variance in the multiplicative case. As we have done in the additive noise case, we will also investigate briefl y the behavior of this equation under different initial conditions, and in par ticular with a non vanishing component along the z-axis. This is illustrated by the c ontent of figure (5). It is observed that for both figures (4) and (5), th e average spin converges to the same final equilibrium state, which depends ultimat ely on the value of the noise amplitude, as shown by equation (27). 3. Discussion Magnetic systems describe vector degrees of freedom, whose Ha miltonian dynamics implies constraints. These constraints can be naturally ta ken into account within Nambu mechanics, that generalizes Hamiltonian mecha nics to phase spaces of odd number of dimensions. In this framework, diss ipation can be described by gradients that are not single–valued and thus do no t define scalar baths, but vector baths, that, when coupled to external torques, can lead to chaotic dynamics. The vector baths can, also, describe non-tr ivial geometries and, in that case, as we have shown by direct numerical study, the stochastic description leads to a coupling between longitudinal and transverse relaxation. This can be, intuitively, understood within Nambu mechanics, in the fo llowing way: 15-1-0.500.51 0 1 2 3 4 5 t (ns)-1-0.500.51 sxsy sz(a) (b) Figure 5: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con- nected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments of the averaged magnetization vector over 104realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param- eters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆t= 10−4ns}. Initial conditions: s(0) =/parenleftbig 1/√ 2,0,1/√ 2/parenrightbig ,/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 except for /an}bracketle{ts1(0)s1(0)/an}bracketri}ht=/an}bracketle{ts1(0)s3(0)/an}bracketri}ht=/an}bracketle{ts3(0)s3(0)/an}bracketri}ht= 1/2. 16The dynamics consists in rendering one of the Hamiltonians, H1≡ω·s, stochastic, since ωbecomes a stochastic process, as it is sensitive to the noise terms–whether these are described by Gilbert dissipation or couplin g to an external bath. Through the Nambu equations, this dependence is “transferred” toH2≡ ||s||2/2. This is one way of realizing the insights the Nambu approach provides. In practice, we may summarize our numerical results as follows: When the amplitude of the noise is small, in the context of Langevin- dynamics formalism for linear systems and for the numerical modeling ofsmall thermal fluctuations in micromagnetic systems, as for a linearized s tochastic LLG equation, the rigorous method of Lyberatos, Berkov and Cha ntrell might be thought to apply [32] and be expected to be equivalent to the app roach presented here. Because this method expresses the approach t o equilibrium of every moment, separately, however, it is restricted to the limit of s mall fluctua- tions around an equilibrium state and, as expected, cannot captur e the transient regime of average magnetization dynamics, even for low temperatu re. This is a useful check. We have also investigated the behaviour of this system under differe nt sets of initial conditions as it is well-known and has been thoroughly studied in [ 1] that in the multiplicative noise case (where the norm is constant) this syst em can show strong sensitivity to initial conditions and it is possible, using ste reographic coordinates to represent the dynamics of this system in 2D. In our additive noise case however, as the norm of the spin is not conserved, it is not eas y to get long run behavior of our system and in particular equilibrium solutions. Mor eover as we no longer have only two independent components of spin, it is not p ossible to obtain a 2D representation of our system and makes it more comp licated to study maps displaying limit cycles, attractors and so on. Thus under standing the dynamics under different initial conditions would require somethin g more and, as it is beyond the scope of this work, will be done elsewhere. Therefore, we have focused on studying the effects of the prese nce of an initial longitudinal component and of additional, diagonal, correlation s. No 17differences have been observed so far. Another issue, that deserves further study, is how the probabilit y density of the initial conditions is affected by the stochastic evolution. In th e present study we have taken the initial probability density to be a δ−function; so it will be of interest to study the evolution of other initial distributions in d etail, in particular, whether the averaging procedures commute–or not. In general, we expect that they won’t. This will be reported in future work. Finally, our study can be readily generalized since any vielbein can be ex - pressed in terms of a diagonal, symmetrical and anti-symmetrical m atrices, whose elements are functions of the dynamical variable s. Because ˙sis a pseu- dovector (and we do not consider that this additional property is a cquired by the noise vector), this suggests that the anti-symmetric part of the vielbein should be the “dominant” one. Interestingly, by numerical investigations , it appears that there are no effects, that might depend on the choice of the n oise connection for the stochastic vortex dynamics in two-dimensional easy-plane ferromagnets [33], even if it is known that for Hamiltonian dynamics, multiplicative and a d- ditive noises usually modify the dynamics quite differently, a point that also deserves further study. References [1] Giorgio Bertotti, Isaak D. Mayergoyz, and Claudio Serpico. Nonlinear Magnetization Dynamics in Nanosystems . Elsevier, April 2009. Google- Books-ID: QH4ShV3mKmkC. [2] Amikam Aharoni. Introduction to the Theory of Ferromagnetism . Claren- don Press, 2000. [3] Jacques Miltat, Gon¸ calo Albuquerque, Andr´ e Thiaville, and Caro le Vouille. Spin transfer into an inhomogeneous magnetization distribution. Journal of Applied Physics , 89(11):6982, 2001. [4] Dmitry V. Berkov and Jacques Miltat. Spin-torque driven magnet ization 18dynamics: Micromagnetic modeling. Journal of Magnetism and Magnetic Materials , 320(7):1238–1259, April 2008. [5] Janusz A. Holyst and Lukasz A. Turski. Dissipative dynamics of qu antum spin systems. Physical Review A , 45(9):6180–6184, May 1992. [6] /suppress Lukasz A. Turski. Dissipative quantum mechanics. Metriplectic d ynamics in action. In Zygmunt Petru, Jerzy Przystawa, and Krzysztof Ra pcewicz, editors,From Quantum Mechanics to Technology , number 477 in Lecture Notes in Physics, pages 347–357. Springer Berlin Heidelberg, 1996. [7] Sonnet Q. H. Nguyen and /suppress Lukasz A. Turski. On the Dirac approa ch to constrained dissipative dynamics. Journal of Physics A: Mathematical and General , 34(43):9281–9302, November 2001. [8] Josi A. de Azc´ arraga and Josi M. Izquierdo. N-ary algebras: A review with applications. Journal of Physics A: Mathematical and Theoretical , 43(29):293001, July 2010. [9] Yoichiro Nambu. Generalized Hamiltonian Dynamics. Physical Review D , 7(8):2405–2412, April 1973. [10] Ra´ ul Ib´ a˜ nez, Manuel de Le´ on, Juan C. Marrero, and Dav id Martı´ n de Diego. Dynamics of generalized Poisson and Nambu–Poisson bracket s. Journal of Mathematical Physics , 38(5):2332, 1997. [11] Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and Symmetry , volume 17 of Texts in Applied Mathematics . Springer New York, New York, NY, 1999. [12] Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry . John Wiley & Sons, August 2014. [13] Minos Axenides and Emmanuel Floratos. Strange attractors in dissipative Nambu mechanics: Classical and quantum aspects. Journal of High Energy Physics , 2010(4), April 2010. 19[14] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Quantum Mag- nets and Matrix Lorenz Systems. 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Functionals and the Random-force Method in Tur- bulence Theory. Soviet Physics Journal of Experimental and Theoretical Physics , 20(5):1290–1294, May 1964. [21] Valery I. Klyatskin. Stochastic Equations through the Eye of the Physicist: Basic Concepts, Exact Results and Asymptotic Approximatio ns. Elsevier, Amsterdam, 1 edition, 2005. OCLC: 255242261. [22] Hannes Risken. The Fokker-Planck Equation , volume 18 of Springer Series in Synergetics . Springer-Verlag, Berlin, Heidelberg, 1989. [23] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Closing th e hier- archy for non-Markovian magnetization dynamics. Physica B: Condensed Matter , 486:57–59, April 2016. 20[24] Dmitry A. Garanin. Fokker-Planck and Landau-Lifshitz-Bloch e quations for classical ferromagnets. Physical Review B , 55(5):3050–3057, February 1997. [25] Pui-Wai Ma and Sergei L. Dudarev. Langevin spin dynamics. Physical Review B , 83:134418, April 2011. [26] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. A functio nal calculus for the magnetization dynamics. arXiv:1606.02137 [cond-mat, physics:nlin, physics:physics] , June 2016. [27] William Fuller Brown. Thermal Fluctuations of a Single-Domain Partic le. Physical Review , 130(5):1677–1686, June 1963. [28] Jean Zinn-Justin. QuantumField Theory and Critical Phenomena . Number 113 in International series of monographs on physics. Clarendon P ress, Oxford, 4. ed., reprinted edition, 2011. OCLC: 767915024. [29] Jos´ e Luis Garc´ ıa-Palacios and Francisco J. L´ azaro. Langev in-dynamics study of the dynamical properties of small magnetic particles. Physical Review B , 58(22):14937–14958, December 1998. [30] Pascal Thibaudeau, Julien Tranchida, and Stam Nicolis. Non-Mar kovian Magnetization Dynamics for Uniaxial Nanomagnets. IEEE Transactions on Magnetics , 52(7):1–4, July 2016. [31] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Colored- noise magnetization dynamics: From weakly to strongly correlated noise. 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1708.02008v2.Chiral_damping__chiral_gyromagnetism_and_current_induced_torques_in_textured_one_dimensional_Rashba_ferromagnets.pdf

arXiv:1708.02008v2 [cond-mat.mes-hall] 31 Aug 2017Chiral damping, chiral gyromagnetism and current-induced torques in textured one-dimensional Rashba ferromagnets Frank Freimuth,∗Stefan Bl¨ ugel, and Yuriy Mokrousov Peter Gr¨ unberg Institut and Institute for Advanced Simula tion, Forschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y (Dated: May 17, 2018) We investigate Gilbert damping, spectroscopic gyromagnet ic ratio and current-induced torques in the one-dimensional Rashba model with an additional nonc ollinear magnetic exchange field. We find that the Gilbert damping differs between left-handed and right-handed N´ eel-type magnetic domain walls due to the combination of spatial inversion asy mmetry and spin-orbit interaction (SOI), consistent with recent experimental observations o f chiral damping. Additionally, we find that also the spectroscopic gfactor differs between left-handed and right-handed N´ eel- type domain walls, which we call chiral gyromagnetism. We also investig ate the gyromagnetic ratio in the Rashba model with collinear magnetization, where we find that scatt ering corrections to the gfactor vanish for zero SOI, become important for finite spin-orbit couplin g, and tend to stabilize the gyromagnetic ratio close to its nonrelativistic value. I. INTRODUCTION In magnetic bilayer systems with structural inversion asymmetry the energies of left-handed and right-handed N´ eel-type domain walls differ due to the Dzyaloshinskii- Moriya interaction (DMI) [1–4]. DMI is a chiral interac- tion, i.e., it distinguishes between left-handed and right- handed spin-spirals. Not only the energy is sensitive to the chirality of spin-spirals. Recently, it has been re- ported that the orbital magnetic moments differ as well between left-handed and right-handed cycloidal spin spi- rals in magnetic bilayers [5, 6]. Moreover, the experi- mental observation of asymmetry in the velocity of do- main walls driven by magnetic fields suggests that also the Gilbert damping is sensitive to chirality [7, 8]. In this work we show that additionally the spectro- scopic gyromagnetic ratio γis sensitive to the chirality of spin-spirals. The spectroscopic gyromagnetic ratio γ can be defined by the equation dm dt=γT, (1) whereTis the torque that acts on the magnetic moment mand dm/dtis the resulting rate of change. γenters the Landau-Lifshitz-Gilbert equation (LLG): dˆM dt=γˆM×Heff+αGˆM×dˆM dt,(2) whereˆMis a normalized vector that points in the direc- tionofthemagnetizationandthetensor αGdescribesthe Gilbert damping. The chiralityofthe gyromagneticratio provides another mechanism for asymmetries in domain- wall motion between left-handed and right-handed do- main walls. Not only the damping and the gyromagnetic ratio exhibit chiral corrections in inversion asymmetric sys- tems but also the current-induced torques. Amongthese torques that act on domain-walls are the adia- batic and nonadiabatic spin-transfer torques [9–12] and the spin-orbit torque [13–16]. Based on phenomenologi- cal grounds additional types of torques have been sug- gested [17]. Since this large number of contributions are difficult to disentangle experimentally, current-driven domain-wall motion in inversion asymmetric systems is not yet fully understood. The two-dimensionalRashbamodel with an additional exchange splitting has been used to study spintronics effects associated with the interfaces in magnetic bi- layer systems [18–22]. Recently, interest in the role of DMI in one-dimensional magnetic chains has been trig- gered [23, 24]. For example, the magnetic moments in bi-atomic Fe chains on the Ir surface order in a 120◦ spin-spiral state due to DMI [25]. Apart from DMI, also other chiral effects, such as chiral damping and chiral gyromagnetism, are expected to be important in one- dimensional magnetic chains on heavy metal substrates. The one-dimensional Rashba model [26, 27] with an ad- ditional exchange splitting can be used to simulate spin- orbit driven effects in one-dimensional magnetic wires on substrates [28–30]. While the generalized Bloch theo- rem[31]usuallycannotbeusedtotreatspin-spiralswhen SOI is included in the calculation, the one-dimensional Rashba model has the advantage that it can be solved with the help of the generalized Bloch theorem, or with a gauge-field approach [32], when the spin-spiral is of N´ eel- type. WhenthegeneralizedBlochtheoremcannotbeem- ployed one needs to resort to a supercell approach [33], use open boundary conditions [34, 35], or apply pertur- bation theory [6, 9, 36–39] in order to study spintronics effects in noncollinear magnets with SOI. In the case of the one-dimensional Rashba model the DMI and the ex- changeparameterswerecalculatedbothdirectlybasedon agauge-fieldapproachandfromperturbationtheory[38]. The results from the two approaches were found to be in perfect agreement. Thus, the one-dimensional Rashba2 model provides also an excellent opportunity to verify expressions obtained from perturbation theory by com- parisonto the resultsfromthe generalizedBlochtheorem or from the gauge-field approach. In this work we study chiral gyromagnetism and chi- ral damping in the one-dimensional Rashba model with an additional noncollinear magnetic exchange field. The one-dimensional Rashba model is very well suited to study these SOI-driven chiral spintronics effects, because it can be solved in a very transparent way without the need for a supercell approach, open boundary conditions or perturbation theory. We describe scattering effects by the Gaussian scalar disorder model. To investigate the role of disorder for the gyromagnetic ratio in general, we studyγalso in the two-dimensional Rashba model with collinear magnetization. Additionally, we compute the current-induced torques in the one-dimensional Rashba model. This paper is structured as follows: In section IIA we introduce the one-dimensional Rashba model. In sec- tion IIB we discuss the formalism for the calculation of the Gilbert damping and of the gyromagnetic ratio. In section IIC we present the formalism used to calcu- late the current-induced torques. In sections IIIA, IIIB, and IIIC we discuss the gyromagnetic ratio, the Gilbert damping, and the current-induced torques in the one- dimensionalRashbamodel, respectively. Thispaperends with a summary in section IV. II. FORMALISM A. One-dimensional Rashba model The two-dimensional Rashba model is given by the Hamiltonian [19] H=−/planckover2pi12 2me∂2 ∂x2−/planckover2pi12 2me∂2 ∂y2+ +iαRσy∂ ∂x−iαRσx∂ ∂y+∆V 2σ·ˆM(r),(3) where the first line describes the kinetic energy, the first twotermsin thesecondline describethe RashbaSOI and the last term in the second line describes the exchange splitting. ˆM(r) is the magnetization direction, which may depend on the position r= (x,y), andσis the vector of Pauli spin matrices. By removing the terms with the y-derivatives from Eq. (3), i.e., −/planckover2pi12 2me∂2 ∂y2and −iαRσx∂ ∂y, one obtains a one-dimensional variant of the Rashba model with the Hamiltonian [38] H=−/planckover2pi12 2me∂2 ∂x2+iαRσy∂ ∂x+∆V 2σ·ˆM(x).(4) Eq. (4) is invariant under the simultaneous rotation ofσand of the magnetization ˆMaround the yaxis.Therefore, if ˆM(x) describes a flat cycloidal spin-spiral propagating into the xdirection, as given by ˆM(x) = sin(qx) 0 cos(qx) , (5) we can use the unitary transformation U(x) =/parenleftBigg cos(qx 2)−sin(qx 2) sin(qx 2) cos(qx 2)/parenrightBigg (6) in order to transform Eq. (4) into a position-independent effective Hamiltonian [38]: H=1 2m/parenleftbig px+eAeff x/parenrightbig2−m(αR)2 2/planckover2pi12+∆V 2σz,(7) wherepx=−i/planckover2pi1∂/∂xis thexcomponent of the momen- tum operator and Aeff x=−m e/planckover2pi1/parenleftbigg αR+/planckover2pi12 2mq/parenrightbigg σy (8) is thex-component of the effective magnetic vector po- tential. Eq. (8) shows that the noncollinearity described byqacts like an effective SOI in the special case of the one-dimensional Rashba model. This suggests to intro- duce the concept of effective SOI strength αR eff=αR+/planckover2pi12 2mq. (9) Based on this concept of the effective SOI strength one can obtain the q-dependence of the one-dimensional Rashba model from its αR-dependence at q= 0. That a noncollinear magnetic texture provides a nonrelativistic effective SOI has been found also in the context of the intrinsic contribution to the nonadiabatic torque in the absence of relativistic SOI, which can be interpreted as a spin-orbit torque arising from this effective SOI [40]. While the Hamiltonian in Eq. (4) depends on position xthrough the position-dependence of the magnetization ˆM(x) in Eq. (5), the effective Hamiltonian in Eq. (7) is not dependent on xand therefore easy to diagonalize. B. Gilbert damping and gyromagnetic ratio In collinear magnets damping and gyromagnetic ratio can be extracted from the tensor [16] Λij=−1 Vlim ω→0ImGR Ti,Tj(/planckover2pi1ω) /planckover2pi1ω, (10) whereVis the volume of the unit cell and GR Ti,Tj(/planckover2pi1ω) =−i∞/integraldisplay 0dteiωt/angbracketleft[Ti(t),Tj(0)]−/angbracketright(11)3 is the retarded torque-torque correlation function. Tiis thei-th component of the torque operator [16]. The dc- limitω→0 in Eq. (10) is only justified when the fre- quency of the magnetization dynamics, e.g., the ferro- magnetic resonance frequency, is smaller than the relax- ationrateoftheelectronicstates. In thin magneticlayers and monoatomicchains on substratesthis is typically the case due to the strong interfacial disorder. However, in very pure crystalline samples at low temperatures the relaxation rate may be smaller than the ferromagnetic resonance frequency and one needs to assume ω >0 in Eq. (10) [41, 42]. The tensor Λdepends on the mag- netization direction ˆMand we decompose it into the tensorS, which is even under magnetization reversal (S(ˆM) =S(−ˆM)), and the tensor A, which is odd un- der magnetization reversal ( A(ˆM) =−A(−ˆM)), such thatΛ=S+A, where Sij(ˆM) =1 2/bracketleftBig Λij(ˆM)+Λij(−ˆM)/bracketrightBig (12) and Aij(ˆM) =1 2/bracketleftBig Λij(ˆM)−Λij(−ˆM)/bracketrightBig .(13) One can show that Sis symmetric, i.e., Sij(ˆM) = Sji(ˆM), while Ais antisymmetric, i.e., Aij(ˆM) = −Aji(ˆM). The Gilbert damping may be extracted from the sym- metric component Sas follows [16]: αG ij=|γ|Sij Mµ0, (14) whereMis the magnetization. The gyromagnetic ratio γis obtained from Λ according to the equation [16] 1 γ=1 2µ0M/summationdisplay ijkǫijkΛijˆMk=1 2µ0M/summationdisplay ijkǫijkAijˆMk. (15) It is convenient to discuss the gyromagnetic ratio in terms of the dimensionless g-factor, which is related to γthrough γ=gµ0µB//planckover2pi1. Consequently, the g-factor is given by 1 g=µB 2/planckover2pi1M/summationdisplay ijkǫijkΛijˆMk=µB 2/planckover2pi1M/summationdisplay ijkǫijkAijˆMk.(16) Due to the presence of the Levi-Civita tensor ǫijkin Eq. (15) and in Eq. (16) the gyromagnetic ratio and the g-factoraredetermined solelyby the antisymmetriccom- ponentAofΛ. Various different conventions are used in the literature concerning the sign of the g-factor [43]. Here, we define the sign of the g-factor such that γ >0 forg >0 and γ <0 forg <0. According to Eq. (1) the rate of change ofthemagneticmomentisthereforeparalleltothetorqueforpositive gandantiparalleltothetorquefornegative g. While we are interested in this work in the spectroscopic g-factor, and hence in the relation between the rate of change of the magnetic moment and the torque, Ref. [43] discusses the relation between the magnetic moment m andtheangularmomentum Lthatgeneratesit, i.e., m= γstaticL. Since differentiation with respect to time and use ofT= dL/dtleads to Eq. (1) our definition of the signs ofgandγagrees essentially with the one suggested in Ref. [43], which proposes to use a positive gwhen the magnetic moment is parallel to the angular momentum generatingitandanegative gwhenthemagneticmoment is antiparallel to the angular momentum generating it. Combining Eq. (14) and Eq. (15) we can express the Gilbert damping in terms of AandSas follows: αG xx=Sxx |Axy|. (17) IntheindependentparticleapproximationEq.(10)can be written as Λij= ΛI(a) ij+ΛI(b) ij+ΛII ij, where ΛI(a) ij=1 h/integraldisplayddk (2π)dTr/angbracketleftbig TiGR k(EF)TjGA k(EF)/angbracketrightbig ΛI(b) ij=−1 h/integraldisplayddk (2π)dReTr/angbracketleftbig TiGR k(EF)TjGR k(EF)/angbracketrightbig ΛII ij=1 h/integraldisplayddk (2π)d/integraldisplayEF −∞dEReTr/angbracketleftbigg TiGR k(E)TjdGR k(E) dE − TidGR k(E) dETjGR k(E)/angbracketrightbigg .(18) Here,dis the dimension ( d= 1 ord= 2 ord= 3),GR k(E) is the retarded Green’s function and GA k(E) = [GR k(E)]†. EFis the Fermi energy. ΛI(b) ijis symmetric under the interchange of the indices iandjwhile ΛII ijis antisym- metric. The term ΛI(a) ijcontains both symmetric and antisymmetric components. Since the Gilbert damping tensor is symmetric, both ΛI(b) ijand ΛI(a) ijcontribute to it. Since the gyromagnetic tensor is antisymmetric, both ΛII ijand ΛI(a) ijcontribute to it. In order to account for disorder we use the Gaus- sian scalardisordermodel, wherethe scatteringpotential V(r) satisfies /angbracketleftV(r)/angbracketright= 0 and /angbracketleftV(r)V(r′)/angbracketright=Uδ(r−r′). This model is frequently used to calculate transport properties in disordered multiband model systems [44], but it has also been combined with ab-initio electronic structure calculations to study the anomalous Hall ef- fect [45, 46] and the anomalous Nernst effect [47] in tran- sition metals and their alloys. In the clean limit, i.e., in the limit U→0, the an- tisymmetric contribution to Eq. (18) can be written as4 Aij=Aint ij+Ascatt ij, where the intrinsic part is given by Aint ij=/planckover2pi1/integraldisplayddk (2π)d/summationdisplay n,m[fkn−fkm]ImTi knmTj kmn (Ekn−Ekm)2 = 2/planckover2pi1/integraldisplayddk (2π)d/summationdisplay n/summationdisplay ll′fknIm/bracketleftbigg∂/angbracketleftukn| ∂ˆMl∂|ukn/angbracketright ∂ˆMl′/bracketrightbigg × ×/summationdisplay mm′ǫilmǫjl′m′ˆMmˆMm′. (19) The second line in Eq. (19) expresses Aint ijin terms of the Berry curvature in magnetization space [48]. The scattering contribution is given by Ascatt ij=/planckover2pi1/summationdisplay nm/integraldisplayddk (2π)dδ(EF−Ekn)Im/braceleftBigg −/bracketleftbigg Mi knmγkmn γknnTj knn−Mj knmγkmn γknnTi knn/bracketrightbigg +/bracketleftBig Mi kmn˜Tj knm−Mj kmn˜Ti knm/bracketrightBig −/bracketleftbigg Mi knmγkmn γknn˜Tj knn−Mj knmγkmn γknn˜Ti knn/bracketrightbigg +/bracketleftBigg ˜Ti knnγknm γknn˜Tj kmn Ekn−Ekm−˜Tj knnγknm γknn˜Ti kmn Ekn−Ekm/bracketrightBigg +1 2/bracketleftbigg ˜Ti knm1 Ekn−Ekm˜Tj kmn−˜Tj knm1 Ekn−Ekm˜Ti kmn/bracketrightbigg +/bracketleftBig Tj knnγknm γknn1 Ekn−Ekm˜Ti kmn −Ti knnγknm γknn1 Ekn−Ekm˜Tj kmn/bracketrightBig/bracerightBigg . (20) Here,Ti knm=/angbracketleftukn|Ti|ukm/angbracketrightare the matrix elements of the torque operator. ˜Ti knmdenotes the vertex corrections of the torque, which solve the equation ˜Ti knm=/summationdisplay p/integraldisplaydnk′ (2π)n−1δ(EF−Ek′p) 2γk′pp× ×/angbracketleftukn|uk′p/angbracketright/bracketleftBig ˜Ti k′pp+Ti k′pp/bracketrightBig /angbracketleftuk′p|ukm/angbracketright.(21) The matrix γknmis given by γknm=−π/summationdisplay p/integraldisplayddk′ (2π)dδ(EF−Ek′p)/angbracketleftukn|uk′p/angbracketright/angbracketleftuk′p|ukm/angbracketright (22) and the Berry connection in magnetization space is de- fined as iMj knm=iTj knm Ekm−Ekn. (23) The scattering contribution Eq. (20) formally resembles the side-jump contribution to the AHE [44] as obtainedfrom the scalar disorder model: It can be obtained by replacing the velocity operators in Ref. [44] by torque operators. We find thatin collinearmagnetswithoutSOI this scattering contribution vanishes. The gyromagnetic ratio is then given purely by the intrinsic contribution Eq. (19). This is an interesting difference to the AHE: Without SOI all contributions to the AHE are zero in collinear magnets, while both the intrinsic and the side- jump contributions are generally nonzero in the presence of SOI. In the absence of SOI Eq. (19) can be expressed in terms of the magnetization [48]: Aint ij=−/planckover2pi1 2µB/summationdisplay kǫijkMk. (24) Inserting Eq. (24) into Eq. (16) yields g=−2, i.e., the expected nonrelativistic value of the g-factor. Theg-factor in the presence of SOI is usually assumed to be given by [49] g=−2Mspin+Morb Mspin=−2M Mspin,(25) whereMorbis the orbital magnetization, Mspinis the spin magnetization and M=Morb+Mspinis the total magnetization. The g-factor obtained from Eq. (25) is usually in good agreementwith experimental results [50]. When SOI is absent, the orbital magnetization is zero, Morb= 0, and consequently Eq. (25) yields g=−2 in that case. Eq. (16) can be rewritten as 1 g=Mspin MµB 2/planckover2pi1Mspin/summationdisplay ijkǫijkAijˆMk=Mspin M1 g1,(26) with 1 g1=µB 2/planckover2pi1Mspin/summationdisplay ijkǫijkAijˆMk. (27) From the comparison of Eq. (26) with Eq. (25) it follows that Eq. (25) holds exactly if g1=−2 is satisfied. How- ever, Eq. (27) usually yields g1=−2 only in collinear magnets when SOI is absent, otherwise g1/negationslash=−2. In the one-dimensionalRashbamodel the orbitalmagnetization is zero,Morb= 0, and consequently 1 g=µB 2/planckover2pi1Mspin/summationdisplay ijkǫijkAijˆMk. (28) The symmetric contribution can be written as Sij= Sint ij+SRR−vert ij+SRA−vert ij, where Sint ij=1 h/integraldisplayddk (2π)dTr/braceleftbig TiGR k(EF)Tj/bracketleftbig GA k(EF)−GR k(EF)/bracketrightbig/bracerightbig (29)5 and SRR−vert ij=−1 h/integraldisplayddk (2π)dTr/braceleftBig ˜TRR iGR k(EF)TjGR k(EF)/bracerightBig (30) and SAR−vert ij=1 h/integraldisplayddk (2π)dTr/braceleftBig ˜TAR iGR k(EF)TjGA k(EF)/bracerightBig , (31) whereGR k(EF) =/planckover2pi1[EF−Hk−ΣR k(EF)]−1is the retarded Green’s function, GA k(EF) =/bracketleftbig GR k(EF)/bracketrightbig†is the advanced Green’s function and ΣR(EF) =U /planckover2pi1/integraldisplayddk (2π)dGR k(EF) (32) is the retarded self-energy. The vertex corrections are determined by the equations ˜TAR=T+U /planckover2pi12/integraldisplayddk (2π)dGA k(EF)˜TAR kGR k(EF) (33) and ˜TRR=T+U /planckover2pi12/integraldisplayddk (2π)dGR k(EF)˜TRR kGR k(EF).(34) In contrast to the antisymmetric tensor A, which be- comes independent of the scattering strength Ufor suf- ficiently small U, i.e., in the clean limit, the symmetric tensorSdepends strongly on Uin metallic systems in the clean limit. Sint ijandSscatt ijdepend therefore on U through the self-energy and through the vertex correc- tions. In the case of the one-dimensional Rashba model, the equations Eq. (19) and Eq. (20) for the antisymmet- ric tensor Aand the equations Eq. (29), Eq. (30) and Eq. (31) for the symmetric tensor Scan be used both for the collinear magnetic state as well as for the spin- spiral of Eq. (5). To obtain the g-factor for the collinear magnetic state, we plug the eigenstates and eigenvalues of Eq. (4) (with ˆM=ˆez) into Eq. (19) and into Eq. (20). In the case of the spin-spiral of Eq. (5) we use instead the eigenstates and eigenvalues of Eq. (7). Similarly, to ob- tain the Gilbert damping in the collinear magnetic state, we evaluate Eq. (29), Eq. (30) and Eq. (31) based on the Hamiltonian in Eq. (4) and for the spin-spiral we use instead the effective Hamiltonian in Eq. (7). C. Current-induced torques The current-induced torque on the magnetization can be expressed in terms of the torkance tensor tijas [15] Ti=/summationdisplay jtijEj, (35)whereEjis thej-th component of the applied elec- tric field and Tiis thei-th component of the torque per volume [51]. tijis the sum of three terms, tij= tI(a) ij+tI(b) ij+tII ij, where [15] tI(a) ij=e h/integraldisplayddk (2π)dTr/angbracketleftbig TiGR k(EF)vjGA k(EF)/angbracketrightbig tI(b) ij=−e h/integraldisplayddk (2π)dReTr/angbracketleftbig TiGR k(EF)vjGR k(EF)/angbracketrightbig tII ij=e h/integraldisplayddk (2π)d/integraldisplayEF −∞dEReTr/angbracketleftbigg TiGR k(E)vjdGR k(E) dE − TidGR k(E) dEvjGR k(E)/angbracketrightbigg .(36) We decompose the torkance into two parts that are, respectively, even and odd with respect to magnetiza- tion reversal, i.e., te ij(ˆM) = [tij(ˆM) +tij(−ˆM)]/2 and to ij(ˆM) = [tij(ˆM)−tij(−ˆM)]/2. In the clean limit, i.e., for U→0, the even torkance can be written as te ij=te,int ij+te,scatt ij, where [15] te,int ij= 2e/planckover2pi1/integraldisplayddk (2π)d/summationdisplay n/negationslash=mfknImTi knmvj kmn (Ekn−Ekm)2(37) is the intrinsic contribution and te,scatt ij=e/planckover2pi1/summationdisplay nm/integraldisplayddk (2π)dδ(EF−Ekn)Im/braceleftBigg /bracketleftBig −Mi knmγkmn γknnvj knn+Aj knmγkmn γknnTi knn/bracketrightBig +/bracketleftBig Mi kmn˜vj knm−Aj kmn˜Ti knm/bracketrightBig −/bracketleftBig Mi knmγkmn γknn˜vj knn−Aj knmγkmn γknn˜Ti knn/bracketrightBig +/bracketleftBig ˜vj kmnγknm γknn˜Ti nn Ekn−Ekm−˜Ti kmnγknm γknn˜vj knn Ekn−Ekm/bracketrightBig +1 2/bracketleftBig ˜vj knm1 Ekn−Ekm˜Ti kmn−˜Ti knm1 Ekn−Ekm˜vj kmn/bracketrightBig +/bracketleftBig vj knnγknm γknn1 Ekn−Ekm˜Ti kmn −Ti knnγknm γknn1 Ekn−Ekm˜vj kmn/bracketrightBig/bracerightBigg . (38) is the scattering contribution. Here, iAj knm=ivj knm Ekm−Ekn=i /planckover2pi1/angbracketleftukn|∂ ∂kj|ukm/angbracketright(39) is the Berry connection in kspace and the vertex correc- tions of the velocity operator solve the equation ˜vi knm=/summationdisplay p/integraldisplaydnk′ (2π)n−1δ(EF−Ek′p) 2γk′pp× ×/angbracketleftukn|uk′p/angbracketright/bracketleftbig ˜vi k′pp+vi k′pp/bracketrightbig /angbracketleftuk′p|ukm/angbracketright.(40)6 The odd contribution can be written as to ij=to,int ij+ tRR−vert ij+tAR−vert ij, where to,int ij=e h/integraldisplayddk (2π)dTr/braceleftbig TiGR k(EF)vj/bracketleftbig GA k(EF)−GR k(EF)/bracketrightbig/bracerightbig (41) and tRR−vert ij=−e h/integraldisplayddk (2π)dTr/braceleftBig ˜TRR iGR k(EF)vjGR k(EF)/bracerightBig (42) and tAR−vert ij=e h/integraldisplayddk (2π)dTr/braceleftBig ˜TAR iGR k(EF)vjGA k(EF)/bracerightBig .(43) The vertex corrections ˜TAR iand˜TRR iof the torque op- erator are given in Eq. (33) and in Eq. (34), respectively. While the even torkance, Eq. (37) and Eq. (38), be- comes independent of the scattering strength Uin the clean limit, i.e., for U→0, the odd torkance to ijdepends strongly on Uin metallic systems in the clean limit [15]. In the case of the one-dimensional Rashba model, the equations Eq. (37) and Eq. (38) for the even torkance te ijand the equations Eq. (41), Eq. (42) and Eq. (43) for the odd torkance to ijcan be used both for the collinear magnetic state as well as for the spin-spiral of Eq. (5). To obtain the even torkance for the collinear magnetic state, we plug the eigenstates and eigenvalues of Eq. (4) (withˆM=ˆez) into Eq. (37) and into Eq. (38). In the case of the spin-spiral of Eq. (5) we use instead the eigen- states and eigenvalues of Eq. (7). Similarly, to obtain the odd torkance in the collinear magnetic state, we evaluate Eq. (41), Eq. (42) and Eq. (43) based on the Hamilto- nian in Eq. (4) and for the spin-spiral we use instead the effective Hamiltonian in Eq. (7). III. RESULTS A. Gyromagnetic ratio We first discuss the g-factor in the collinear case, i.e., whenˆM(r) =ˆez. Inthis casetheenergybandsaregiven by E=/planckover2pi12k2 x 2m±/radicalbigg 1 4(∆V)2+(αRkx)2.(44) When ∆ V/negationslash= 0 orαR/negationslash= 0 the energy Ecan become negative. The band structure of the one-dimensional Rashba model is shown in Fig. 1 for the model param- etersαR=2eV˚A and ∆ V= 0.5eV. For this choice of parameters the energy minima are not located at kx= 0 but instead at kmin x=±/radicalBig (αR)4m2−1 4/planckover2pi14(∆V)2 /planckover2pi12αR,(45)-0.4 -0.2 0 0.2 0.4 k-Point kx [Å-1]00.511.5Band energy [eV] FIG. 1: Band structure of theone-dimensional Rashbamodel. and the corresponding minimum of the energy is given by Emin=−m(αR)4+1 4/planckover2pi14 m(∆V)2 2/planckover2pi12(αR)2. (46) The inverse g-factor is shown as a function of the SOI strength αRin Fig. 2 for the exchange splitting ∆ V= 1eV and Fermi energy EF= 1.36eV. At αR= 0 the scattering contribution is zero, i.e., the g-factor is de- termined completely by the intrinsic Berry curvature ex- pression, Eq. (24). Thus, at αR= 0 it assumes the value 1/g=−0.5, which is the expected nonrelativistic value (see the discussion below Eq. (24)). With increasing SOI strength αRthe intrinsic contribution to 1 /gis more and more suppressed. However, the scattering contribution compensates this decrease such that the total 1 /gis close to its nonrelativistic value of −0.5. The neglect of the scattering corrections at large values of αRwould lead in this case to a strong underestimation of the magnitude of 1/g, i.e., a strong overestimation of the magnitude of g. However, at smaller values of the Fermi energy, the gfactor can deviate substantially from its nonrelativis- tic value of −2. To show this we plot in Fig. 3 the in- verseg-factor as a function of the Fermi energy when the exchange splitting and the SOI strength are set to ∆V= 1eV and αR=2eV˚A, respectively. As discussed in Eq. (44) the minimal Fermi energyis negativ in this case. The intrinsic contribution to 1 /gdeclines with increas- ing Fermi energy. At large values of the Fermi energy this decline is compensated by the increase of the vertex corrections and the total value of 1 /gis close to −0.5. Previous theoretical works on the g-factor have not considered the scattering contribution [52]. It is there- fore important to find out whether the compensation of the decrease of the intrinsic contribution by the in-7 00.511.52 SOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/gscattering intrinsic total FIG. 2: Inverse g-factor vs. SOI strength αRin the one- dimensional Rashba model. 0 1 2 3 4 5 6 Fermi energy [eV]-0.6-0.4-0.201/gscattering intrinsic total FIG. 3: Inverse g-factor vs. Fermi energy in the one- dimensional Rashba model. crease of the extrinsic contribution as discussed in Fig. 2 and Fig. 3 is peculiar to the one-dimensional Rashba model or whether it can be found in more general cases. For this reason we evaluate g1for the two-dimensional Rashba model. In Fig. 4 we show the inverse g1-factor in the two-dimensional Rashba model as a function of SOI strength αRfor the exchange splitting ∆ V= 1eV and the Fermi energy EF= 1.36eV. Indeed for αR< 0.5eV˚A the scattering corrections tend to stabilize g1at its non-relativistic value. However, in contrast to the one-dimensional case (Fig. 2), where gdoes not deviate much from its nonrelativistic value up to αR= 2eV˚A, g1starts to be affected by SOI at smaller values of αR in the two-dimensional case. According to Eq. (26) the fullgfactor is given by g=g1(1+Morb/Mspin). There- fore, when the scattering corrections stabilize g1at its00.511.52 SOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/g1 scattering intrinsic total FIG. 4: Inverse g1-factor vs. SOI strength αRin the two- dimensional Rashba model. nonrelativistic value the Eq. (25) is satisfied. In the two- dimensional Rashba model Morb= 0 when both bands are occupied. For the Fermi energy EF= 1.36eV both bands are occupied and therefore g=g1for the range of parameters used in Fig. 4. The inverse g1of the two-dimensional Rashba model is shown in Fig. 5 as a function of Fermi energy for the parameters ∆ V= 1eV and αR= 2eV˚A. The scattering correction is as large as the intrinsic contribution when EF>1eV. While the scattering correction is therefore important, it is not sufficiently large to bring g1close to its nonrelativistic value in the energy range shown in the figure, which is a major difference to the one-dimensional case illustrated in Fig. 3. According to Eq. (26) the g factor is related to g1byg=g1M/Mspin. Therefore, we show in Fig. 6 the ratio M/Mspinas a function of Fermi energy. AthighFermienergy(whenbothbandsareoccu- pied) the orbital magnetization is zeroand M/Mspin= 1. At low Fermi energy the sign of the orbital magnetiza- tionis oppositeto the signofthe spin magnetizationsuch that the magnitude of Mis smaller than the magnitude ofMspinresulting in the ratio M/Mspin<1. Next, we discuss the g-factor of the one-dimensional Rashba model in the noncollinear case. In Fig. 7 we plot the inverse g-factor and its decomposition into the intrinsic and scattering contributions as a function of the spin-spiral wave vector q, where exchange splitting, SOI strength and Fermi energy are set to ∆ V= 1eV, αR= 2eV˚A andEF= 1.36eV, respectively. Since the curves are not symmetric around q= 0, the g- factor at wave number qdiffers from the one at −q, i.e., thegyromagnetism in the Rashba model is chiral . At q=−2meαR//planckover2pi12theg-factorassumesthevalueof g=−2 and the scattering corrections are zero. Moreover, the curves are symmetric around q=−2meαR//planckover2pi12. These8 0 2 4 6 Fermi energy [eV]-0.5-0.4-0.3-0.2-0.101/g1 scattering intrinsic total FIG. 5: Inverse g1-factor 1 /g1vs. Fermi energy in the two- dimensional Rashba model. -2 0 2 4 6 Fermi energy [eV]00.511.52M/Mspin FIG. 6: Ratio of total magnetization and spin magnetization , M/Mspin, vs. Fermi energy in the two-dimensional Rashba model. observationscan be explained by the concept of the effec- tive SOI introduced in Eq. (9): At q=−2meαR//planckover2pi12the effective SOI is zero and consequently the noncollinear magnet behaves like a collinear magnet without SOI at this value of q. As we have discussed above in Fig. 2, the g-factor of collinear magnets is g=−2 when SOI is ab- sent, which explains why it is also g=−2 in noncollinear magnets with q=−2meαR//planckover2pi12. If only the intrinsic con- tribution is considered and the scattering corrections are neglected, 1 /gvaries much stronger around the point of zero effective SOI q=−2meαR//planckover2pi12, i.e., the scattering corrections stabilize gat its nonrelativistic value close to the point of zero effective SOI.-2 -1 0 1 Wave vector q [Å-1]-0.8-0.6-0.4-0.201/g scattering intrinsic total FIG. 7: Inverse g-factor 1 /gvs. wave number qin the one- dimensional Rashba model. 0 1 2 3 4 Scattering strength U [(eV)2Å]-0.4-0.200.20.4Gilbert Damping αG xx RR-Vertex AR-Vertex intrinsic total FIG. 8: Gilbert damping αG xxvs. scattering strength Uin the one-dimensional Rashba model without SOI. In this case the vertex corrections and the intrinsic contribution sum up to zero. B. Damping We first discuss the Gilbert damping in the collinear case, i.e., we set ˆM(r) =ˆezin Eq. (4). The xxcom- ponent of the Gilbert damping is shown in Fig. 8 as a function of scattering strength Ufor the following model parameters: exchange splitting ∆ V=1eV, Fermi energyEF= 2.72eV and SOI strength αR= 0. All three contributions are individually non-zero, but the contribution from the RR-vertex correction (Eq. (30)) is muchsmallerthanthe onefromthe AR-vertexcorrection (Eq. (31)) and much smaller than the intrinsic contribu- tion (Eq. (29)). However, in this case the total damping is zero, because a non-zero damping in periodic crystals with collinear magnetization is only possible when SOI is present [53].9 1 2 3 4 Scattering strength U [(eV)2Å]050100150200250300Gilbert Damping αG xx RR-Vertex AR-Vertex intrinsic total FIG. 9: Gilbert damping αG xxvs. scattering strength Uin the one-dimensional Rashba model with SOI. In Fig. 9 we show the xxcomponent of the Gilbert damping αG xxas a function of scattering strength Ufor the model parameters ∆ V= 1eV, EF= 2.72eV and αR= 2eV˚A. ThedominantcontributionistheAR-vertex correction. The damping as obtained based on Eq. (10) diverges like 1 /Uin the limit U→0, i.e., proportional to the relaxation time τ[53]. However, once the relax- ation time τis larger than the inverse frequency of the magnetization dynamics the dc-limit ω→0 in Eq. (10) is not appropriate and ω >0 needs to be used. It has beenshownthattheGilbertdampingisnotinfinite inthe ballistic limit τ→ ∞whenω >0 [41, 42]. In the one- dimensional Rashba model the effective magnetic field exerted by SOI on the electron spins points in ydirec- tion. Since a magnetic field along ydirection cannot lead toatorquein ydirectionthe yycomponentoftheGilbert damping αG yyis zero (not shown in the Figure). Next, we discuss the Gilbert damping in the non- collinear case. In Fig. 10 we plot the xxcomponent of the Gilbert damping as a function of spin spiral wave number qfor the model parameters ∆ V= 1eV, EF= 1.36eV,αR= 2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The curves are symmetric around q=−2meαR//planckover2pi12, because the damping is determined by the effective SOI defined in Eq. (9). At q=−2meαR//planckover2pi12 the effective SOI is zero and therefore the total damp- ing is zero as well. The damping at wave number qdif- fers from the one at wave number −q, i.e.,the damp- ing is chiral in the Rashba model . Around the point q=−2meαR//planckover2pi12the damping is described by aquadratic parabola at first. In the regions -2 ˚A−1< q <-1.2˚A−1 and 0.2˚A−1< q <1˚A−1this trend is interrupted by a W- shape behaviour. In the quadratic parabola region the lowest energy band crosses the Fermi energy twice. As shown in Fig. 1 the lowest band has a local maximum at-2-1.5-1-0.500.51 Wave vector q [Å-1]05101520Gilbert damping αxxG RR-Vertex AR-Vertex intrinsic total FIG. 10: Gilbert damping αG xxvs. spin spiral wave number q in the one-dimensional Rashba model. q= 0. In the W-shape region this local maximum shifts upwards, approaches the Fermi level and finally passes it such that the lowest energy band crosses the Fermi level four times. This transition in the band structure leads to oscillations in the density of states, which results in the W-shape behaviour of the Gilbert damping. Since the damping is determined by the effective SOI, we can use Fig. 10 to draw conclusions about the damp- ing in the noncollinear case with αR= 0: We only need to shift all curves in Fig. 10 to the right such that they are symmetric around q= 0 and shift the Fermi energy. Thus, for αR= 0 the Gilbert damping does not vanish ifq/negationslash= 0. Since for αR= 0 angular momentum transfer from the electronic system to the lattice is not possible, the damping is purely nonlocal in this case, i.e., angular momentum is interchanged between electrons at differ- ent positions. This means that for a volume in which the magnetization of the spin-spiral in Eq. (5) performs exactly one revolution between one end of the volume and the other end the total angular momentum change associated with the damping is zero, because the angu- lar momentum is simply redistributed within this volume and there is no net change of the angular momentum. A substantial contribution of nonlocal damping has also been predicted for domain walls in permalloy [35]. In Fig. 11 we plot the yycomponent of the Gilbert damping as a function of spin spiral wave number qfor the model parameters ∆ V= 1eV,EF= 1.36eV,αR= 2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The totaldampingiszerointhiscase. Thiscanbeunderstood from the symmetry properties of the one-dimensional Rashba Hamiltonian, Eq. (4): Since this Hamiltonian is invariant when both σandˆMare rotated around the yaxis, the damping coefficient αG yydoes not depend on the position within the cycloidal spin spiral of Eq. (5).10 -3 -2 -1 0 1 2 Wave vector q [Å-1]-0.4-0.200.20.4Gilbert Damping αG yyRR-Vertex AR-Vertex intrinsic total FIG. 11: Gilbert damping αG yyvs. spin spiral wave number q in the one-dimensional Rashba model. Therefore, nonlocal damping is not possible in this case andαG yyhas to be zero when αR= 0. It remains to be shown that αG yy= 0 also for αR/negationslash= 0. However, this fol- lows directly from the observation that the damping is determined by the effective SOI, Eq. (9), meaning that any case with q/negationslash= 0 and αR/negationslash= 0 can always be mapped onto a case with q/negationslash= 0 and αR= 0. As an alternative argumentation we can also invoke the finding discussed abovethat αG yy= 0 in the collinearcase. Since the damp- ing is determined by the effective SOI, it follows that αG yy= 0 also in the noncollinear case. C. Current-induced torques We first discuss the yxcomponent of the torkance. In Fig. 12 we show the torkance tyxas a function of the Fermi energy EFfor the model parameters ∆ V= 1eV andαR= 2eV˚A when the magnetization is collinear and points in zdirection. We specify the torkance in units of the positive elementary charge e, which is a convenient choice for the one-dimensional Rashba model. When the torkance is multiplied with the electric field, we ob- tain the torque per length (see Eq. (35) and Ref. [51]). Since the effective magnetic field from SOI points in ydirection, it cannot give rise to a torque in ydirec- tion and consequently the total tyxis zero. Interest- ingly, the intrinsic and scattering contributions are indi- vidually nonzero. The intrinsic contribution is nonzero, because the electric field accelerates the electrons such that/planckover2pi1˙kx=−eEx. Therefore, the effective magnetic fieldBSOI y=αRkx/µBchanges as well, i.e., ˙BSOI y= αR˙kx/µB=−αRExe/(/planckover2pi1µB). Consequently, the electron spin is no longer aligned with the total effective magnetic field (the effective magnetic field resulting from both SOI-1 0 1 2 3 4 5 6 Fermi energy [eV]-0.2-0.100.10.2Torkance tyx [e]scattering intrinsic total FIG. 12: Torkance tyxvs. Fermi energy EFin the one- dimensional Rashba model. and from the exchange splitting ∆ V), when an electric field is applied. While the total effective magnetic field lies in the yzplane, the electron spin acquires an xcom- ponent, because it precesses around the total effective magnetic field, with which it is not aligned due to the applied electric field [54]. The xcomponent of the spin density results in a torque in ydirection, which is the reason why the intrinsic contribution to tyxis nonzero. The scattering contribution to tyxcancels the intrinsic contribution such that the total tyxis zero and angular momentum conservation is satisfied. Using the concept of effective SOI, Eq. (9), we con- clude that tyxis also zero for the noncollinear spin-spiral described by Eq. (5). Thus, both the ycomponent of the spin-orbit torque and the nonadiabatic torque are zero for the one-dimensional Rashba model. To show that tyx= 0 is a peculiarity of the one- dimensional Rashba model, we plot in Fig. 13 the torkance tyxin the two-dimensional Rashba model. The intrinsic and scattering contributions depend linearly on αRfor small values of αR, but the slopes are opposite such that the total tyxis zero for sufficiently small αR. However, for largervalues of αRthe intrinsic and scatter- ing contributions do not cancel each other and therefore the total tyxbecomes nonzero, in contrast to the one- dimensional Rashba model, where tyx= 0 even for large αR. Several previous works determined the part of tyx that is proportionalto αRin the two-dimensionalRashba model and found it to be zero [21, 22] for scalar disor- der, which is consistent with our finding that the linear slopes of the intrinsic and scattering contributions to tyx are opposite for small αR. Next, we discuss the xxcomponent of the torkance in the collinear case ( ˆM=ˆez). In Fig. 14 we plot the torkance txxvs. scattering strength Uin the one-11 00.511.52 SOI strength αR [eVÅ]-0.00500.0050.01Torkance tyx [e/Å] scattering intrinsic total FIG. 13: Nonadiabatic torkance tyxvs. SOI parameter αRin the two-dimensional Rashba model. dimensional Rashba model for the parameters ∆ V= 1eV,EF= 2.72eV and αR= 2eV˚A. The dominant con- tribution is the AR-type vertex correction (see Eq. (43)). txxdiverges like 1 /Uin the limit U→0 as expected for the odd torque in metallic systems [15]. In Fig. 15 and Fig. 16 we plot txxas a function of spin-spiral wave number qfor the model parameters ∆V= 1eV,EF= 2.72eV and U= 0.18(eV)2˚A. In Fig. 15 the case with αR= 2eV˚A is shown, while Fig. 16 illus- trates the case with αR= 0. In the case αR= 0 the torkance txxdescribes the spin-transfer torque (STT). In the case αR/negationslash= 0 the torkance txxis the sum of contribu- tions from STT and spin-orbit torque (SOT). The curves withαR= 0 andαR/negationslash= 0 are essentially related by a shift of ∆q=−2meαR//planckover2pi12, which can be understood based on the concept of the effective SOI, Eq. (9). Thus, in the special case of the one-dimensional Rashba model STT and SOT are strongly related. IV. SUMMARY We study chiral damping, chiral gyromagnetism and current-induced torques in the one-dimensional Rashba model with an additional N´ eel-type noncollinear mag- netic exchange field. In order to describe scattering ef- fects we use a Gaussian scalar disorder model. Scat- tering contributions are generally important in the one- dimensional Rashba model with the exception of the gy- romagnetic ratio in the collinear case with zero SOI, where the scattering correctionsvanish in the clean limit. In the one-dimensional Rashba model SOI and non- collinearity can be combined into an effective SOI. Us- ing the concept of effective SOI, results for the mag- netically collinear one-dimensional Rashba model can be used to predict the behaviour in the noncollinear case.1 2 3 4 Scattering strength U [(eV)2Å]-6-4-20Torkance txx [e] RR-Vertex AR-Vertex intrinsic total FIG. 14: Torkance txxvs. scattering strength Uin the one- dimensional Rashba model. -2 -1 0 1 Wave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex AR-Vertex intrinsic total FIG. 15: Torkance txxvs. wave vector qin the one- dimensional Rashba model with SOI. In the noncollinear Rashba model the Gilbert damp- ing is nonlocal and does not vanish for zero SOI. 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2005.12756v1.A_transmission_problem_for_the_Timoshenko_system_with_one_local_Kelvin_Voigt_damping_and_non_smooth_coefficient_at_the_interface.pdf

arXiv:2005.12756v1 [math.AP] 24 May 2020A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM WITH ONE LO CAL KELVIN-VOIGT DAMPING AND NON-SMOOTH COEFFICIENT AT THE INT ERFACE MOUHAMMAD GHADER AND ALI WEHBE LEBANESE UNIVERSITY, FACULTY OF SCIENCES 1, KHAWARIZMI LAB ORATORY OF MATHEMATICS AND APPLICATIONS-KALMA, HADATH-BEIRUT. EMAILS: MHAMMADGHADER@HOTMAIL.COM AND ALI.WEHBE@UL.EDU .LB. Abstract. In this paper, we study the indirect stability of Timoshenko system with local or global Kelvin–Voigt damping, under fully Dirichlet or mixed boundary conditions. Unlike [ 43] and [ 39], in this paper, we consider the Timoshenko system with only one locally or globally distributed Kelvin-Voigt damping D(see System ( 1.1)). Indeed, we prove that the energy of the system decays polynomially of type t−1and that this decay rate is in some sense optimal. The method i s based on the frequency domain approach combining with multiplier method. MSC Classification. 35B35; 35B40; 93D20. Keywords. Timoshenko beam; Kelvin-Voigt damping; Semigroup; Stabil ity. 1.Introduction In this paper, we study the indirect stability of a one-dimen sional Timoshenko system with only one local or global Kelvin-Voigt damping. This system consists of two co upled hyperbolic equations: (1.1)ρ1utt−k1(ux+y)x= 0, (x,t)∈(0,L)×R+, ρ2ytt−(k2yx+Dyxt)x+k1(ux+y) = 0,(x,t)∈(0,L)×R+. System ( 1.1) is subject to the following initial conditions: (1.2)u(x,0) =u0(x), ut(x,0) =u1(x), x∈(0,L), y(x,0) =y0(x), yt(x,0) =y1(x), x∈(0,L), in addition to the following boundary conditions: (1.3) u(0,t) =y(0,t) =u(L,t) =y(L,t) = 0, t∈R+, or (1.4) u(0,t) =yx(0,t) =u(L,t) =yx(L,t) = 0, t∈R+. Here the coefficients ρ1, ρ2, k1, andk2are strictly positive constant numbers. The function D∈L∞(0,L), such thatD(x)≥0,∀x∈[0,L]. We assume that there exist D0>0,α, β∈R,0≤α<β≤L,such that (H) D∈C([α,β])andD(x)≥D0>0∀x∈(α,β). The hypothesis (H) means that the control Dcan be locally near the boundary (see Figures 1aand1b), or locally internal (see Figure 2a), or globally (see Figure 2b). Indeed, in the case when Dis local damping (i.e., α/ne}ationslash= 0orβ/ne}ationslash=L), we see that Dis not necessary continuous over (0,L)(see Figures 1a,1b, and 2a). The Timoshenko system is usually considered in describing t he transverse vibration of a beam and ignoring damping effects of any nature. Indeed, we have the following m odel, see in [ 40], /braceleftigg ρϕtt= (K(ϕx−ψ))x Iρψtt= (EIψx)x−K(ϕx−ψ), whereϕis the transverse displacement of the beam and ψis the rotation angle of the filament of the beam. The coefficients ρ, Iρ, E, I, andKare respectively the density (the mass per unit length), the polar moment 1STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING x 0αβ=LD(x) D(x) Figure 1ax 0 =αβLD(x) D(x) Figure 1b x 0αβLD(x)D(x) D(x) Figure 2ax 0 =αL=βD(x) Figure 2b of inertia of a cross section, Young’s modulus of elasticity , the moment of inertia of a cross section and the shear modulus respectively. The stabilization of the Timoshenko system with different ki nds of damping has been studied in number of publications. For the internal stabilization, Raposo and a l. in [ 34] showed that the Timoshenko system with two internal distributed dissipation is exponentially sta ble. Messaoudi and Mustafa in [ 27] extended the results to nonlinear feedback laws. Soufyane and Wehbe in [ 37] showed that Timoshenko system with one internal distributed dissipation law is exponentially stable if and only if the wave propagation speeds are equal (i.e., k1 ρ1=ρ2 k2), otherwise, only the strong stability holds. Indeed, Rive ra and Racke in [ 30] they improved the results of [ 37], where an exponential decay of the solution of the system ha s been established, allowing the coefficient of the feedback to be with an indefinite sign. Wehbe and Youssef in [ 41] proved that the Timoshenko system with one locally distributed viscous feedback is exp onentially stable if and only if the wave propagation speeds are equal (i.e.,k1 ρ1=ρ2 k2), otherwise, only the polynomial stability holds. Tebou in [38] showed that the Timoshenko beam with same feedback control in both equat ions is exponentially stable. The stability of the Timoshenko system with thermoelastic dissipation ha s been studied in [ 36], [12], [13], and [ 15]. The stability of Timoshenko system with memory type has been stu died in [ 3], [36], [14], [28], and [ 1]. For the boundary stabilization of the Timoshenko beam. Kim and Rena rdy in [ 19] showed that the Timoshenko beam under two boundary controls is exponentially stable. Ammar -Khodja and al. in [ 4] studied the decay rate of the energy of the nonuniform Timoshenko beam with two boun dary controls acting in the rotation-angle equation. In fact, under the equal speed wave propagation co ndition, they established exponential decay results up to an unknown finite dimensional space of initial data. In a ddition, they showed that the equal speed wave propagation condition is necessary for the exponential sta bility. However, in the case of non-equal speed, no decay rate has been discussed. This result has been recently improved by Wehbe and al. in [ 7]; i.e., the authors in [7], proved nonuniform stability and an optimal polynomial en ergy decay rate of the Timoshenko system with only one dissipation law on the boundary. For the stabil ization of the Timoshenko beam with nonlinear term, we mention [ 29], [2], [5], [27], [10], and [ 15]. Kelvin-Voigt material is a viscoelastic structure having p roperties of both elasticity and viscosity. There are a number of publications concerning the stabilization o f wave equation with global or local Kelvin-Voigt damping. For the global case, the authors in [ 16,21], proved the analyticity and the exponential stability of t he semigroup. When the Kelvin-Voigt damping is localized on an interval of the string, the regularity and stability 2STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING of the solution depend on the properties of the damping coeffic ient. Notably, the system is more effectively controlled by the local Kelvin-Voigt damping when the coeffic ient changes more smoothly near the interface (see [22,35,42,25,23]). Last but not least, in addition to the previously cited paper s, the stability of the Timoshenko system with Kelvin-Voigt damping has been studied in few papers. Zhao an d al. in [ 43] they considered the Timoshenko system with local distributed Kelvin–Voigt damping: (1.5)ρ1utt−[k1(ux+y)x+D1(uxt−yt)]x= 0, (x,t)∈(0,L)×R+, ρ2ytt−(k2yx+D2yxt)x+k1(ux+y)x+D1(uxt−yt) = 0,(x,t)∈(0,L)×R+. They proved that the energy of the System ( 1.5) subject to Dirichlet-Neumann boundary conditions has an exponential decay rate when coefficient functions D1, D2∈C1,1([0,L])and satisfy D1≤cD2(c >0).Tian and Zhang in [ 39] considered the Timoshenko System ( 1.5) under fully Dirichlet boundary conditions with locally or globally distributed Kelvin-Voigt damping when coefficient functions D1, D2∈C([0,L]). First, when the Kelvin-Voigt damping is globally distributed, the y showed that the Timoshenko System ( 1.5) under fully Dirichlet boundary conditions is analytic. Next, for their system with local Kelvin-Voigt damping, they analyzed the exponential and polynomial stability accordi ng to the properties of coefficient functions D1, D2. Unlike [ 43] and [ 39], in this paper, we consider the Timoshenko system with only one locally or globally distributed Kelvin-Voigt damping D(see System ( 1.1)). Indeed, in this paper, under hypothesis (H), we show that the energy of the Timoshenko System ( 1.1) subject to initial state ( 1.2) to either the boundary conditions (1.3) or (1.4) has a polynomial decay rate of type t−1and that this decay rate is in some sense optimal. This paper is organized as follows: In Section 2, first, we show that the Timoshenko System ( 1.1) subject to initial state ( 1.2) to either the boundary conditions ( 1.3) or (1.4) can reformulate into an evolution equation and we deduce the well-posedness property of the problem by t he semigroup approach. Second, using a criteria of Arendt-Batty [ 6], we show that our system is strongly stable. In Section 3, we show that the Timoshenko System ( 1.1)-(1.2) with the boundary conditions ( 1.4) is not uniformly exponentially stable. In Section 4, we prove the polynomial energy decay rate of type t−1for the System ( 1.1)-(1.2) to either the boundary conditions (1.3) or (1.4). Moreover, we prove that this decay rate is in some sense opt imal. 2.Well-Posedness and Strong Stability 2.1.Well-posedness of the problem. In this part, under condition (H), using a semigroup approac h, we establish well-posedness result for the Timoshenko System (1.1)-(1.2) to either the boundary conditions ( 1.3) or (1.4). The energy of solutions of the System ( 1.1) subject to initial state ( 1.2) to either the boundary conditions (1.3) or (1.4) is defined by: E(t) =1 2/integraldisplayL 0/parenleftig ρ1|ut|2+ρ2|yt|2+k1|ux+y|2+k2|yx|2/parenrightig dx. Let(u,y)be a regular solution for the System ( 1.1). Multiplying the first and the second equation of ( 1.1) by utandyt,respectively, then using the boundary conditions ( 1.3) or (1.4), we get E′(t) =−/integraldisplayL 0D(x)|yxt|2dx≤0. Thus System ( 1.1) subject to initial state ( 1.2) to either the boundary conditions ( 1.3) or (1.4) is dissipative in the sense that its energy is non increasing with respect to th e timet. Let us define the energy spaces H1and H2by: H1=H1 0(0,L)×L2(0,L)×H1 0(0,L)×L2(0,L) and H2=H1 0(0,L)×L2(0,L)×H1 ∗(0,L)×L2(0,L), such that H1 ∗(0,L) =/braceleftigg f∈H1(0,L)|/integraldisplayL 0fdx= 0/bracerightigg . 3STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING It is easy to check that the space H1 ∗is Hilbert spaces over Cequipped with the norm /bardblu/bardbl2 H1 ∗(0,L)=/bardblux/bardbl2, where/bardbl· /bardbldenotes the usual norm of L2(0,L). Both energy spaces H1andH2are equipped with the inner product defined by: /an}bracketle{tU,U1/an}bracketri}htHj=ρ1/integraldisplayL 0vv1dx+ρ2/integraldisplayL 0zz1dx+k1/integraldisplayL 0(ux+y)((u1)x+y1)dx+k2/integraldisplayL 0yx(y1)xdx for allU= (u,v,y,z)andU1= (u1,v1,y1,z1)inHj,j= 1,2. We use /bardblU/bardblHjto denote the corresponding norms. We now define the following unbounded linear operator sAjinHjby D(A1) =/braceleftbig U= (u,v,y,z)∈ H1|v, z∈H1 0(0,L), u∈H2(0,L),(k2yx+Dzx)x∈L2(0,L)/bracerightbig , D(A2) =/braceleftbigg U= (u,v,y,z)∈ H2|v∈H1 0(0,L), z∈H1 ∗(0,L), u∈H2(0,L), (k2yx+Dzx)x∈L2(0,L), yx(0) =yx(L) = 0/bracerightbigg and forj= 1,2, AjU=/parenleftbigg v,k1 ρ1(ux+y)x,z,1 ρ2(k2yx+Dzx)x−k1 ρ2(ux+y)/parenrightbigg ,∀U= (u,v,y,z)∈D(Aj). IfU= (u,ut,y,yt)is the state of System ( 1.1)-(1.2) to either the boundary conditions ( 1.3) or (1.4), then the Timoshenko system is transformed into a first order evolutio n equation on the Hilbert space Hj: (2.1)/braceleftigg Ut(x,t) =AjU(x,t), U(x,0) =U0(x), where U0(x) = (u0(x),u1(x),y0(x),y1(x)). Proposition 2.1. Under hypothesis (H), for j= 1,2,the unbounded linear operator Ajis m-dissipative in the energy space Hj. Proof. Letj= 1,2, forU= (u,v,y,z)∈D(Aj), one has ℜ/an}bracketle{tAjU,U/an}bracketri}htHj=−/integraldisplayL 0D(x)|zx|2dx≤0, which implies that Ajis dissipative under hypothesis (H). Here ℜis used to denote the real part of a complex number. We next prove the maximality of Aj. ForF= (f1,f2,f3,f4)∈ Hj, we prove the existence of U= (u,v,y,z)∈D(Aj), unique solution of the equation −AjU=F. Equivalently, one must consider the system given by −v=f1, (2.2) −k1(ux+y)x=ρ1f2, (2.3) −z=f3, (2.4) −(k2yx+Dzx)x+k1(ux+y) =ρ2f4, (2.5) with the boundary conditions (2.6) u(0) =u(L) =v(0) =v(L) = 0 and/braceleftigg y(0) =y(L) =z(0) =z(L) = 0,forj= 1, yx(0) =yx(L) = 0, forj= 2. Let(ϕ,ψ)∈ Vj(0,L), whereV1(0,L) =H1 0(0,L)×H1 0(0,L)andV2(0,L) =H1 0(0,L)×H1 ∗(0,L). Multiplying Equations ( 2.3) and ( 2.5) byϕandψrespectively, integrating in (0,L), taking the sum, then using Equation (2.4) and the boundary condition ( 2.6), we get (2.7)/integraldisplayL 0/parenleftig k1(ux+y)(ϕx+ψ)+k2yxψx/parenrightig dx=/integraldisplayL 0/parenleftbig ρ1f1¯ϕ+ρ2f4¯ψ+D(f3)xψx/parenrightbig dx,∀(ϕ,ψ)∈ Vj(0,L). 4STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING The left hand side of ( 2.7) is a bilinear continuous coercive form on Vj(0,L)×Vj(0,L), and the right hand side of (2.7) is a linear continuous form on Vj(0,L). Then, using Lax-Milligram theorem (see in [ 32]), we deduce that there exists (u,y)∈ Vj(0,L)unique solution of the variational Problem ( 2.7). Thus, using ( 2.2), (2.4), and classical regularity arguments, we conclude that −AjU=Fadmits a unique solution U∈D(Aj)and consequently 0∈ρ(Aj), whereρ(Aj)denotes the resolvent set of Aj. Then, Ajis closed and consequently ρ(Aj)is open set of C(see Theorem 6.7 in [ 18]). Hence, we easily get λ∈ρ(Aj)for sufficiently small λ>0. This, together with the dissipativeness of Aj, imply that D(Aj)is dense in Hjand that Ajis m-dissipative in Hj(see Theorems 4.5, 4.6 in [ 32]). Thus, the proof is complete. /square Thanks to Lumer-Phillips theorem (see [ 26,32]), we deduce that Ajgenerates a C0-semigroup of contraction etAjinHjand therefore Problem ( 2.1) is well-posed. Then, we have the following result. Theorem 2.2. Under hypothesis (H), for j= 1,2,for anyU0∈ Hj, the Problem ( 2.1) admits a unique weak solutionU(x,t) =etAjU0(x), such that U∈C(R+;Hj). Moreover, if U0∈D(Aj),then U∈C(R+;D(Aj))∩C1(R+;Hj). /square Before starting the main results of this work, we introduce h ere the notions of stability that we encounter in this work. Definition 2.3. LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0onH. The C0-semigroup/parenleftbig etA/parenrightbig t≥0is said to be 1. strongly stable if lim t→+∞/bardbletAx0/bardblH= 0,∀x0∈H; 2. exponentially (or uniformly) stable if there exist two po sitive constants Mandǫsuch that /bardbletAx0/bardblH≤Me−ǫt/bardblx0/bardblH,∀t>0,∀x0∈H; 3. polynomially stable if there exists two positive constan tsCandαsuch that /bardbletAx0/bardblH≤Ct−α/bardblAx0/bardblH,∀t>0,∀x0∈D(A). In that case, one says that solutions of ( 2.1) decay at a rate t−α. TheC0-semigroup/parenleftbig etA/parenrightbig t≥0is said to be polynomially stable with optimal decay rate t−α(withα >0) if it is polynomially stable with decay ratet−αand, for any ε>0small enough, there exists solutions of ( 2.1) which do not decay at a ratet−(α+ε). /square We now look for necessary conditions to show the strong stabi lity of theC0-semigroup/parenleftbig etA/parenrightbig t≥0. We will rely on the following result obtained by Arendt and Batty in [ 6]. Theorem 2.4 (Arendt and Batty in [ 6]).LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0onH. If 1.Ahas no pure imaginary eigenvalues, 2.σ(A)∩iRis countable, whereσ(A)denotes the spectrum of A, then theC0-semigroup/parenleftbig etA/parenrightbig t≥0is strongly stable. /square Our subsequent findings on polynomial stability will rely on the following result from [ 9,24,8], which gives necessary and sufficient conditions for a semigroup to be poly nomially stable. For this aim, we recall the following standard result (see [ 9,24,8] for part (i) and [ 17,33] for part (ii)). Theorem 2.5. LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0onH. Assume thatiλ∈ρ(A),∀λ∈R. Then, the C0-semigroup/parenleftbig etA/parenrightbig t≥0is 5STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING (i) Polynomially stable of order1 ℓ(ℓ>0)if and only if lim sup λ∈R,|λ|→∞|λ|−ℓ/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble L(H)<+∞. (ii) Exponentially stable if and only if lim sup λ∈R,|λ|→∞/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble L(H)<+∞. /square 2.2.Strong stability. In this part, we use general criteria of Arendt-Batty in [ 6] (see Theorem 2.4) to show the strong stability of the C0-semigroup etAjassociated to the Timoshenko System ( 2.1). Our main result is the following theorem. Theorem 2.6. Assume that (H) is true. Then, for j= 1,2,theC0−semigroupetAjis strongly stable in Hj; i.e., for allU0∈ Hj, the solution of ( 2.1) satisfies lim t→+∞/vextenddouble/vextenddoubleetAjU0/vextenddouble/vextenddouble Hj= 0. The argument for Theorem 2.6relies on the subsequent lemmas. Lemma 2.7. Under hypothesis (H), for j= 1,2,one has ker(iλI−Aj) ={0},∀λ∈R. Proof. Forj= 1,2, from Proposition 2.1, we deduce that 0∈ρ(Aj). We still need to show the result for λ∈R∗. Suppose that there exists a real number λ/ne}ationslash= 0andU= (u,v,y,z)∈D(Aj)such that AjU=iλU. Equivalently, we have (2.8) v=iλu, k1(ux+y)x=iρ1λv, z=iλy, (k2yx+Dzx)x−k1(ux+y) =iρ2λz. First, a straightforward computation gives 0 =ℜ/an}bracketle{tiλU,U/an}bracketri}htHj=ℜ/an}bracketle{tAjU,U/an}bracketri}htHj=−/integraldisplayL 0D(x)|zx|2dx, using hypothesis (H), we deduce that (2.9) Dzx= 0 over(0,L)andzx= 0 over(α,β). Inserting ( 2.9) in (2.8), we get u=yx= 0,over(α,β), (2.10) k1uxx+ρ1λ2u+k1yx= 0,over(0,L), (2.11) −k1ux+k2yxx+/parenleftbig ρ2λ2−k1/parenrightbig y= 0,over(0,L), (2.12) with the following boundary conditions (2.13) u(0) =u(L) =y(0) =y(L) = 0,ifj= 1 oru(0) =u(L) =yx(0) =yx(L) = 0,ifj= 2. In fact, System ( 2.11)-(2.13) admits a unique solution (u,y)∈C2((0,L)). From ( 2.10) and by the uniqueness of solutions, we get (2.14) u=yx= 0,over(0,L). 1. Ifj= 1, from ( 2.14) and the fact that y(0) = 0,we get u=y= 0,over(0,L), hence,U= 0. In this case the proof is complete. 6STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING 2. Ifj= 2, from ( 2.14) and the fact that y∈H1 ∗(0,L)(i.e.,/integraltextL 0ydx= 0),we get u=y= 0,over(0,L), therefore,U= 0, also in this case the proof is complete. /square Lemma 2.8. Under hypothesis (H), for j= 1,2,for allλ∈R, theniλI−Ajis surjective. Proof. LetF= (f1,f2,f3,f4)∈ Hj, we look for U= (u,v,y,z)∈D(Aj)solution of (iλU−Aj)U=F. Equivalently, we have v=iλu−f1, (2.15) z=iλy−f3, (2.16) λ2u+k1 ρ1(ux+y)x=F1, (2.17) λ2y+ρ2−1[(k2+iλD)yx]x−k1 ρ2(ux+y) =F2, (2.18) with the boundary conditions (2.19) u(0) =u(L) =v(0) =v(L) = 0 and/braceleftigg y(0) =y(L) =z(0) =z(L) = 0,forj= 1, yx(0) =yx(L) = 0, forj= 2. Such that /braceleftigg F1=−f2−iλf1∈L2(0,L), F2=−f4−iλf3+ρ2−1(D(f3)x)x∈H−1(0,L). We define the operator Ljby LjU=/parenleftbigg −k1 ρ1(ux+y)x,−ρ−1 2[(k2+iλD)yx]x+k1 ρ2(ux+y)/parenrightbigg ,∀ U= (u,y)∈ Vj(0,L), where V1(0,L) =H1 0(0,L)×H1 0(0,L)andV2(0,L) =H1 0(0,L)×H1 ∗(0,L). Using Lax-Milgram theorem, it is easy to show that Ljis an isomorphism from Vj(0,L)onto(H−1(0,L))2. LetU= (u,y)andF= (−F1,−F2), then we transform System ( 2.17)-(2.18) into the following form (2.20) U −λ2L−1 jU=L−1F. Using the compactness embeddings from L2(0,L)intoH−1(0,L)and fromH1 0(0,L)intoL2(0,L), and from H1 L(0,L)intoL2(0,L), we deduce that the operator L−1 jis compact from L2(0,L)×L2(0,L)intoL2(0,L)× L2(0,L). Consequently, by Fredholm alternative, proving the exist ence ofUsolution of ( 2.20) reduces to provingker/parenleftbig I−λ2L−1 j/parenrightbig = 0. Indeed, if (ϕ,ψ)∈ker(I−λ2L−1 j), then we have λ2(ϕ,ψ)− Lj(ϕ,ψ) = 0. It follows that (2.21) λ2ϕ+k1 ρ1(ϕx+ψ)x= 0, λ2ψ+ρ2−1[(k2+iλD)ψx]x−k1 ρ2(ϕx+ψ) = 0, with the following boundary conditions (2.22) ϕ(0) =ϕ(L) =ψ(0) =ψ(L) = 0,ifj= 1 orϕ(0) =ϕ(L) =ψx(0) =ψx(L) = 0,ifj= 2. It is now easy to see that if (ϕ,ψ)is a solution of System ( 2.21)-(2.22), then the vector Vdefined by V= (ϕ,iλϕ,ψ,iλψ ) belongs toD(Aj)andiλV−AjV= 0.Therefore,V∈ker(iλI−Aj). Using Lemma 2.7, we getV= 0, and so ker(I−λ2L−1 j) ={0}. 7STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Thanks to Fredholm alternative, the Equation ( 2.20) admits a unique solution (u,v)∈ Vj(0,L). Thus, using (2.15), (2.17) and a classical regularity arguments, we conclude that (iλ−Aj)U=Fadmits a unique solution U∈D(Aj). Thus, the proof is complete. /square We are now in a position to conclude the proof of Theorem 2.6. Proof of Theorem 2.6.Using Lemma 2.7, we directly deduce that Ajha non pure imaginary eigenvalues. According to Lemmas 2.7,2.8and with the help of the closed graph theorem of Banach, we ded uce that σ(Aj)∩iR={∅}. Thus, we get the conclusion by applying Theorem 2.4of Arendt and Batty. /square 3.Lack of exponential stability of A2 In this section, our goal is to show that the Timoshenko Syste m (1.1)-(1.2) with Dirichlet-Neumann boundary conditions ( 1.4) is not exponentially stable. 3.1.Lack of exponential stability of A2with global Kelvin–Voigt damping. In this part, assume that (3.1) D(x) =D0>0,∀x∈(0,L), whereD0∈R+ ∗. We prove the following theorem. Theorem 3.1. Under hypothesis ( 3.1), forǫ>0(small enough ), we cannot expect the energy decay rate t−2 2−ǫ for all initial data U0∈D(A2)and for allt>0. Proof. Following to Borichev [ 9] (see Theorem 2.4part (i)), it suffices to show the existence of sequences (λn)n⊂Rwithλn→+∞,(Un)n⊂D(A2), and(Fn)n⊂ H2such that (iλnI−A2)Un=Fnis bounded in H2andλ−2+ǫ n/bardblUn/bardbl →+∞. Set Fn=/parenleftig 0,sin/parenleftignπx L/parenrightig ,0,0/parenrightig , Un=/parenleftig Ansin/parenleftignπx L/parenrightig ,iλnAnsin/parenleftignπx L/parenrightig ,Bncos/parenleftignπx L/parenrightig ,iλnBncos/parenleftignπx L/parenrightig/parenrightig and (3.2) λn=nπ L/radicaligg k1 ρ1, An=−inπD0 k1L/radicalbiggρ1 k1+k2 k1/parenleftbiggρ2 k2−ρ1 k1/parenrightbigg −ρ1L2 k1π2n2, Bn=ρ1L k1nπ. Clearly that Un∈D(A2),andFnis bounded in H2. Let us show that (iλnI−A2)Un=Fn. Detailing (iλnI−A2)Un, we get (iλnI−A2)Un=/parenleftig 0,C1,nsin/parenleftignπx L/parenrightig ,0,C2,ncos/parenleftignπx L/parenrightig/parenrightig , where (3.3)C1,n=/parenleftbiggk1 ρ1/parenleftignπ L/parenrightig2 −λ2 n/parenrightbigg An+k1nπ ρ1LBn, C2,n=nπk1 ρ2LAn+/parenleftbigg −λ2 n+k1 ρ2+k2+iλnD0 ρ2/parenleftignπ L/parenrightig2/parenrightbigg Bn. Inserting ( 3.2) in (3.3), we get C1,n= 1 andC2,n= 0, hence we obtain (iλnI−A2)Un=/parenleftig 0,sin/parenleftignπx L/parenrightig ,0,0/parenrightig =Fn. Now, we have /bardblUn/bardbl2 H2≥ρ1/integraldisplayL 0/vextendsingle/vextendsingle/vextendsingleiλnAnsin/parenleftignπx L/parenrightig/vextendsingle/vextendsingle/vextendsingle2 dx=ρ1Lλ2 n 2|An|2∼λ4 n. Therefore, for ǫ>0(small enough ), we have λ−2+ǫ n/bardblUn/bardblH2∼λǫ n→+∞. Finally, following to Borichev [ 9] (see Theorem 2.4part (i)) we cannot expect the energy decay rate t−2 2−ǫ./square Note that Theorem 3.1also implies that our system is non-uniformly stable. 8STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING 3.2.Lack of exponential stability of A2with local Kelvin–Voigt damping. In this part, under the equal speed wave propagation condition (i.e.,ρ1 k1=ρ2 k2), we use the classical method developed by Littman and Markus in [ 20] (see also [ 11]), to show that the Timoshenko System ( 1.1)-(1.2) with local Kelvin–Voigt damping, and with Dirichlet-Neumann boundary conditions ( 1.4) is not exponentially stable. For this aim, assume that (3.4)ρ1 k1=ρ2 k2andD(x) =/braceleftigg 0,0<x≤α, D0α<x≤L, whereD0∈R+ ∗andα∈(0,L). For simplicity and without loss of generality, in this part , we takeρ1 k1= 1, D0=k2,L= 1, andα=1 2, then hypothesis ( 3.4) becomes (3.5)ρ1 k1=ρ2 k2= 1andD(x) =/braceleftigg 0,0<x≤1 2, k21 2<x≤1. Our main result in this part is following theorem. Theorem 3.2. Under hypothesis ( 3.5). The semigroup generated by the operator A2is not exponentially stable in the energy space H2. For the proof of Theorem 3.2, we recall the following definitions: the growth bound ω0(A2)and the the spectral bounds(A2)ofA2are defined respectively as ω(A2) = lim t→∞log/vextenddouble/vextenddoubleetA2/vextenddouble/vextenddouble L(H2) tands(A2) = sup{ℜ(λ) :λ∈σ(A2)}. From the Hille-Yoside theorem (see also Theorem 2.1.6 and Le mma 2.1.11 in [ 11]), one has that s(A2)≤ω0(A2). By the previous results, one clearly has that s(A2)≤0and the theorem would follow if equality holds in the previous inequality. It therefore amounts to show the exist ence of a sequence of eigenvalues of A2whose real parts tend to zero. SinceA2is dissipative, we fix α0>0small enough and we study the asymptotic behavior of the eige nvaluesλ ofA2in the strip S={λ∈C:−α0≤ ℜ(λ)≤0}. First, we determine the characteristic equation satisfied b y the eigenvalues of A2. For this aim, let λ∈C∗be an eigenvalue of A2and letU= (u,λu,y,λy,ω )∈D(A2)be an associated eigenvector. Then the eigenvalue problem is given by λ2u−uxx−yx= 0, x∈(0,1), (3.6) c2ux+/parenleftbig λ2+c2/parenrightbig y−/parenleftbigg 1+D k2λ/parenrightbigg yxx= 0, x∈(0,1), (3.7) with the boundary conditions (3.8) u(0) =yx(0) =u(1) =yx(1) = 0, wherec=/radicalig k1k−1 2. We define /braceleftigg u−(x) :=u(x), y−(x) :=y(x), x∈(0,1/2), u+(x) :=u(x), y+(x) :=y(x), x∈[1/2,1), then System ( 3.6)-(3.8) becomes λ2u−−u− xx−y− x= 0, x∈(0,1/2), (3.9) c2u− x+/parenleftbig λ2+c2/parenrightbig y−−y− xx= 0, x∈(0,1/2), (3.10) λ2u+−u+ xx−y+ x= 0, x∈[1/2,1), (3.11) c2u+ x+/parenleftbig λ2+c2/parenrightbig y+−(1+λ)y+ xx= 0, x∈[1/2,1), (3.12) 9STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING with the boundary conditions u−(0) =y− x(0) = 0, (3.13) u+(1) =y+ x(1) = 0, (3.14) and the continuity conditions u−(1/2) =u+(1/2), (3.15) u− x(1/2) =u+ x(1/2), (3.16) y−(1/2) =y+(1/2), (3.17) y− x(1/2) = (1+λ)y+ x(1/2). (3.18) In order to proceed, we set the following notation. Here and b elow, in the case where zis a non zero non-real number, we define (and denote) by√zthe square root of z; i.e., the unique complex number with positive real part whose square is equal to z. Our aim is to study the asymptotic behavior of the large eige nvaluesλofA2 inS. By taking λlarge enough, the general solution of System ( 3.9)-(3.10) with boundary condition ( 3.13) is given by u−(x) =α1sinh(r1x)+α2sinh(r2x), y−(x) =α1λ2−r2 1 r1cosh(r1x)+α2λ2−r2 2 r2cosh(r2x), and the general solution of Equation ( 3.11)-(3.12) with boundary condition ( 3.14) is given by u+(x) =β1sinh(s1(1−x))+β2sinh(s2(1−x)), y+(x) =−β1λ2−s2 1 s1cosh(s1(1−x))−β2λ2−s2 2 s2cosh(s2(1−x)), whereα1, α2, β1, β2∈C, (3.19) r1=λ/radicalbigg 1+ic λ, r 2=λ/radicalbigg 1−ic λ and (3.20) s1=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtλ+λ2 2/parenleftbigg 1+/radicalig 1−4c2 λ3−4c2 λ4/parenrightbigg 1+1 λ, s 2=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtλ+λ2 2/parenleftbigg 1−/radicalig 1−4c2 λ3−4c2 λ4/parenrightbigg 1+1 λ. The boundary conditions in ( 3.15)-(3.18) can be expressed by M α1 α2 β1 β2 = 0, where M= sinh/parenleftbigr1 2/parenrightbig sinh/parenleftbigr2 2/parenrightbig −sinh/parenleftbigs1 2/parenrightbig −sinh/parenleftbigs2 2/parenrightbig r1 icλ2cosh/parenleftbigr1 2/parenrightbigr2 icλ2cosh/parenleftbigr2 2/parenrightbigs1 icλ2cosh/parenleftbigs1 2/parenrightbigs2 icλ2cosh/parenleftbigs2 2/parenrightbig r2 1sinh/parenleftbigr1 2/parenrightbig r2 2sinh/parenleftbigr2 2/parenrightbig /parenleftbig λ3−(λ+1)s2 1/parenrightbig sinh/parenleftbigs1 2/parenrightbig /parenleftbig λ3−(λ+1)s2 2/parenrightbig sinh/parenleftbigs2 2/parenrightbig r−1 1cosh/parenleftbigr1 2/parenrightbig r−1 2cosh/parenleftbigr2 2/parenrightbig s−1 1cosh/parenleftbigs1 2/parenrightbig s−1 2cosh/parenleftbigs2 2/parenrightbig . Denoting the determinant of a matrix Mbydet(M), consequently, System ( 3.9)-(3.18) admits a non trivial solution if and only if det(M) = 0. Using Gaussian elimination, det(M) = 0 is equivalent to det/parenleftig ˜M/parenrightig = 0, 10STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING where˜Mis given by ˜M= sinh/parenleftbigr1 2/parenrightbig sinh/parenleftbigr2 2/parenrightbig −sinh/parenleftbigs1 2/parenrightbig −1−e−s2 r1 icλ2cosh/parenleftbigr1 2/parenrightbigr2 icλ2cosh/parenleftbigr2 2/parenrightbigs1 icλ2cosh/parenleftbigs1 2/parenrightbigs2 icλ2(1+e−s2) r2 1sinh/parenleftbigr1 2/parenrightbig r2 2sinh/parenleftbigr2 2/parenrightbig /parenleftbig λ3−(λ+1)s2 1/parenrightbig sinh/parenleftbigs1 2/parenrightbig /parenleftbig λ3−(λ+1)s2 2/parenrightbig (1−e−s2) r−1 1cosh/parenleftbigr1 2/parenrightbig r−1 2cosh/parenleftbigr2 2/parenrightbig s−1 1cosh/parenleftbigs1 2/parenrightbig s−1 2(1+e−s2) . One gets that (3.21)det/parenleftig ˜M/parenrightig =g1cosh/parenleftigr1 2/parenrightig cosh/parenleftigr2 2/parenrightig sinh/parenleftigs1 2/parenrightig +g2sinh/parenleftigr1 2/parenrightig cosh/parenleftigr2 2/parenrightig cosh/parenleftigs1 2/parenrightig +g3cosh/parenleftigr1 2/parenrightig sinh/parenleftigr2 2/parenrightig cosh/parenleftigs1 2/parenrightig +g4sinh/parenleftigr1 2/parenrightig sinh/parenleftigr2 2/parenrightig cosh/parenleftigs1 2/parenrightig +g5cosh/parenleftigr1 2/parenrightig sinh/parenleftigr2 2/parenrightig sinh/parenleftigs1 2/parenrightig +g6sinh/parenleftigr1 2/parenrightig cosh/parenleftigr2 2/parenrightig sinh/parenleftigs1 2/parenrightig /parenleftbigg −g1cosh/parenleftigr1 2/parenrightig cosh/parenleftigr2 2/parenrightig sinh/parenleftigs1 2/parenrightig −g2sinh/parenleftigr1 2/parenrightig cosh/parenleftigr2 2/parenrightig cosh/parenleftigs1 2/parenrightig −g3cosh/parenleftigr1 2/parenrightig sinh/parenleftigr2 2/parenrightig cosh/parenleftigs1 2/parenrightig +g4sinh/parenleftigr1 2/parenrightig sinh/parenleftigr2 2/parenrightig cosh/parenleftigs1 2/parenrightig +g5cosh/parenleftigr1 2/parenrightig sinh/parenleftigr2 2/parenrightig sinh/parenleftigs1 2/parenrightig +g6sinh/parenleftigr1 2/parenrightig cosh/parenleftigr2 2/parenrightig sinh/parenleftigs1 2/parenrightig/parenrightbigg e−s2, where (3.22) g1=(λ+1)/parenleftbig r2 1−r2 2/parenrightbig/parenleftbig s2 1−s2 2/parenrightbig icr1r2λ2, g2=/parenleftbig r2 2−s2 1/parenrightbig/parenleftbig (λ+1)s2 2−λ3−r2 1/parenrightbig ics1r2λ2, g3=−/parenleftbig r2 1−s2 1/parenrightbig/parenleftbig (λ+1)s2 2−λ3−r2 2/parenrightbig icr1s1λ2, g4=/parenleftbig r2 1−r2 2/parenrightbig/parenleftbig s2 1−s2 2/parenrightbig ics1s2λ2, g5=/parenleftbig r2 1−s2 2/parenrightbig/parenleftbig (λ+1)s2 1−λ3−r2 2/parenrightbig ics2r1λ2, g6=−/parenleftbig r2 2−s2 2/parenrightbig/parenleftbig (λ+1)s2 1−λ3−r2 1/parenrightbig icr2s2λ2. Proposition 3.3. Under hypothesis ( 3.5), there exist n0∈Nsufficiently large and two sequences (λ1,n)|n|≥n0 and(λ2,n)|n|≥n0of simple roots of det(˜M)(that are also simple eigenvalues of A2) satisfying the following asymptotic behavior: Case 1. If there exist no integers κ∈Nsuch thatc= 2κπ(i.e.,sin/parenleftbigc 4/parenrightbig /ne}ationslash= 0andcos/parenleftbigc 4/parenrightbig /ne}ationslash= 0), then λ1,n= 2inπ−2(1−isign(n))sin/parenleftbigc 4/parenrightbig2 /parenleftbig 3+cos/parenleftbigc 2/parenrightbig/parenrightbig/radicalbig π|n|+O/parenleftbig n−1/parenrightbig , (3.23) λ2,n= 2inπ+πi+iarccos/parenleftig cos/parenleftigc 4/parenrightig/parenrightig −(1−isign(n))cos/parenleftbigc 4/parenrightbig2 /parenleftig 1+cos/parenleftbigc 4/parenrightbig2/parenrightig/radicalbig π|n|+O/parenleftbig n−1/parenrightbig . (3.24) Case 2. If there exists κ0∈Nsuch thatc= 2(2κ0+1)π, (i.e.,cos/parenleftbigc 4/parenrightbig = 0), then λ1,n= 2inπ−1−isign(n)/radicalbig π|n|+O/parenleftbig n−1/parenrightbig , (3.25) λ2,n= 2inπ+3πi 2+ic2 32πn−(8+i(3π−2))c2 128π2n2+O/parenleftig |n|−5/2/parenrightig . (3.26) Case 3. If there exists κ1∈Nsuch thatc= 4κ1π, (i.e.,sin/parenleftbigc 4/parenrightbig = 0), then λ1,n= 2inπ+ic2 32πn−c2 16π2n2+O/parenleftig |n|−5/2/parenrightig , (3.27) λ2,n= 2inπ+πi+ic2 32πn−(4+iπ)c2 64π2n2+O/parenleftig |n|−5/2/parenrightig . (3.28) Heresignis used to denote the sign function or signum function. 11STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING The argument for Proposition 3.3relies on the subsequent lemmas. Lemma 3.4. Under hypothesis ( 3.5), letλbe a large eigenvalue of A2, thenλis large root of the following asymptotic equation: (3.29) F(λ) :=f0(λ)+f1(λ) λ1/2+f2(λ) 8λ+f3(λ) 8λ3/2+f4(λ) 128λ2+f5(λ) 128λ5/2+O/parenleftbig λ−3/parenrightbig = 0, where (3.30) f0(λ) = sinh/parenleftbigg3λ 2/parenrightbigg +sinh/parenleftbiggλ 2/parenrightbigg cos/parenleftigc 2/parenrightig , f1(λ) = cosh/parenleftbigg3λ 2/parenrightbigg −cosh/parenleftbiggλ 2/parenrightbigg cos/parenleftigc 2/parenrightig , f2(λ) =c2cosh/parenleftbigg3λ 2/parenrightbigg −4ccosh/parenleftbiggλ 2/parenrightbigg sin/parenleftigc 2/parenrightig , f3(λ) =c2sinh/parenleftbigg3λ 2/parenrightbigg −4cosh/parenleftbigg3λ 2/parenrightbigg +12csinh/parenleftbiggλ 2/parenrightbigg sin/parenleftigc 2/parenrightig +4cosh/parenleftbiggλ 2/parenrightbigg cos/parenleftigc 2/parenrightig , f4(λ) =c2/parenleftbig c2−56/parenrightbig sinh/parenleftbigg3λ 2/parenrightbigg −32c2cosh/parenleftbigg3λ 2/parenrightbigg +8c2/parenleftig csin/parenleftigc 2/parenrightig −8cos/parenleftigc 2/parenrightig +1/parenrightig sinh/parenleftbiggλ 2/parenrightbigg −32c/parenleftig 8sin/parenleftigc 2/parenrightig +ccos/parenleftigc 2/parenrightig/parenrightig cos/parenleftigc 2/parenrightig , f5(λ) =−40c2sinh/parenleftbigg3λ 2/parenrightbigg +/parenleftbig c4−88c2+48/parenrightbig cosh/parenleftbigg3λ 2/parenrightbigg +32c/parenleftig 5sin/parenleftigc 2/parenrightig +ccos/parenleftigc 2/parenrightig/parenrightig sinh/parenleftbiggλ 2/parenrightbigg −/parenleftig 8c3sin/parenleftigc 2/parenrightig −16(4c2−3)cos/parenleftigc 2/parenrightig −24c2/parenrightig cos/parenleftigc 2/parenrightig . Proof. Letλbe a large eigenvalue of A2, thenλis root ofdet/parenleftig ˜M/parenrightig . In this lemma, we furnish an asymptotic development of the function det/parenleftig ˜M/parenrightig for largeλ. First, using the asymptotic expansion in ( 3.19) and ( 3.20), we get (3.31) r1=λ+ic 2+c2 8λ−ic3 16λ2+O/parenleftbig λ−3/parenrightbig , r2=λ−ic 2+c2 8λ+ic3 16λ2+O/parenleftbig λ−3/parenrightbig , s1=λ−c2 2λ2+O/parenleftbig λ−5/parenrightbig , s2=λ1/2−1 2λ1/2+4c2+3 8λ3/2+O/parenleftig λ−5/2/parenrightig . Inserting ( 3.31) in (3.22), we get (3.32) g1= 2−c2 λ2+O/parenleftbig λ−3/parenrightbig , g2= 1+ic 2λ−(3c−16i)c 8λ2+O/parenleftbig λ−3/parenrightbig , g3= 1−ic 2λ−(3c+16i)c 8λ2+O/parenleftbig λ−3/parenrightbig , g4= 2λ1/2−1 λ3/2−4c2−3 4λ5/2+O/parenleftig λ−7/2/parenrightig , g5=λ1/2−1−3ic 2λ3/2−7c2−3−10ic 8λ5/2+O/parenleftig λ−7/2/parenrightig , g6=λ1/2−1+3ic 2λ3/2−7c2−3+10ic 8λ5/2+O/parenleftig λ−7/2/parenrightig . 12STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Inserting ( 3.32) in (3.21), then using the fact that real λis bounded in S, we get (3.33)det/parenleftig ˜M/parenrightig = sinh(L1)+sinh(L2)cosh(L3)+cosh(L1)−cosh(L2)cosh(L3) λ1/2 +iccosh(L2)sinh(L3) 2λ−cosh(L1)−cosh(L2)cosh(L3)+3icsinh(L2)sinh(L3) 2λ3/2 −7c2sinh(L1)+8c2sinh(L2)cosh(L3)−32iccosh(L2)sinh(L3)−c2sinh(L4) 16λ3/2 −(11c2−6)cosh(L1)−(8c2−6) cosh(L2)cosh(L3)+20icsinh(L2)sinh(L3)−3c2cosh(L4) 16λ5/2 +/parenleftig sinh(L1)+sinh(L2)cosh(L3)+O/parenleftig λ−1/2/parenrightig/parenrightig e−s2+O/parenleftbig λ−3/parenrightbig , where L1=r1+r2+s1 2, L2=s1 2, L3=r1−r2 2, L4=r1+r2−s1 2. Next, from ( 3.31) and using the fact that real λis bounded S, we get (3.34) sinh(L1) = sinh/parenleftbigg3λ 2/parenrightbigg +c2cosh/parenleftbig3λ 2/parenrightbig 8λ+c2/parenleftbig c2sinh/parenleftbig3λ 2/parenrightbig −32cosh/parenleftbig3λ 2/parenrightbig/parenrightbig 128λ2+O/parenleftbig λ−3/parenrightbig , cosh(L1) = cosh/parenleftbigg3λ 2/parenrightbigg +c2sinh/parenleftbig3λ 2/parenrightbig 8λ+c2/parenleftbig c2cosh/parenleftbig3λ 2/parenrightbig −32sinh/parenleftbig3λ 2/parenrightbig/parenrightbig 128λ2+O/parenleftbig λ−3/parenrightbig , sinh(L2) = sinh/parenleftbiggλ 2/parenrightbigg −c2cosh/parenleftbigλ 2/parenrightbig 4λ2+O/parenleftbig λ−4/parenrightbig , cosh(L2) = cosh/parenleftbiggλ 2/parenrightbigg −c2sinh/parenleftbigλ 2/parenrightbig 4λ2+O/parenleftbig λ−4/parenrightbig , sinh(L3) =isin/parenleftigc 2/parenrightig −ic3cos/parenleftbigc 2/parenrightbig 16λ2+O/parenleftbig λ−3/parenrightbig , cosh(L3) = cos/parenleftigc 2/parenrightig +c3cos/parenleftbigc 2/parenrightbig 16λ2+O/parenleftbig λ−3/parenrightbig , sinh(L4) = sinh/parenleftbiggλ 2/parenrightbigg +O/parenleftbig λ−1/parenrightbig ,cosh(L4) = cosh/parenleftbiggλ 2/parenrightbigg +O/parenleftbig λ−1/parenrightbig . On the other hand, from ( 3.31) and ( 3.34), we obtain (3.35)/parenleftig sinh(L1)+sinh(L2)cosh(L3)+O/parenleftig λ−1/2/parenrightig/parenrightig e−s2=−/parenleftbigg sinh/parenleftbigg3λ 2/parenrightbigg +sinh/parenleftbiggλ 2/parenrightbigg cos/parenleftigc 2/parenrightig/parenrightbigg e−√ λ. Since real part of√ λis positive, then lim |λ|→∞e−√ λ λ3= 0, hence (3.36) e−√ λ=o/parenleftbig λ−3/parenrightbig . Therefore, from ( 3.35) and ( 3.36), we get (3.37)/parenleftig sinh(L1)+sinh(L2)cosh(L3)+O/parenleftig λ−1/2/parenrightig/parenrightig e−s2=o/parenleftbig λ−3/parenrightbig . Finally, inserting ( 3.34) and ( 3.37) in (3.33), we getλis large root of F, whereFdefined in ( 3.29). /square Lemma 3.5. Under hypothesis ( 3.5), there exist n0∈Nsufficiently large and two sequences (λ1,n)|n|≥n0and (λ2,n)|n|≥n0of simple roots of F(that are also simple eigenvalues of A2) satisfying the following asymptotic behavior: (3.38) λ1,n= 2iπn+iπ+ǫ1,n,such that lim |n|→+∞ǫ1,n= 0 13STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING and (3.39) λ2,n= 2nπi+iπ+iarccos/parenleftig cos2/parenleftigc 4/parenrightig/parenrightig +ǫ2,n,such that lim |n|→+∞ǫ2,n= 0. Proof. First, we look at the roots of f0. From ( 3.30), we deduce that f0can be written as (3.40) f0(λ) = 2sinh/parenleftbiggλ 2/parenrightbigg/parenleftig cosh(λ)+cos2/parenleftigc 4/parenrightig/parenrightig . The roots of f0are given by µ1,n= 2nπi, n ∈Z, µ2,n= 2nπi+iπ+iarccos/parenleftig cos2/parenleftigc 4/parenrightig/parenrightig , n∈Z. Now with the help of Rouché’s theorem, we will show that the ro ots ofFare close to f0. Let us start with the first family µ1,n. LetBn=B(2nπi,rn)be the ball of centrum 2nπiand radiusrn=1 |n|1 4 andλ∈∂Bn; i.e.,λ= 2nπi+rneiθ, θ∈[0,2π). Then (3.41) sinh/parenleftbiggλ 2/parenrightbigg = (−1)nsinh/parenleftbiggrneiθ 2/parenrightbigg =(−1)nrneiθ 2+O(r2 n),cosh(λ) = cosh/parenleftbig rneiθ/parenrightbig = 1+O(r2 n). Inserting ( 3.41) in (3.40), we get f0(λ) = (−1)nrneiθ/parenleftig 1+cos2/parenleftigc 4/parenrightig/parenrightig +O(r3 n). It follows that there exists a positive constant Csuch that ∀λ∈∂Bn,|f0(λ)| ≥Crn=C |n|1 4. On the other hand, from ( 3.29), we deduce that |F(λ)−f0(λ)|=O/parenleftbigg1√ λ/parenrightbigg =O/parenleftigg 1/radicalbig |n|/parenrightigg . It follows that, for |n|large enough ∀λ∈∂Bn,|F(λ)−f0(λ)|<|f0(λ)|. Hence, with the help of Rouché’s theorem, there exists n0∈N∗large enough, such that ∀ |n| ≥n0(n∈Z∗), the first branch of roots of F, denoted by λ1,nare close to µ1,n, hence we get ( 3.38). The same procedure yields (3.39). Thus, the proof is complete. /square Remark 3.6. From Lemma 3.5, we deduce that the real part of the eigenvalues of A2tends to zero, and this is enough to get Theorem 3.2. But we look forward to knowing the real part of λ1,nandλ2,n. Since in the next section, we will use the real part of λ1,nandλ2,nfor the optimality of polynomial stability. /square We are now in a position to conclude the proof of Proposition 3.3. Proof of Proposition 3.3.The proof is divided into two steps. Step 1. Calculation of ǫ1,n. From ( 3.38), we have (3.42) cosh/parenleftbigg3λ1,n 2/parenrightbigg = (−1)ncosh/parenleftbigg3ǫ1,n 2/parenrightbigg ,sinh/parenleftbigg3λ1,n 2/parenrightbigg = (−1)nsinh/parenleftbigg3ǫ1,n 2/parenrightbigg , cosh/parenleftbiggλ1,n 2/parenrightbigg = (−1)ncosh/parenleftigǫ1,n 2/parenrightig ,sinh/parenleftbiggλ1,n 2/parenrightbigg = (−1)nsinh/parenleftigǫ1,n 2/parenrightig , 14STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING and (3.43) 1 λ1,n=−i 2πn+O/parenleftbig ǫ1,nn−2/parenrightbig +O/parenleftbig n−3/parenrightbig ,1 λ2 1,n=−1 4π2n2+O/parenleftbig n−3/parenrightbig , 1/radicalbig λ1,n=1−isign(n) 2/radicalbig π|n|+O/parenleftig ǫ1,n|n|−3/2/parenrightig +O/parenleftig |n|−5/2/parenrightig , 1/radicalig λ3 1,n=−1−isign(n) 4/radicalbig π3|n|3+O/parenleftig |n|−5/2/parenrightig ,1/radicalig λ5 1,n=O/parenleftig |n|−5/2/parenrightig . On the other hand, since lim|n|→+∞ǫ1,n= 0, we have the asymptotic expansion (3.44) cosh/parenleftbigg3ǫ1,n 2/parenrightbigg = 1+9ǫ2 1,n 8+O(ǫ4 1,n),sinh/parenleftbigg3ǫ1,n 2/parenrightbigg =3ǫ1,n 2+O(ǫ3 1,n), cosh/parenleftigǫ1,n 2/parenrightig = 1+ǫ2 1,n 8+O(ǫ4 1,n),sinh/parenleftigǫ1,n 2/parenrightig =ǫ1,n 2+O(ǫ3 1,n). Inserting ( 3.44) in (3.42), we get (3.45) cosh/parenleftbigg3λ1,n 2/parenrightbigg = (−1)n+9(−1)nǫ1,n 8+O(ǫ4 1,n),sinh/parenleftbigg3λ1,n 2/parenrightbigg =3(−1)nǫ1,n 2+O(ǫ3 1,n), cosh/parenleftbiggλ1,n 2/parenrightbigg = (−1)n+(−1)nǫ1,n 8+O(ǫ4 1,n),sinh/parenleftbiggλ1,n 2/parenrightbigg =(−1)nǫ1,n 2+O(ǫ3 1,n). Inserting ( 3.43) and ( 3.45) in (3.29), we get (3.46)ǫ1,n 2/parenleftig 3+cos/parenleftigc 2/parenrightig/parenrightig +(1−isign(n))/parenleftbig 1−cos/parenleftbigc 2/parenrightbig/parenrightbig 2/radicalbig π|n|+ic/parenleftbig 4sin/parenleftbigc 2/parenrightbig −c/parenrightbig 16πn +(1+isign(n))/parenleftbig 1−cos/parenleftbigc 2/parenrightbig/parenrightbig 8/radicalbig π3|n|3+8csin/parenleftbigc 2/parenrightbig +/parenleftbig 1+cos/parenleftbigc 2/parenrightbig/parenrightbig c2 16π2n2 +O/parenleftig |n|−5/2/parenrightig +O/parenleftig ǫ1,n|n|−3/2/parenrightig +O/parenleftig ǫ2 1,n|n|−1/2/parenrightig +O/parenleftbig ǫ3 1,n/parenrightbig = 0. We distinguish two cases: Case 1. Ifsin/parenleftbigc 4/parenrightbig /ne}ationslash= 0,then 1−cos/parenleftigc 2/parenrightig = 2sin2/parenleftigc 4/parenrightig /ne}ationslash= 0, therefore, from ( 3.46), we get ǫ1,n 2/parenleftig 3+cos/parenleftigc 2/parenrightig/parenrightig +sin2/parenleftbigc 4/parenrightbig (1−isign(n))/radicalbig |n|π+O/parenleftbig ǫ3 1,n/parenrightbig +O/parenleftig |n|−1/2ǫ2 1,n/parenrightig +O/parenleftbig n−1/parenrightbig = 0, hence, we get (3.47) ǫ1,n=−2sin2/parenleftbigc 4/parenrightbig (1−isign(n))/parenleftbig 3+cos/parenleftbigc 2/parenrightbig/parenrightbig/radicalbig |n|π+O/parenleftbig n−1/parenrightbig . Inserting ( 3.47) in (3.38), we get ( 3.23) and ( 3.25). Case 2. Ifsin/parenleftbigc 4/parenrightbig = 0,then 1−cos/parenleftigc 2/parenrightig = 2sin2/parenleftigc 4/parenrightig = 0,sin/parenleftigc 2/parenrightig = 2sin/parenleftigc 4/parenrightig cos/parenleftigc 4/parenrightig = 0, therefore, from ( 3.46), we get (3.48) 2ǫ1,n−ic2 16πn+c2 8π2n2+O/parenleftig |n|−5/2/parenrightig +O/parenleftig ǫ1,n|n|−3/2/parenrightig +O/parenleftig ǫ2 1,n|n|−1/2/parenrightig +O/parenleftbig ǫ3 1,n/parenrightbig = 0. Solving Equation ( 3.48), we get (3.49) ǫ1,n=ic2 32πn−c2 16π2n2+O/parenleftig |n|−5/2/parenrightig . 15STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Inserting ( 3.49) in (3.38), we get ( 3.27). Step 2. Calculation of ǫ2,n. We distinguish three cases: Case 1. Ifsin/parenleftbigc 4/parenrightbig /ne}ationslash= 0andcos/parenleftbigc 4/parenrightbig /ne}ationslash= 0, then0<cos2/parenleftbigc 4/parenrightbig <1.Therefore ζ:= arccos/parenleftig cos2/parenleftigc 4/parenrightig/parenrightig ∈/parenleftig 0,π 2/parenrightig . From ( 3.39), we have (3.50)1/radicalbig λ2,n=1−isign(n) 2/radicalbig π|n|+O/parenleftig |n|−3/2/parenrightig and1 λ2,n=O(n−1). Inserting ( 3.39) and ( 3.50) in (3.29), we get (3.51)2sinh/parenleftbiggλ2,n 2/parenrightbigg/parenleftig cosh(λ2,n)+cos2/parenleftigc 4/parenrightig/parenrightig +cosh/parenleftig λ2,n 2/parenrightig/parenleftbig cosh(λ2,n)−cos2/parenleftbigc 4/parenrightbig/parenrightbig (1−isign(n)) /radicalbig π|n|+O(n−1) = 0. From ( 3.39), we obtain (3.52) cosh(λ2,n) =−cos2/parenleftigc 4/parenrightig cosh(ǫ2,n)−isin(ζ)sinh(ǫ2,n), cosh/parenleftbiggλ2,n 2/parenrightbigg = (−1)n/parenleftbigg −sin/parenleftbiggζ 2/parenrightbigg cosh/parenleftigǫ2,n 2/parenrightig +icos/parenleftbiggζ 2/parenrightbigg sinh/parenleftigǫ2,n 2/parenrightig/parenrightbigg , sinh/parenleftbiggλ2,n 2/parenrightbigg = (−1)n/parenleftbigg −sin/parenleftbiggζ 2/parenrightbigg sinh/parenleftigǫ2,n 2/parenrightig +icos/parenleftbiggζ 2/parenrightbigg cosh/parenleftigǫ2,n 2/parenrightig/parenrightbigg . Sinceζ= arccos/parenleftbig cos2/parenleftbigc 4/parenrightbig/parenrightbig ∈/parenleftbig 0,π 2/parenrightbig , we have (3.53) sin(ζ) =/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc 4/parenrightig/vextendsingle/vextendsingle/vextendsingle/radicalbigg 1+cos2/parenleftigc 4/parenrightig ,cos/parenleftbiggζ 2/parenrightbigg =/radicalig 1+cos2/parenleftbigc 4/parenrightbig √ 2,sin/parenleftbiggζ 2/parenrightbigg =/vextendsingle/vextendsinglesin/parenleftbigc 4/parenrightbig/vextendsingle/vextendsingle √ 2. On the other hand, since lim|n|→+∞ǫ2,n= 0, we have the asymptotic expansion (3.54) cosh(ǫ2,n) = 1+O(ǫ2 2,n),sinh(ǫ2,n) =ǫ2,n+O(ǫ3 2,n), cosh/parenleftigǫ2,n 2/parenrightig = 1+O(ǫ2 2,n),sinh/parenleftigǫ2,n 2/parenrightig =ǫ2,n 2+O(ǫ3 2,n). Inserting ( 3.53) and ( 3.54) in (3.52), we get (3.55) cosh(λ2,n) =−cos2/parenleftigc 4/parenrightig −iǫ2,n/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc 4/parenrightig/vextendsingle/vextendsingle/vextendsingle/radicalbigg 1+cos2/parenleftigc 4/parenrightig +O(ǫ2 2,n), cosh/parenleftbiggλ2,n 2/parenrightbigg =(−1)n √ 2 iǫ2,n/radicalig 1+cos2/parenleftbigc 4/parenrightbig 2−/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc 4/parenrightig/vextendsingle/vextendsingle/vextendsingle +O(ǫ2 2,n), sinh/parenleftbiggλ2,n 2/parenrightbigg =−(−1)n 2√ 2/parenleftbigg/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc 4/parenrightig/vextendsingle/vextendsingle/vextendsingleǫ2,n−2i/radicalbigg 1+cos2/parenleftigc 4/parenrightig/parenrightbigg +O(ǫ2 2,n). Inserting ( 3.55) in (3.51), we get √ 2(−1)n/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc 4/parenrightig/vextendsingle/vextendsingle/vextendsingle/parenleftig 1+cos2/parenleftigc 4/parenrightig/parenrightig/parenleftigg ǫ2,n+cos2/parenleftbigc 4/parenrightbig (1−isign(n))/parenleftbig 1+cos2/parenleftbigc 4/parenrightbig/parenrightbig/radicalbig π|n|/parenrightigg +O(n−1)+O(ǫ2 2,n)+O/parenleftig |n|−1/2ǫ2,n/parenrightig = 0, 16STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING since in this case sin/parenleftbigc 4/parenrightbig /ne}ationslash= 0, then we get (3.56) ǫ2,n=−cos2/parenleftbigc 4/parenrightbig (1−isign(n))/parenleftbig 1+cos2/parenleftbigc 4/parenrightbig/parenrightbig/radicalbig π|n|+O(n−1). Inserting ( 3.56) in (3.46), we get ( 3.24). Case 2. Ifcos/parenleftbigc 4/parenrightbig = 0, then (3.57) cos/parenleftigc 2/parenrightig =−1,sin/parenleftigc 2/parenrightig = 0. In this case λ2,nbecomes (3.58) λ2,n= 2inπ+3πi 2+ǫ2,n. Therefore, we have (3.59) cosh/parenleftbigg3λ2,n 2/parenrightbigg =(−1)n √ 2/parenleftbigg cosh/parenleftbigg3ǫ2,n 2/parenrightbigg +isinh/parenleftbigg3ǫ2,n 2/parenrightbigg/parenrightbigg , sinh/parenleftbigg3λ2,n 2/parenrightbigg =(−1)n √ 2/parenleftbigg icosh/parenleftbigg3ǫ2,n 2/parenrightbigg +sinh/parenleftbigg3ǫ2,n 2/parenrightbigg/parenrightbigg , cosh/parenleftbiggλ2,n 2/parenrightbigg =(−1)n √ 2/parenleftig −cosh/parenleftigǫ2,n 2/parenrightig +isinh/parenleftigǫ2,n 2/parenrightig/parenrightig , sinh/parenleftbiggλ2,n 2/parenrightbigg =(−1)n √ 2/parenleftig icosh/parenleftigǫ2,n 2/parenrightig −sinh/parenleftigǫ2,n 2/parenrightig/parenrightig . On the other hand, since lim|n|→+∞ǫ2,n= 0, we have the asymptotic expansion (3.60) cosh/parenleftbigg3ǫ2,n 2/parenrightbigg = 1+9ǫ2 2,n 8+O(ǫ4 2,n),sinh/parenleftbigg3ǫ2,n 2/parenrightbigg =3ǫ2,n 2+O(ǫ3 2,n), cosh/parenleftigǫ2,n 2/parenrightig = 1+ǫ2 2,n 8+O(ǫ4 2,n),sinh/parenleftigǫ2,n 2/parenrightig =ǫ2,n 2+O(ǫ3 2,n). Inserting ( 3.60) in (3.59), we get (3.61) cosh/parenleftbigg3λ2,n 2/parenrightbigg =(−1)n √ 2/parenleftigg 1+3iǫ2,n 2+9ǫ2 2,n 8+O(ǫ3 2,n)/parenrightigg , sinh/parenleftbigg3λ2,n 2/parenrightbigg =(−1)n √ 2/parenleftigg i+3ǫ2,n 2+9iǫ2 2,n 8+O(ǫ3 2,n)/parenrightigg , cosh/parenleftbiggλ2,n 2/parenrightbigg =(−1)n √ 2/parenleftigg −1+iǫ2,n 2−ǫ2 2,n 8+O(ǫ3 2,n)/parenrightigg , sinh/parenleftbiggλ2,n 2/parenrightbigg =(−1)n √ 2/parenleftigg i−ǫ2,n 2+iǫ2 2,n 8+O(ǫ3 2,n)/parenrightigg . Moreover, from ( 3.58), we get (3.62) 1 λ2,n=−i 2πn+3iπ 8π2n2+O/parenleftbig ǫ2,nn−2/parenrightbig +O/parenleftbig n−3/parenrightbig ,1 λ2 2,n=−1 4π2n2+O/parenleftbig n−3/parenrightbig , 1/radicalbig λ2,n=1−isign(n) 2/radicalbig π|n|+3(−sign(n)+i) 16/radicalbig π|n|3+O/parenleftig ǫ2,n|n|−3/2/parenrightig +O/parenleftig |n|−5/2/parenrightig , 1/radicalig λ3 2,n=−1−isign(n) 4/radicalbig π3|n|3+O/parenleftig |n|−5/2/parenrightig ,1/radicalig λ5 2,n=O/parenleftig |n|−5/2/parenrightig . 17STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Inserting ( 3.57), (3.61), and ( 3.62) in (3.29), we get (3.63)iǫ2 2,n 2+/parenleftigg 1+sign(n)+i 2/radicalbig π|n|+3c2 64πn/parenrightigg ǫ2,n−ic2 32πn+(sign(n)−i)c2 64/radicalbig π3|n|3+/parenleftbig 64−i/parenleftbig c2−24π+16/parenrightbig/parenrightbig c2 1024π2n2 +O/parenleftig |n|−5/2/parenrightig +O/parenleftig ǫ2,n|n|−3/2/parenrightig +O/parenleftig ǫ2 2,n|n|−1/2/parenrightig +O/parenleftbig ǫ3 2,n/parenrightbig = 0. From ( 3.63), we get ǫ2,n−ic2 32πn+O/parenleftig ǫ2,n|n|−1/2/parenrightig +O/parenleftbig ǫ2 2,n/parenrightbig = 0, hence (3.64) ǫ2,n=ic2 32πn+ξn n,such that lim |n|→+∞ξn= 0. Inserting ( 3.64) in (3.63), we get ξn n+(8+i(3π−2))c2 128π2n2+O/parenleftig ξn|n|−3/2/parenrightig +O/parenleftig |n|−5/2/parenrightig = 0, therefore (3.65) ξn=−(8+i(3π−2))c2 128π2n+O(n−3/2). Inserting ( 3.64) in (3.65), we get (3.66) ǫ2,n=ic2 32πn−(8+i(3π−2))c2 128π2n2+O(n−5/2). Finally, inserting ( 3.66) in (3.58), we get ( 3.26). Case 3. Ifsin/parenleftbigc 4/parenrightbig = 0, then (3.67) cos/parenleftigc 2/parenrightig = 1,sin/parenleftigc 2/parenrightig = 0. In this case λ2,nbecomes (3.68) λ2,n= 2inπ+iπ+ǫ2,n. Similar to case 2, from ( 3.68) and using the fact that lim|n|→+∞ǫ2,n= 0, we have the asymptotic expansion (3.69) cosh/parenleftbigg3λ2,n 2/parenrightbigg =−3i(−1)nǫ2,n 2+O/parenleftbig ǫ3 2,n/parenrightbig ,sinh/parenleftbigg3λ2,n 2/parenrightbigg =−i(−1)n/parenleftigg 1+9ǫ2 2,n 8/parenrightigg +O(ǫ4 2,n), cosh/parenleftbigg3λ2,n 2/parenrightbigg =i(−1)nǫ2,n 2+O/parenleftbig ǫ3 2,n/parenrightbig ,sinh/parenleftbigg3λ2,n 2/parenrightbigg =i(−1)n/parenleftigg 1+ǫ2 2,n 8/parenrightigg +O(ǫ4 2,n). Moreover, from ( 3.68), we get (3.70) 1 λ2,n=−i 2πn+iπ 4π2n2+O/parenleftbig ǫ2,nn−2/parenrightbig +O/parenleftbig n−3/parenrightbig ,1 λ2 2,n=−1 4π2n2+O/parenleftbig n−3/parenrightbig , 1/radicalbig λ2,n=1−isign(n) 2/radicalbig π|n|+(1+isign(n))ǫ2,n+(−sign(n)+i)π 8/radicalbig π|n|3 +3(1−isign(n)) 64/radicalbig π|n|5+O/parenleftig ǫ2,n|n|−5/2/parenrightig +O/parenleftig |n|−7/2/parenrightig , 1/radicalig λ3 2,n=−1−isign(n) 4/radicalbig π3|n|3+3(sign(n)+i) 16/radicalbig π3|n|5+O/parenleftig ǫ2,n|n|−5/2/parenrightig +O/parenleftig |n|−7/2/parenrightig , 1/radicalig λ5 2,n=−1+isign(n) 8/radicalbig π5|n|5+O/parenleftig |n|−7/2/parenrightig ,1 λ3 2,n=O/parenleftbig n−3/parenrightbig . 18STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Inserting ( 3.67), (3.69), and ( 3.70) in (3.29), we get (3.71)−iǫ2 2,n+/parenleftigg −sign(n)+i/radicalbig π|n|−3c2 32πn+sign(n)−i+(1+isign(n))π 4/radicalbig π3|n|3/parenrightigg ǫ2,n−(sign(n)−i)c2 32/radicalbig π3|n|3 +ic4 512π2n2−3(3(sign(n)+i)−(1−isign(n))π)c2 128/radicalbig π5|n|5 +O/parenleftbig n−3/parenrightbig +O/parenleftbig ǫ2,nn−2/parenrightbig +O/parenleftbig ǫ2 2,nn−1/parenrightbig +O/parenleftbig ǫ3 2,n/parenrightbig = 0. Similar to case 2, solving Equation ( 3.71), we get (3.72) ǫ2,n=ic2 32πn−(4+iπ)c2 64π2n2+O/parenleftig |n|−5/2/parenrightig . Finally, inserting ( 3.72) in (3.68), we get ( 3.28). Thus, the proof is complete. /square Proof of Theorem 3.2.From Proposition 3.3the operator A2has two branches of eigenvalues with eigenvalues admitting real parts tending to zero. Hence, the energy corr esponding to the first and second branch of eigenvalues has no exponential decaying. Therefore the tot al energy of the Timoshenko System ( 1.1)-(1.2) with local Kelvin–Voigt damping, and with Dirichlet-Neuma nn boundary conditions ( 1.4), has no exponential decaying in the equal speed case. /square 4.Polynomial stability In this section, we use the frequency domain approach method to show the polynomial stability of/parenleftbig etAj/parenrightbig t≥0 associated with the Timoshenko System ( 2.1). We prove the following theorem. Theorem 4.1. Under hypothesis (H), for j= 1,2,there exists C >0such that for every U0∈D(Aj), we have (4.1) E(t)≤C t/bardblU0/bardbl2 D(Aj), t>0. SinceiR⊆ρ(Aj),then for the proof of Theorem 4.1, according to Theorem 2.5, we need to prove that (H3) sup λ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−Aj)−1/vextenddouble/vextenddouble/vextenddouble L(Hj)=O/parenleftbig λ2/parenrightbig . We will argue by contradiction. Therefore suppose there exi sts{(λn,Un= (un,vn,yn,zn))}n≥1⊂R×D(Aj), withλn>1and (4.2) λn→+∞,/bardblUn/bardblHj= 1, such that (4.3) λ2 n(iλnUn−AjUn) = (f1,n,f2,n,f3,n,f4,n)→0inHj. Equivalently, we have iλnun−vn=λ−2 nf1,n→0inH1 0(0,L), (4.4) iλnvn−k1 ρ1((un)x+yn)x=λ−2 nf2,n→0inL2(0,L), (4.5) iλnyn−zn=λ−2 nf3,n→0inWj(0,L), (4.6) iλnzn−k2 ρ2/parenleftbigg (yn)x+D k2(zn)x/parenrightbigg x+k1 ρ2((un)x+yn) =λ−2 nf4,n→0inL2(0,L), (4.7) where Wj(0,L) =/braceleftigg H1 0(0,L),ifj= 1, H1 ∗(0,L),ifj= 2. 19STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING In the following, we will check the condition (H3) by finding a contradiction with ( 4.2) such as /bardblUn/bardblHj=o(1). For clarity, we divide the proof into several lemmas. From no w on, for simplicity, we drop the index n. Since Uis uniformly bounded in H,we get from ( 4.4) and ( 4.6) respectively that (4.8)/integraldisplayL 0|u|2dx=O/parenleftbig λ−2/parenrightbig and/integraldisplayL 0|y|2dx=O/parenleftbig λ−2/parenrightbig , Lemma 4.2. Under hypothesis (H), for j= 1,2,we have /integraldisplayL 0D(x)|zx|2dx=o/parenleftbig λ−2/parenrightbig ,/integraldisplayβ α|zx|2dx=o/parenleftbig λ−2/parenrightbig , (4.9) /integraldisplayβ α|yx|2dx=o/parenleftbig λ−4/parenrightbig . (4.10) Proof. First, taking the inner product of ( 4.3) withUinHj, then using the fact that Uis uniformly bounded inHj, we get /integraldisplayL 0D(x)|zx|2dx=−λ−2ℜ/parenleftig/angbracketleftbig λ2AjU,U/angbracketrightbig Hj/parenrightig =λ−2ℜ/parenleftig/angbracketleftbig λ2(iλU−AjU),U/angbracketrightbig Hj/parenrightig =o/parenleftbig λ−2/parenrightbig , hence, we get the first asymptotic estimate of ( 4.9). Next, using hypothesis (H) and the first asymptotic estimate of ( 4.9), we get the second asymptotic estimate of ( 4.9). Finally, from ( 4.3), (4.6), and ( 4.9), we get the asymptotic estimate of ( 4.10). /square Letg∈C1([α,β])such that g(β) =−g(α) = 1,max x∈[α,β]|g(x)|=cgandmax x∈[α,β]|g′(x)|=cg′, wherecgandcg′are strictly positive constant numbers. Remark 4.3. It is easy to see the existence of g(x). For example, we can take g(x) = cos/parenleftig (β−x)π β−α/parenrightig to get g(β) =−g(α) = 1,g∈C1([α,β]),|g(x)| ≤1and|g′(x)| ≤π β−α. Also, we can take g(x) =x2−/parenleftig β+α−2 (β−α)−1/parenrightig x+αβ−(β+α)(β−α)−1. /square Lemma 4.4. Under hypothesis (H), for j= 1,2,we have |z(β)|2+|z(α)|2≤/parenleftigg ρ2λ1 2 2k2+2cg′/parenrightigg/integraldisplayβ α|z|2dx+o/parenleftig λ−5 2/parenrightig , (4.11) /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≤ρ2λ3 2 2k2/integraldisplayβ α|z|2dx+o/parenleftbig λ−1/parenrightbig . (4.12) Proof. The proof is divided into two steps. Step 1. In this step, we prove the asymptotic behavior estimate of ( 4.11). For this aim, first, from ( 4.6), we have (4.13) zx=iλyx−λ−2(f3)xinL2(α,β). Multiplying ( 4.13) by2gzand integrating over (α,β),then taking the real part, we get /integraldisplayβ αg(x)(|z|2)xdx=ℜ/braceleftigg 2iλ/integraldisplayβ αg(x)yxzdx/bracerightigg −ℜ/braceleftigg 2λ−2/integraldisplayβ αg(x)(f4)xzdx/bracerightigg , using by parts integration in the left hand side of above equa tion, we get /bracketleftbig g(x)|z|2/bracketrightbigβ α=/integraldisplayβ αg′(x)|z|2dx+ℜ/braceleftigg 2iλ/integraldisplayβ αg(x)yxzdx/bracerightigg −ℜ/braceleftigg 2λ−2/integraldisplayβ αg(x)(f3)xzdx/bracerightigg , 20STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING consequently, (4.14) |z(β)|2+|z(α)|2≤cg′/integraldisplayβ α|z|2dx+2λcg/integraldisplayβ α|yx||z|dx+2λ−2cg/integraldisplayβ α|(f3)x||z|dx. On the other hand, we have 2λcg|yx||z| ≤ρ2λ1 2|z|2 2k2+2k2λ3 2c2 g ρ2|yx|2and2λ−2|(f3)x||z| ≤cg′|z|2+c2 gλ−4 cg′|(f3)x|2. Inserting the above equation in ( 4.14), then using ( 4.10) and the fact that (f3)x→0inL2(α,β), we get |z(β)|2+|z(α)|2≤/parenleftigg ρ2λ1 2 2k2+2cg′/parenrightigg/integraldisplayβ α|z|2dx+o/parenleftig λ−5 2/parenrightig , hence, we get ( 4.11). Step 2. In this step, we prove the following asymptotic behavior est imate of ( 4.12). For this aim, first, multiplying ( 4.7) by−2ρ2 k2g/parenleftig yx+D(x) k2zx/parenrightig and integrating over (α,β),then taking the real part, we get /integraldisplayβ αg(x)/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg xdx=2ρ2λ k2ℜ/braceleftigg i/integraldisplayβ αg(x)z/parenleftbigg yx+D(x) k2zx/parenrightbigg dx/bracerightigg +2k1 k2ℜ/braceleftigg/integraldisplayβ αg(x)(ux+y)/parenleftbigg yx+D(x) k2zx/parenrightbigg dx/bracerightigg −2ρ2λ−2 k2ℜ/braceleftigg/integraldisplayβ αg(x)f4/parenleftbigg yx+D(x) k2zx/parenrightbigg dx/bracerightigg , using by parts integration in the left hand side of above equa tion, we get /bracketleftigg g(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracketrightiggβ α=/integraldisplayβ αg′(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dx+2ρ2λ k2ℜ/braceleftigg i/integraldisplayβ αg(x)z/parenleftbigg yx+D(x) k2zx/parenrightbigg dx/bracerightigg +2k1 k2ℜ/braceleftigg/integraldisplayβ αg(x)(ux+y)/parenleftbigg yx+D(x) k2zx/parenrightbigg dx/bracerightigg −2ρ2λ−2 k2ℜ/braceleftigg/integraldisplayβ αg(x)f4/parenleftbigg yx+D(x) k2zx/parenrightbigg dx/bracerightigg , consequently, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≤2ρ2cgλ k2/integraldisplayβ α|z|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx cg′/integraldisplayβ α/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dx+2k1cg k2/integraldisplayβ α|ux+y|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx+2ρ2cgλ−2 k2/integraldisplayβ α|f4|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx. Now, using Cauchy Schwarz inequality, Equations ( 4.9)-(4.10), the fact that f5→0inL2(α,β)and the fact thatux+yis uniformly bounded in L2(α,β)in the right hand side of above equation, we get (4.15)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≤2ρ2cgλ k2/integraldisplayβ α|z|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx+o/parenleftbig λ−1/parenrightbig . On the other hand, we have 2ρ2cgλ k2|z|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ρ2λ3 2 2k2|z|2+2ρ2λ1 2c2 g k2/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x) k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . Inserting the above equation in ( 4.15), then using Equations ( 4.9)-(4.10), we get /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≤ρ2λ3 2 2k2/integraldisplayβ α|z|2dx+o/parenleftbig λ−1/parenrightbig , hence, we get ( 4.12). Thus, the proof is complete. /square 21STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Lemma 4.5. Under hypothesis (H), for j= 1,2,we have |ux(α)+y(α)|2=O(1),|ux(β)+y(β)|2=O(1). (4.16) |u(α)|2=O/parenleftbig λ−2/parenrightbig ,|u(β)|2=O/parenleftbig λ−2/parenrightbig , (4.17) |v(α)|2=O(1),|v(β)|2=O(1). (4.18) Proof. Multiplying Equation ( 4.5) by−2ρ1 k1g(ux+y)and integrating over (α,β),then taking the real part and using the fact that ux+yis uniformly bounded in L2(α,β),f2→0inL2(α,β), we get (4.19)/integraldisplayβ αg(x)/parenleftig |ux+y|2/parenrightig xdx−2ρ1λ k1ℜ/braceleftigg i/integraldisplayβ αg(x)uxvdx/bracerightigg =2ρ1λ k1ℜ/braceleftigg i/integraldisplayβ αg(x)yvdx/bracerightigg +o/parenleftbig λ−2/parenrightbig . Now, we divided the proof into two steps. Step 1. In this step, we prove the asymptotic behavior estimates of ( 4.16)-(4.17). First, from ( 4.4), we have −iλv=λ2u+iλ−1f1. Inserting the above equation in the second term in left of ( 4.19), then using the fact that uxis uniformly bounded in L2(α,β)andf1→0inL2(α,β), we get /integraldisplayβ αg(x)/parenleftig |ux+y|2/parenrightig xdx+ρ1λ2 k1/integraldisplayβ αg(x)/parenleftig |u|2/parenrightig xdx=−2ρ1λ2 k1ℜ/braceleftigg/integraldisplayβ αg(x)uydx/bracerightigg +o/parenleftbig λ−1/parenrightbig . Using by parts integration and the fact that g(β) =−g(α) = 1 in the above equation, we get |ux(β)+y(β)|2+ρ1λ2 k1|u(β)|2+|ux(α)+y(α)|2+ρ1λ2 k1|u(α)|2=/integraldisplayβ αg′(x)|ux+y|2dx +ρ1λ2 k1/integraldisplayβ αg′(x)|u|2dx−2ρ1λ2 k1ℜ/braceleftigg/integraldisplayβ αg(x)uydx/bracerightigg +o/parenleftbig λ−1/parenrightbig , consequently, |ux(β)+y(β)|2+ρ1λ2 k1|u(β)|2+|ux(α)+y(α)|2+ρ1λ2 k1|u(α)|2≤cg′/integraldisplayβ α|ux+y|2dx +ρ1cg′λ2 k1/integraldisplayβ α|u|2dx+2ρ1cgλ2 k1/integraldisplayβ α|u||y|dx+o/parenleftbig λ−1/parenrightbig . Next, since λu, λy andux+yare uniformly bounded, then from the above equation, we get ( 4.16)-(4.17). Step 2. In this step, we prove the asymptotic behavior estimates of ( 4.18). First, from ( 4.4), we have −iλux=vx−λ−2(f1)x. Inserting the above equation in the second term in left of ( 4.19), then using the fact that vis uniformly bounded inL2(α,β)and(f1)x→0inL2(α,β), we get /integraldisplayβ αg(x)/parenleftig |ux+y|2/parenrightig xdx+ρ1 k1/integraldisplayβ αg(x)/parenleftig |v|2/parenrightig xdx=−2ρ1λ2 k1ℜ/braceleftigg/integraldisplayβ αg(x)uydx/bracerightigg +o/parenleftbig λ−1/parenrightbig . Similar to step 1, by using by parts integration and the fact t hatg(β) =−g(α) = 1 in the above equation, then using the fact that v, λu, λy andux+yare uniformly bounded in L2(α,β), we get ( 4.18). Thus, the proof is complete. /square Lemma 4.6. Under hypothesis (H), for j= 1,2,forλlarge enough, we have /integraldisplayβ α|z|2dx=o/parenleftig λ−5 2/parenrightig ,/integraldisplayβ α|y|2dx=o/parenleftig λ−9 2/parenrightig , (4.20) /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =o/parenleftbig λ−1/parenrightbig ,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D(x) k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =o/parenleftbig λ−1/parenrightbig . (4.21) 22STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Proof. The proof is divided into two steps. Step 1. In this step, we prove the following asymptotic behavior est imate (4.22)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ α(ux+y)zdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbigg1 4+k2cg′ ρ2λ1 2+k1 ρ2λ2/parenrightbigg/integraldisplayβ α|z|2dx+o/parenleftbig λ−3/parenrightbig . For this aim, first, we have (4.23)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ α(ux+y)zdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ αyzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ αuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Now, from ( 4.6) and using the fact that f3→0inL2(α,β)andzis uniformly bounded in L2(α,β), we get (4.24)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ αyzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤k1 ρ2λ2/integraldisplayβ α|z|2dx+o/parenleftbig λ−4/parenrightbig . Next, by using by parts integration, we get/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ αuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−ik1 ρ2λ/integraldisplayβ αuzxdx+ik1 ρ2λu(β)z(β)−ik1 ρ2λu(α)z(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, consequently, (4.25)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ αuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤k1 ρ2λ/integraldisplayβ α|u||zx|dx+k1 ρ2λ(|u(β)||z(β)|+|u(α)||z(α)|), On the other hand, we have k1 ρ2λ(|u(β)||z(β)|+|u(α)||z(α)|)≤k2 1 2k2ρ2λ3 2/parenleftig |u(α)|2+|u(β)|2/parenrightig +k2 2ρ2λ1 2/parenleftig |z(α)|2+|z(β)|2/parenrightig . Inserting ( 4.11) and ( 4.17) in the above equation, we get k1 ρ2λ(|u(β)||z(β)|+|u(α)||z(α)|)≤/parenleftbigg1 4+k2cg′ ρ2λ1 2/parenrightbigg/integraldisplayβ α|z|2dx+o/parenleftbig λ−3/parenrightbig . Inserting the above equation in ( 4.25), then using ( 4.9) and the fact that λuis bounded in L2(α,β), we get /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ αuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbigg1 4+k2cg′ ρ2λ1 2/parenrightbigg/integraldisplayβ α|z|2dx+o/parenleftbig λ−3/parenrightbig . Finally, inserting the above equation and Equation ( 4.24) in (4.23), we get ( 4.22). Step 2. In this step, we prove the asymptotic behavior estimates of ( 4.20)-(4.21). For this aim, first, multiplying (4.7) by−iλ−1ρ−1 2zand integrating over (α,β),then taking the real part, we get /integraldisplayβ α|z|2dx=−k2 ρ2λℜ/braceleftigg i/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg xzdx/bracerightigg +k1 ρ2λℜ/braceleftigg i/integraldisplayβ α(ux+y)zdx/bracerightigg −λ−3ℜ/braceleftigg i/integraldisplayβ αf4zdx/bracerightigg , consequently, (4.26)/integraldisplayβ α|z|2dx≤k2 ρ2λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1 ρ2λ/integraldisplayβ α(ux+y)zdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+λ−3/integraldisplayβ α|f4||z|dx. From the fact that zis uniformly bounded in L2(α,β)andf5→0inL2(α,β), we get (4.27) λ−3/integraldisplayβ α|f4||z|dx=o/parenleftbig λ−3/parenrightbig . Inserting ( 4.22) and ( 4.27) in (4.26), we get (4.28)/integraldisplayβ α|z|2dx≤k2 ρ2λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/parenleftbigg1 4+k2cg′ ρ2λ1 2+k1 ρ2λ2/parenrightbigg/integraldisplayβ α|z|2dx+o/parenleftbig λ−3/parenrightbig . 23STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Now, using by parts integration and ( 4.9)-(4.10), we get (4.29)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg/parenleftbigg yx+D k2zx/parenrightbigg z/bracketrightbiggβ α−/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg zxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(β)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(α)|+/integraldisplayβ α/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle|zx|dx ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(β)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(α)|+o/parenleftbig λ−2/parenrightbig . Inserting ( 4.29) in (4.28), we get (4.30)/parenleftbigg3 4−k2cg′ ρ2λ1 2−k1 ρ2λ2/parenrightbigg/integraldisplayβ α|z|2dx ≤k2 ρ2λ/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(β)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(α)|/parenrightbigg +o/parenleftbig λ−3/parenrightbig . Now, forζ=βorζ=α, we have k2 ρ2λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (ζ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(ζ)| ≤k2λ−1 2 2ρ2|z(ζ)|2+k2λ−3 2 2ρ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (ζ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . Inserting the above equation in ( 4.30), we get /parenleftbigg3 4−k2cg′ ρ2λ1 2−k1 ρ2λ2/parenrightbigg/integraldisplayβ α|z|2dx≤k2λ−3 2 2ρ2/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg +k2λ−1 2 2ρ2/parenleftbig |z(α)|2+|z(β)|2/parenrightbig +o/parenleftbig λ−3/parenrightbig . Inserting Equations ( 4.11) and ( 4.12) in the above inequality, we obtain /parenleftbigg3 4−k2cg′ ρ2λ1 2−k1 ρ2λ2/parenrightbigg/integraldisplayβ α|z|2dx≤/parenleftbigg1 2+k2cg′ ρ2λ1 2/parenrightbigg/integraldisplayβ α|z|2dx+o/parenleftig λ−5 2/parenrightig , consequently,/parenleftbigg1 4−2k2cg′ ρ2λ1 2−k1 ρ2λ2/parenrightbigg/integraldisplayβ α|z|2dx≤o/parenleftig λ−5 2/parenrightig , sinceλ→+∞, forλlarge enough, we get 0</parenleftbigg1 4−2k2cg′ ρ2λ1 2−k1 ρ2λ2/parenrightbigg/integraldisplayβ α|z|2dx≤o/parenleftig λ−5 2/parenrightig , hence, we get the first asymptotic estimate of ( 4.20). Then, inserting the first asymptotic estimate of ( 4.20) in (4.6), we get the second asymptotic estimate of ( 4.20). Finally, inserting ( 4.20) in (4.12), we get ( 4.21). Thus, the proof is complete. /square Lemma 4.7. Under hypothesis (H), for j= 1,2,forλlarge enough, we have (4.31)/integraldisplayβ α|ux|2dx=o(1)and/integraldisplayβ α|v|2dx=o(1). Proof. The proof is divided into two steps. Step 1. In this step, we prove the first asymptotic behavior estimate of (4.31). First, multiplying Equation (4.7) byρ2 k1(ux+y)and integrating over (α,β), we get /integraldisplayβ α|ux+y|2dx−k2 k1/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg x(ux+y)dx=−iρ2λ k1/integraldisplayβ αz(ux+y)dx+ρ2 k1λ2/integraldisplayβ αf4(ux+y)dx, 24STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING using by parts integration in the second term in the left hand side of above equation, we get (4.32)/integraldisplayβ α|ux+y|2dx+k2 k1/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg (ux+y)xdx=k2 k1/bracketleftbigg/parenleftbigg yx+D k2zx/parenrightbigg (ux+y)/bracketrightbiggβ α −iρ2λ k1/integraldisplayβ αz(ux+y)dx+ρ2 k1λ2/integraldisplayβ αf4(ux+y)dx. Next, multiplying Equation ( 4.5) byρ1k2 k2 1/parenleftig yx+D k2zx/parenrightig and integrating over (α,β), then using the fact that f2→0inL2(0,L)and Equations ( 4.9)-(4.10), we get −k2 k1/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg (ux+y)xdx=−iρ1k2λ k2 1/integraldisplayβ αv/parenleftbigg yx+D k2zx/parenrightbigg dx+ρ1k2 k2 1λ2/integraldisplayβ αf2/parenleftbigg yx+D k2zx/parenrightbigg dx, consequently, (4.33)−k2 k1/integraldisplayβ α/parenleftbigg yx+D k2zx/parenrightbigg (ux+y)xdx=iρ1k2λ k2 1/integraldisplayβ αv/parenleftbigg yx+D k2zx/parenrightbigg dx+ρ1k2 k2 1λ2/integraldisplayβ αf2/parenleftbigg yx+D k2zx/parenrightbigg dx. Adding ( 4.32) and ( 4.33), we obtain /integraldisplayβ α|ux+y|2dx=−iρ2λ k1/integraldisplayβ αz(ux+y)dx+k2 k1/bracketleftbigg/parenleftbigg yx+D k2zx/parenrightbigg (ux+y)/bracketrightbiggβ α +iρ1k2λ k2 1/integraldisplayβ αv/parenleftbigg yx+D k2zx/parenrightbigg dx+ρ2 k1λ2/integraldisplayβ αf4(ux+y)dx+ρ1k2 k2 1λ2/integraldisplayβ αf2/parenleftbigg yx+D k2zx/parenrightbigg dx, therefore (4.34)/integraldisplayβ α|ux+y|2dx≤ρ2λ k1/integraldisplayβ α|z||ux+y|dx+k2 k1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(β)+y(β)| +k2 k1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(α)+y(α)|+ρ1k2λ k2 1/integraldisplayβ α|v|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx +ρ2 k1λ2/integraldisplayβ α|f4||ux+y|dx+ρ1k2 k2 1λ2/integraldisplayβ α|f2|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx. From ( 4.3), (4.9), (4.10), (4.16), (4.20), (4.21) and the fact that v, ux+yare uniformly bounded in L2(α,β), we obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(β)+y(β)|=o/parenleftig λ−1 2/parenrightig ,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg yx+D k2zx/parenrightbigg (α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(α)+y(α)|=o/parenleftig λ−1 2/parenrightig , λ/integraldisplayβ α|z||ux+y|dx=o/parenleftig λ−1 4/parenrightig , λ/integraldisplayβ α|v|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx=o(1), λ−2/integraldisplayβ α|f4||ux+y|dx=o/parenleftbig λ−2/parenrightbig , λ−2/integraldisplayβ α|f2|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx=o/parenleftbig λ−3/parenrightbig . Inserting the above equation in ( 4.34), we get /integraldisplayβ α|ux+y|2dx=o(1). From the above equation and ( 4.20), we get the first asymptotic estimate of ( 4.31). Step 2. In this step, we prove the second asymptotic behavior estima te of ( 4.31). Multiplying ( 4.5) by−iλ−1v and integrating over (α,β),then taking the real part, we get /integraldisplayβ α|v|2dx=−k1 ρ1λℜ/braceleftigg i/integraldisplayβ α(ux+y)xvdx/bracerightigg −λ−3ℜ/braceleftigg i/integraldisplayβ αf2vdx/bracerightigg , using by parts integration in the second term in the right han d side of above equation, we get /integraldisplayβ α|v|2dx=k1 ρ1λℜ/braceleftigg i/integraldisplayβ α(ux+y)vxdx/bracerightigg −k1 ρ1λℜ/braceleftig i[(ux+y)v]β α/bracerightig −λ−3ℜ/braceleftigg i/integraldisplayβ αf2vdx/bracerightigg . 25STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING consequently, (4.35)/integraldisplayβ α|v|2dx≤k1 ρ1λ/integraldisplayβ α|ux+y||vx|dx+k1 ρ1λ(|ux(β)+y(β)||v(β)|+|ux(α)+y(α)|v(α)|) +λ−3/integraldisplayβ α|f2||v|dx. Finally, from ( 4.16), (4.18), (4.20), the first asymptotic behavior estimate of ( 4.31), the fact that λ−1vx, vare uniformly bounded in L2(α,β)and the fact that f2→0inL2(α,β), we get the second asymptotic behavior estimate of ( 4.20). Thus, the proof is complete. /square From what precedes, under hypothesis (H), for j= 1,2,from Lemmas 4.5,4.6and4.7, we deduce that (4.36) /bardblU/bardblHj=o(1),over(α,β). Lemma 4.8. Under hypothesis (H), for j= 1,2,we have /bardblU/bardblHj=o(1),over(0,L). Proof. Letφ∈H1 0(0,L)be a given function. We proceed the proof in two steps. Step 1. Multiplying Equation ( 4.5) by2ρ1φuxand integrating over (α,β),then using the fact that uxis bounded in L2(0,L),f2→0inL2(0,L), and use Dirichlet boundary conditions to get (4.37) ℜ/braceleftigg 2iρ1λ/integraldisplayL 0φvuxdx/bracerightigg +k1/integraldisplayL 0φ′|ux|2dx−ℜ/braceleftigg 2k1/integraldisplayL 0φuxyxdx/bracerightigg =o(λ−2). From ( 4.4), we have iλux=−vx−λ−2(f1)x. Inserting the above equation in ( 4.37), then using the fact that (f1)x→0inL2(0,L)and the fact that vis bounded in L2(0,L), we get (4.38) ρ1/integraldisplayL 0φ′|v|2dx+k1/integraldisplayL 0φ′|ux|2dx−ℜ/braceleftigg 2k1/integraldisplayL 0φuxyxdx/bracerightigg =o(λ−2). Similarly, multiplying Equation ( 4.7) by2ρ2φ/parenleftig yx+D k1zx/parenrightig and integrating over (α,β),then using by parts integration and Dirichlet boundary conditions to get (4.39)ℜ/braceleftigg 2iρ2λ/integraldisplayL 0φzyxdx/bracerightigg +k2/integraldisplayL 0φ′/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dx+ℜ/braceleftigg 2k1/integraldisplayL 0φyxuxdx/bracerightigg =−λ−1ℜ/braceleftigg 2k1/integraldisplayL 0φλyyxdx/bracerightigg −ℜ/braceleftigg 2iρ2 k1λ/integraldisplayL 0D(x)φzzxdx/bracerightigg −ℜ/braceleftigg 2/integraldisplayL 0D(x)φzxuxdx/bracerightigg −ℜ/braceleftigg 2/integraldisplayL 0D(x)φzxydx/bracerightigg +ℜ/braceleftigg 2ρ2λ−2/integraldisplayL 0φf3yxdx/bracerightigg +ℜ/braceleftigg 2ρ2 k1λ−2/integraldisplayL 0D(x)φf3zxdx/bracerightigg . For all bounded h∈L2(0,L), using Cauchy-Schwarz inequality, the first estimation of ( 4.9), and the fact that D∈L∞(0,L), to obtain (4.40) ℜ/braceleftigg/integraldisplayL 0D(x)hzxdx/bracerightigg ≤/parenleftigg sup x∈(0,L)D1/2(x)/parenrightigg/parenleftigg/integraldisplayL 0D(x)|zx|2dx/parenrightigg1/2/parenleftigg/integraldisplayL 0|h|2dx/parenrightigg1/2 =o(λ−1). From ( 4.39) and using ( 4.40), the fact that z, λy, y xare bounded in L2(0,L), the fact that f3→0inL2(0,L), we get (4.41) ℜ/braceleftigg 2iρ2λ/integraldisplayL 0φzyxdx/bracerightigg +k2/integraldisplayL 0φ′/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dx+ℜ/braceleftigg 2k1/integraldisplayL 0φyxuxdx/bracerightigg =o(1). On the other hand, from ( 4.6), we have iλyx=−zx−λ−2(f3)x. 26STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Inserting the above equation in ( 4.41), then using the fact that (f3)x→0inL2(0,L)and the fact that zis bounded in L2(0,L), we get (4.42) ρ2/integraldisplayL 0φ′|z|2dx+k2/integraldisplayL 0φ′/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dx+ℜ/braceleftigg 2k1/integraldisplayL 0φyxuxdx/bracerightigg =o(1). Adding ( 4.38) and ( 4.42), we get (4.43)/integraldisplayL 0φ′/parenleftigg ρ1|v|2+ρ2|z|2+k1|ux|2+k2/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg dx=o(1). Step 2. Letǫ>0such thatα+ǫ<β and define the cut-off function ς1inC1([0,L])by 0≤ς1≤1, ς1= 1on[0,α]andς1= 0on[α+ǫ,L]. Takeφ=xς1in (4.43), then use the fact that /bardblU/bardblHj=o(1)on(α,β)(i.e., ( 4.36)), the fact that α<α+ǫ<β, and (4.9)-(4.10), we get (4.44)/integraldisplayα 0/parenleftigg ρ1|v|2+ρ2|z|2+k1|ux|2+k2/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg dx=o(1). Moreover, using Cauchy-Schwarz inequality, the first estim ation of ( 4.9), the fact that D∈L∞(0,L), and (4.44), we get (4.45)/integraldisplayα 0|yx|2dx≤2/integraldisplayα 0/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dx+2 k2 2/integraldisplayα 0D(x)2|zx|2dx, ≤2/integraldisplayα 0/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D k2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dx+2/parenleftig supx∈(0,α)D(x)/parenrightig k2 2/integraldisplayα 0D(x)|zx|2dx, =o(1). Using ( 4.44) and ( 4.45), we get /bardblU/bardblHj=o(1)on(0,α). Similarly, by symmetry, we can prove that /bardblU/bardblHj=o(1)on(β,L)and therefore /bardblU/bardblHj=o(1)on(0,L). Thus, the proof is complete. /square Proof of Theorem 4.1.Under hypothesis (H), for j= 1,2,from Lemma 4.8, we have /bardblU/bardblHj=o(1),over (0,L), which contradicts ( 4.2). This implies that sup λ∈R/vextenddouble/vextenddouble/vextenddouble(iλId−Aj)−1/vextenddouble/vextenddouble/vextenddouble L(Hj)=O/parenleftbig λ2/parenrightbig . The result follows from Theorem 2.5part (i). /square It is very important to ask the question about the optimality of (4.1). For the optimality of ( 4.1), we first recall Theorem 3.4.1 stated in [ 31]. Theorem 4.9. LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0onH. Assume thatiR∈ρ(A). Let(λk,n)1≤k≤k0, n≥1denote the k-th branch of eigenvalues of Aand(ek,n)1≤k≤k0, n≥1the system of normalized associated eigenvectors. Assume that for each 1≤k≤k0there exist a positive sequence µk,n→ ∞ asn→ ∞ and two positive constant αk>0,βk>0such that (4.46) ℜ(λk,n)∼ −βk µαk k,nandℑ(λk,n)∼µk,nasn→ ∞. Hereℑis used to denote the imaginary part of a complex number. Furt hermore, assume that for u0∈D(A), there exists constant M >0independent of u0such that (4.47)/vextenddouble/vextenddoubleetAu0/vextenddouble/vextenddouble2 H≤M t2 ℓk/bardblu0/bardbl2 D(A), ℓk= max 1≤k≤k0αk,∀t>0. 27STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING Then the decay rate ( 4.47) is optimal in the sense that for any ǫ>0we cannot expect the energy decay rate t−2 ℓk−ǫ. /square In the next corollary, we show that the optimality of ( 4.1) in some cases. Corollary 4.10. For everyU0∈D(A2), we have the following two cases: 1. If condition ( 3.1) holds, then the energy decay rate in ( 4.1) is optimal. 2. If condition ( 3.4) holds and if there exists κ1∈Nsuch thatc=/radicalig k1 k2= 2κ1π, then the energy decay rate in ( 4.1) is optimal. Proof. We distinguish two cases: 1. If condition ( 3.1) holds, then from Theorem 3.1, forǫ>0(small enough ), we cannot expect the energy decay ratet−2 2−ǫfor all initial data U0∈D(A2)and for allt>0.Hence the energy decay rate in ( 4.1) is optimal. 2. If condition ( 3.4) holds, first following Theorem 4.1, for all initial data U0∈D(A2)and for all t>0, we get ( 4.47) withℓk= 2. Furthermore, from Proposition 3.3(case 2 and case 3), we remark that: Case 1. If there exists κ0∈Nsuch thatc= 2(2κ0+1)π, we have ℜ(λ1,n)∼ −1 π1/2|n|1/2,ℑ(λ1,n)∼2nπ, ℜ(λ2,n)∼ −c2 16π2n2,ℑ(λ2,n)∼/parenleftbigg 2n+3 2/parenrightbigg π, then ( 4.46) holds with α1=1 2andα2= 2. Therefore, ℓk= 2 = max( α1,α2).Then, applying Theorem 4.9, we get that the energy decay rate in ( 4.1) is optimal. Case 2. 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2210.03865v1.Recover_all_Coefficients_in_Second_Order_Hyperbolic_Equations_from_Finite_Sets_of_Boundary_Measurements.pdf

arXiv:2210.03865v1 [math.AP] 8 Oct 2022RECOVER ALL COEFFICIENTS IN SECOND-ORDER HYPERBOLIC EQUATIONS FROM FINITE SETS OF BOUNDARY MEASUREMENTS SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS Abstract. We consider the inverse hyperbolic problem of recovering all spatial dependent coefficients, which are the wave speed, the damping coe fficient, po- tential coefficient and gradient coefficient, in a second-order hype rbolic equation defined on an open bounded domain with smooth enough boundary. W e show that by appropriately selecting finite pairs of initial conditions we can uniquely and Lipschitz stably recover all those coefficients from the corres ponding bound- ary measurements of their solutions. The proofs are based on sha rp Carleman estimate, continuous observability inequality and regularity theory for general second-order hyperbolic equations. Keywords : Inverse hyperbolic problem, finite sets of measurements, Carlem an estimates, uniqueness and stability 2010 Mathematics Subject Classifications : 35R30; 35L10 1.Introduction and Main Results Let Ω⊂Rn,n≥2, be an open bounded domain with smooth enough (e.g., C2) boundary Γ = ∂Ω =Γ0∪Γ1, where Γ 0∩Γ1=∅. We refer Γ 1as the observed part of the boundary where the measurements are taken, and Γ 0as the unobserved part of the boundary. We consider the following general second-o rder hyperbolic equation for w=w(x,t) defined on Q= Ω×[−T,T], along with initial conditions {w0,w1}and Dirichlet boundary condition hon Σ = Γ ×[−T,T] that are given in appropriate function spaces: (1) wtt−c2(x)∆w+q1(x)wt+q0(x)w+q(x)·∇w= 0 inQ w(x,0) =w0(x);wt(x,0) =w1(x) in Ω w(x,t) =h(x,t) in Σ . Here the wave speed c(x) satisfies c∈C={c∈C1(Ω) :c−1 0≤c(x)≤c0,for some c0>0} Date: October 11, 2022. 12 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS q1∈L∞(Ω),q0∈L∞(Ω), and q∈(L∞(Ω))nare the damping, potential, and gradient coefficients, respectively. We then consider the following inverse problem for the system (1): R ecover all together the wave speed c(x), the damping coefficient q1(x), the potential coefficient q0(x), and the gradient coefficient q(x) from measurements of Neumann boundary traces of the solution w=w(w0,w1,h,c,q 1,q0,q) over the observed part Γ 1of the boundary and over the time interval [ −T,T]. Of course here T >0 should be sufficiently large due to the finite propagation speed of the syste m (1). In addition, to make the observed part Γ 1of the boundary more precise, in this paper we assume the following standard geometrical assumptions on the d omain Ω and the unobserved part of the boundary Γ 0: (A.1) There exists a strictly convex function d:Ω→Rin the metric g= c−2(x)dx2, and of class C3(Ω), such that the following two properties hold true (through translation and rescaling if necessary): (i) The normal derivative of don the unobserved part Γ 0of the boundary is non-positive. Namely, ∂d ∂ν=/a\}⌊∇a⌋ketle{tDd(x),ν(x)/a\}⌊∇a⌋ket∇i}ht ≤0,∀x∈Γ0, whereDd=∇gdis the gradient vector field on Ω with respect to g. (ii) D2d(X,X) =/a\}⌊∇a⌋ketle{tDX(Dd),X/a\}⌊∇a⌋ket∇i}htg≥2|X|2 g,∀X∈Mx,min x∈Ωd(x) =m0>0 whereD2dis the Hessian of d(a second-order tensor) and Mxis the tangent space atx∈Ω. (A.2)d(x) has no critical point on Ω. In other words, inf x∈Ω|Dd|>0,so that we may take inf x∈Ω|Dd|2 d>4. Remark 1.1. The geometrical assumptions above permit the construction of a v ec- tor field that enables a pseudo-convex function necessary for allo wing a Carleman estimate containing no lower-order terms for the general second -order equation (1) (see Section 2). These assumptions are first formulated in [16] und er the framework of a Euclidean metric, with [22] employing them under the more genera l Riemann- ian framework. For examples and detailed illustrations of large gener al classes of domains {Ω,Γ1,Γ0}satisfying the aforementioned assumptions we refer to [22, Ap- pendix B]. One canonical example is to take d(x) =|x−x0|2, withx0being a point outsideΩ, if the wave speed csatisfies/vextendsingle/vextendsingle/vextendsingle∇c(x)·(x−x0) 2c(x)/vextendsingle/vextendsingle/vextendsingle≤rc<1 for some rc∈(0,1). The classical inverse hyperbolic problems usually involve recovering a single un- known coefficient, typically the damping coefficient orthe potential c oefficient, fromRECOVER ALL COEFFICIENTS 3 asingleboundary measurement of the solution. To some extent, those se tup are ex- pectedsince theunknown coefficient, whether itisthedampingorth epotentialone, depends on nindependent variables and the corresponding boundary measurem ent also depends on nfree variables. In fact, under proper conditions it is even possible to recover both potential and damping coefficients in one shot by ju st one single boundary measurement [19]. In the case of a gradient coefficient, t he unknown function is vector-valued and containing ndifferent real-valued functions. Hence a single measurement does not seem to be sufficient to recover all of t hem, which is probably why such problem is much less studied in the literature. Neve rtheless, it is possible to recover the coefficient by properly making nsets of boundary measure- ments [8]. Last, in the case of recovering the variable unknown wave speed, since the unknown function is at the principle order level, one typically need s to rewrite the hyperbolic equation as a Riemannian wave equation so that the pr inciple part becomes constant coefficients on an appropriate Riemannian manifo ld [3, 20]. In this paper, we seek to recover all together the aformentioned coefficients in the second-order hyperbolic equation (1). To the best of our knowled ge, this is the first paper that addresses the uniqueness and stability of recovering a ll these coefficients at once through finitely many boundary measurements. Note that all together these coefficients contain a total of n+3 unknown functions, so naturally one may expect to be able to recover them by making n+3 sets of boundary measurements. This is entirely possible to do following the ideas in this paper (see Remar k (1) in Section 4). Nevertheless, in the following we will show that by appro priately choosing ⌊n+4 2⌋1pairs of initial conditions {w0,w1}and a boundary condition h, we can uniquely and Lipschitz stably recover the coefficients c,q1, q0,andqall at once from the corresponding Neumann boundary measurements of the ir solutions. The precise results are stated in Theorem 1.1 and Theorem 1.2 below. As mentioned above recovering a single coefficient from a single bound ary mea- surement isastandardformulationininversehyperbolicproblemsan dsuchproblem hasbeenstudiedextensively intheliterature. Hereweonlymentiont hemonographs and lecture notes [4, 6, 7, 9, 10, 17, 21] and refer to the substan tial lists of refer- ences therein. The standard approach for this type of inverse hy perbolic problems typically involves using Carleman-type estimates for the second-or der hyperbolic equations. To certain extent, such methods can all be seen as var iations or improve- ments of the so called Bukhgeim–Klibanov (BK) method which was origin ated in the seminal paper [5]. Our approach to solve the present inverse pr oblem also relies on a sharp Carleman estimate for general second-order hyperbo lic equations and in particular a post Carleman estimate route that was introduced by Isakov in [7, Theorem 8.2.2]. Another standard feature of the BK method is the n eed of certain positivity assumptions on the initial conditions. Of course the precis e assumption depends on what coefficient(s) one is trying to recover. 1Here⌊·⌋denotes the usual floor function.4 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS On the other hand, let us also mention that there is another standa rd formula- tion of inverse hyperbolic problems that usually does not require pos itivity on the initial conditions. In this formulation, one tries to recover informat ion of second- order hyperbolic equations from all possible boundary measuremen ts, which are often modeled by the Dirichlet to Neumann or Neumann to Dirichlet ope rator. In particular, in this case it is possible to recover all coefficients in syste m (1) up to natural gauge transformations [12], following the powerful Bound ary Control (BC) method developed by Belishev [1]. For more inverse hyperbolic problem s with in- finitely many measurements and the BC method, we refer to the rev iew paper [2] and the monograph [11]. Let us now state the main theorems in this paper. Theorem 1.1. Under the geometrical assumptions (A.1) and (A.2) and let (2) T > T 0= 2/radicalbigg max x∈Ωd(x). Suppose the initial and boundary conditions are in the follo wing function spaces (3) {w0,w1} ∈Hγ+1(Ω)×Hγ(Ω), h∈Hγ+1(Σ),whereγ >n 2+4 along with all compatibility conditions (trace coincidenc e) which make sense. In ad- dition, dependingon the dimension nof the space, we assume the following positivity condition: There exists r0>0such that Case I: If nis odd, i.e., n= 2m+1for some m∈N, then we choose m+2pairs of initial conditions {w(i) 0,w(i) 1},i= 1,...,m+ 2, and a boundary condition hso that they satisfy (3) and (4) |detW(x)| ≥r0, a.e. x∈Ω whereW(x)is the(n+3)×(n+3)matrix defined by (5) W(x) = w(1) 0(x)w(1) 1(x)∂x1w(1) 0(x)···∂xnw(1) 0(x) ∆w(1) 0(x) w(1) 1(x)w(1) tt(x)∂x1w(1) 1(x)···∂xnw(1) 1(x) ∆w(1) 1(x) .................. w(m+2) 0(x)w(m+2) 1(x)∂x1w(m+2) 0(x)···∂xnw(m+2) 0(x) ∆w(m+2) 0(x) w(m+2) 1(x)w(m+2) tt(x)∂x1w(m+2) 1(x)···∂xnw(m+2) 1(x) ∆w(m+2) 1(x) Case II: If nis even, i.e., n= 2mfor some m∈N, then we choose m+2pairs of initial conditions {w(i) 0,w(i) 1},i= 1,...,m+2, and a boundary condition hso thatRECOVER ALL COEFFICIENTS 5 they satisfy (3) and (6) |det/tildewiderW(x)| ≥r0, a.e. x∈Ω where/tildewiderW(x)is the(n+3)×(n+3)matrix defined by (7) /tildewiderW(x) = w(1) 0(x)w(1) 1(x)∂x1w(1) 0(x)···∂xnw(1) 0(x) ∆w(1) 0(x) w(1) 1(x)w(1) tt(x)∂x1w(1) 1(x)···∂xnw(1) 1(x) ∆w(1) 1(x) .................. w(m+1) 0(x)w(m+1) 1(x)∂x1w(m+1) 0(x)···∂xnw(m+1) 0(x) ∆w(m+1) 0(x) w(m+1) 1(x)w(m+1) tt(x)∂x1w(m+1) 1(x)···∂xnw(m+1) 1(x) ∆w(m+1) 1(x) w(m+2) 0(x)w(m+2) 1(x)∂x1w(m+2) 0(x)···∂xnw(m+2) 0(x) ∆w(m+2) 0(x) Letw(i)(c,q1,q0,q)andw(i)(˜c,p1,p0,p)be the corresponding solutions of equation (1) with different coefficients {c,q1,q0,q}and{˜c,p1,p0,p}, as well as the initial and boundary conditions {w(i) 0,w(i) 1,h},i= 1,···,m+2. If we have the same Neumann boundary traces over the observed part Γ1of the boundary and over the time interval [−T,T], i.e., for i= 1,···,m+2, (8)∂w(i)(c,q1,q0,q) ∂ν(x,t) =∂w(i)(˜c,p1,p0,p) ∂ν(x,t),(x,t)∈Γ1×[−T,T], then we must have that all the coefficients coincide, namely, (9)c(x) = ˜c(x), q1(x) =p1(x), q0(x) =p0(x),q(x) =p(x)a.e. x∈Ω. After proving the above uniqueness theorem, we may also get the f ollowing Lips- chitz stability result for recovering all coefficients {c,q1,q0,q}from the correspond- ing finite sets of boundary measurements. Theorem 1.2. Under the assumptions in Theorem 1.1, again let w(i)(c,q1,q0,q) andw(i)(˜c,p1,p0,p)denote the corresponding solutions of equation (1) with coe ffi- cients{c,q1,q0,q}and{˜c,p1,p0,p}, as well as the initial and boundary conditions {w(i) 0,w(i) 1,h},i= 1,···,m+2(eithernis odd or even). Then there exists C >0 depends on Ω,T,Γ1,c,q1,q0,q,w(i) 0,w(i) 1,hsuch that /⌊a∇d⌊lc2−˜c2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lq1−p1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lq0−p0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lq−p/⌊a∇d⌊l2 L2(Ω) ≤Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂w(i) tt(c,q1,q0,q) ∂ν−∂w(i) tt(˜c,p1,p0,p) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1), (10)6 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS for all such coefficients c,˜c,q1,q0,p1,p0∈H1 0(Ω),q,p∈(H1 0(Ω))n, where/⌊a∇d⌊l·/⌊a∇d⌊lL2(Ω) is defined as /⌊a∇d⌊lr/⌊a∇d⌊lL2(Ω)=/parenleftBigg/integraldisplay Ωn/summationdisplay i=1|ri(x)|2dx/parenrightBigg1 2 ,forr(x) = (r1(x),···,rn(x)). Inverse source problem . The first step to solve the inverse problem above is to convert it into a corresponding inverse source problem. Indeed, if we let f2(x) =c2(x)−˜c2(x), f1(x) =p1(x)−q1(x), f0(x) =p0(x)−q0(x),f(x) =p(x)−q(x); u(x,t) =w(c,q1,q0,q)−w(˜c,p1,p0,p), R(x,t) =w(˜c,p1,p0,p)(x,t),(11) thenu=u(x,t) is readily seen to satisfy the following homogeneous mixed problem (12) utt−c2(x)∆u+q1(x)ut+q0(x)u+q(x)·∇u=S(x,t) inQ u(x,0) =ut(x,0) = 0 in Ω u(x,t) = 0 in Σ, where (13)S(x,t) =f0(x)R(x,t)+f1(x)Rt(x,t)+f(x)·∇R(x,t)+f2(x)∆R(x,t). Here we assume that c∈C,q0,q1∈L∞(Ω) and q∈(L∞(Ω))nare given fixed andR=R(x,t) is a given function that can be suitably chosen. On the other hand, the source coefficients f0,f1,f2∈L2(Ω) and f∈(L2(Ω))nare assumed to be unknown. The inverse source problem is to determine f0,f1,f2andffrom the Neumann boundary measurements of uover the observed part Γ 1of the boundary and over a sufficiently long time interval [ −T,T]. More specifically, corresponding with Theorems 1.1 and 1.2, we will prove the following uniqueness and st ability results. Theorem 1.3. Under geometrical assumptions (A.1) and (A.2) and let Tsatisfy (2). Depending on the dimension n, we assume the following regularity and posi- tivity conditions: Case I: If nis odd, i.e., n= 2m+ 1for some m∈N, then we choose m+ 2 functions R(1),···,R(m+2)such that they satisfy (14) R(i),R(i) t,R(i) tt,R(i) ttt∈W2,∞(Q), i= 1,···,m+2 and there exists r0>0such that (15) |detU(x)| ≥r0, a.e. x∈ΩRECOVER ALL COEFFICIENTS 7 whereU(x)is the(n+3)×(n+3)matrix defined by (16) U(x) = R(1)(x,0)R(1) t(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0) R(1) t(x,0)R(1) tt(x,0)∂x1R(1) t(x,0)···∂xnR(1) t(x,0) ∆R(1) t(x,0) .................. R(m+2)(x,0)R(m+2) t(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0) R(m+2) t(x,0)R(m+2) tt(x,0)∂x1R(m+2) t(x,0)···∂xnR(m+2) t(x,0) ∆R(m+2) t(x,0) Case II: If nis even, i.e., n= 2mfor some m∈N, then we choose m+ 2 functions R(1),···,R(m+2)such that they satisfy (14) and there exists r0>0such that (17) |det/tildewideU(x)| ≥r0, a.e. x∈Ω where/tildewideU(x)is the(n+3)×(n+3)matrix defined by (18) /tildewideU(x) = R(1)(x,0)R(1) t(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0) R(1) t(x,0)R(1) tt(x,0)∂x1R(1) t(x,0)···∂xnR(1) t(x,0) ∆R(1) t(x,0) .................. R(m+1)(x,0)R(m+1) t(x,0)∂x1R(m+1)(x,0)···∂xnR(m+1)(x,0) ∆R(m+1)(x,0) R(m+1) t(x,0)R(m+1) tt(x,0)∂x1R(m+1) t(x,0)···∂xnR(m+1) t(x,0) ∆R(m+1) t(x,0) R(m+2)(x,0)R(m+2) t(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0) Letu(i)(f0,f1,f2,f)be the solutions of equation (12) with the functions R(i),i= 1,···,m+2. If (19)∂u(i)(f0,f1,f2,f) ∂ν(x,t) = 0,(x,t)∈Γ1×[−T,T], i= 1,···,m+2, then we must have (20) f0(x) =f1(x) =f2(x) =f(x) = 0,a.e.x∈Ω. Theorem 1.4. Under the assumptions in Theorem 1.3, again let u(i)(f0,f1,f2,f) denote the solutions of equation (12) with the functions R(i),i= 1,···,m+2(either nis odd or even). Then there exists C >0depends on Ω,T,Γ1,c,q1,q0,q,w(i) 0, w(i) 1,hsuch that (21)/⌊a∇d⌊lf0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2 L2(Ω)≤Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(f0,f1,f2,f) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1)8 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS for allf0,f1,f2∈H1 0(Ω)andf∈(H1 0(Ω))n. The rest of the paper is organized as follows. In the next section we recall some necessary tools to solve the inverse problem. This includes the sharp Carle- man estimate, continuous observability inequality and regularity the ory for general second-order hyperbolic equations with Dirichlet boundary conditio n. In Section 3 we provide the proofs of Theorems 1.1, 1.2, 1.3 and 1.4, and in the las t section we give some examples where the positivity conditions (4), (6), (15) and (17) are satisfied and some concluding remarks. 2.Carleman Estimate, Continuous Observability Inequality a nd Regularity Theory for Second-Order Hyperbolic Equations In this section we recall some key ingredients of the proofs used in t he next sec- tion. This includes Carleman estimate, continuous observability inequ ality, as well as regularity theory for general second-order hyperbolic equat ions with Dirichlet boundary condition. For simplicity here we only state the main results and refer to [22] and [13] for greater details. To begin with, consider a Riemannian metric g(·,·) =/a\}⌊∇a⌋ketle{t·,·/a\}⌊∇a⌋ket∇i}htand squared norm |X|2=g(X,X),on a smooth finite dimensional manifold M. On the Riemannian manifold ( M,g) we define Ω as an open bounded, connected set of Mwith smooth boundary Γ = Γ0∪Γ1, where Γ 0∩Γ1=∅. Letνdenote the unit outward normal field along the boundary Γ. Furthermore, we denote by ∆ gthe Laplace–Beltrami operator on the manifold Mand byDthe Levi–Civita connection on M. Consider the following second-order hyperbolic equation with energ y level terms defined on Q= Ω×[−T,T] for some T >0: (22)wtt(x,t)−∆gw(x,t)+F(w) =G(x,t),(x,t)∈Q= Ω×[−T,T] where the forcing term G∈L2(Q) and the energy level differential term F(w) is given by F(w) =/a\}⌊∇a⌋ketle{tP(x,t),Dw/a\}⌊∇a⌋ket∇i}ht+P1(x,t)wt+P0(x,t)w. HereP0,P1arefunctionsonΩ ×[−T,T],P(x,t)isavectorfieldon Mfort∈[−T,T], and they satisfy the following estimate: there exists a constant CT>0 such that |F(w)| ≤CT[w2+w2 t+|Dw|2],∀(x,t)∈Q. Pseudo-convex function. Having chosen, on the strength of geometrical as- sumption (A.1), a strictly convex function d(x), we can define the function ϕ(x,t) : Ω×R→Rof classC3by setting ϕ(x,t) =d(x)−αt2, x∈Ω, t∈[−T,T], whereT > T 0as in (2). Moreover, α∈(0,1) is selected as follows: Let T > T 0be given, then there exists δ >0 such that T2>4max x∈Ωd(x)+4δ.RECOVER ALL COEFFICIENTS 9 For thisδ >0, there exists a constant α∈(0,1), such that αT2>4max x∈Ωd(x)+4δ. It is easy to check such function ϕ(x,t) carries the following properties: (a) For the constant δ >0 fixed above, we have ϕ(x,−T) =ϕ(x,T)≤max x∈Ωd(x)−αT2≤ −δuniformly in x∈Ω; and ϕ(x,t)≤ϕ(x,0),for anyt∈[−T,T] and any x∈Ω. (b) There are t0andt1, with−T < t0<0< t1< T, say, chosen symmetrically about 0, such that min x∈Ω,t∈[t0,t1]ϕ(x,t)≥σ,where 0< σ < m 0= min x∈Ωd(x). Moreover, let Q(σ) be the subset of Q= Ω×[−T,T] defined by (23) Q(σ) ={(x,t) :ϕ(x,t)≥σ >0,x∈Ω,−T≤t≤T}, Then we have (24) Ω ×[t0,t1]⊂Q(σ)⊂Ω×[−T,T]. Carleman estimate for general second-order hyperbolic equ ations. We now return to the equation (22), and consider solutions w(x,t) in the class (25)/braceleftBigg w∈H1,1(Q) =L2(−T,T;H1(Ω))∩H1(−T,T;L2(Ω)); wt,∂w ∂ν∈L2(−T,T;L2(Γ)). Then for these solutions with geometrical assumptions (A.1) and (A .2) on Ω, the following one-parameter family of estimates hold true, with β >0 being a suitable constant ( βis positive by virtue of (A.2)), for all τ >0 sufficiently large and ǫ >0 small: (26) BT(w)+2/integraldisplay Qe2τϕ|G|2dQ+C1,Te2τσ/integraldisplay Qw2dQ+cTτ3e−2τδ[Ew(−T)+Ew(T)] ≥C1,τ/integraldisplay Qe2τϕ[w2 t+|Dw|2]dQ+C2,τ/integraldisplay Q(σ)e2τϕw2dxdt where (27) C1,τ=τǫ(1−α)−2CT, C2,τ= 2τ3β+O(τ2)−2CT.10 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS Hereδ >0,σ >0 are the constants as in above, CT,cTandC1,Tare positive constants depending on T, as well as d(but not on τ). The energy function Ew(t) is defined as Ew(t) =/integraldisplay Ω[w2(x,t)+w2 t(x,t)+|Dw(x,t)|2]dΩ. In addition, BT(w) stands for boundary terms and can be explicitly calculated as BT(w) = 2τ/integraldisplay Σe2τϕ/parenleftbig w2 t−|Dw|2/parenrightbig /a\}⌊∇a⌋ketle{tDd,ν/a\}⌊∇a⌋ket∇i}htdΣ + 4τ/integraldisplay Σe2τϕ/a\}⌊∇a⌋ketle{tDd,Dw/a\}⌊∇a⌋ket∇i}ht/a\}⌊∇a⌋ketle{tDw,ν/a\}⌊∇a⌋ket∇i}htdΣ+8ατ/integraldisplay Σe2τϕtwt/a\}⌊∇a⌋ketle{tDw,ν/a\}⌊∇a⌋ket∇i}htdΣ + 4τ2/integraldisplay Σe2τϕ/bracketleftbigg |Dd|2−4α2t2+∆d−α−1 2τ/bracketrightbigg w/a\}⌊∇a⌋ketle{tDw,ν/a\}⌊∇a⌋ket∇i}htdΣ + 2τ/integraldisplay Σe2τϕ/bracketleftbig 2τ2/parenleftbig |Dd|2−4α2t2/parenrightbig +τ(3α+1)/bracketrightbig w2/a\}⌊∇a⌋ketle{tDd,ν/a\}⌊∇a⌋ket∇i}htdΣ. Clearly if we have w|Γ×[−T,T]= 0 and∂w ∂ν=/a\}⌊∇a⌋ketle{tDw,ν/a\}⌊∇a⌋ket∇i}ht= 0 on Γ 1×[−T,T], then in view of the geometrical assumption (A.1) we may compute (28) BT(w) = 2τ/integraldisplayT −T/integraldisplay Γ0e2τϕ|Dw|2/a\}⌊∇a⌋ketle{tDd,ν/a\}⌊∇a⌋ket∇i}htdΓ0dt≤0. Continuous observability inequality . As a corollary of the Carleman estimate, we also have the following continuous observability inequality (29) CTEw(0)≤/integraldisplayT −T/integraldisplay Γ1/parenleftbigg∂w ∂ν/parenrightbigg2 dΓdt+/⌊a∇d⌊lG/⌊a∇d⌊l2 L2(Q) for the equation (22) with homogeneous Dirichlet boundary conditio nw|Σ= 0. HereT > T 0as in (2) and Ω satisfies the geometrical assumptions (A.1) and (A.2) . Remark 2.1. The continuous observability inequality (29) may be interpreted as follows: If the second-order hyperbolic equation equation (22) ha s homogeneous Dirichlet boundary condition and nonhomogeneous forcing term G∈L2(Q), and Neumann boundary trace∂w ∂ν∈L2(Σ1), then necessarily the initial conditions {w(·,0),wt(·,0)}must lie in the natural energy space H1 0(Ω)×L2(Ω). This fact will be used in the proofs in Section 3. Regularity theory for general second-order hyperbolic equ ations with Dirichlet boundary condition . Consider the second-order hyperbolic equation (22) with initial conditions w(x,0) =w0(x),wt(x,0) =w1(x) and Dirichlet bound- ary condition w|Σ=h(x,t). Then the following interior and boundary regularityRECOVER ALL COEFFICIENTS 11 results for the solution whold true: For γ≥0 (not necessarily an integer), if the given data satisfy the following regularity assumptions /braceleftBigg G∈L1(0,T;Hγ(Ω)), ∂(γ) tG∈L1(0,T;L2(Ω)), w0∈Hγ+1(Ω), w1∈Hγ(Ω), h∈Hγ+1(Σ) with all compatibility conditions (trace coincidence) which make sense . Then, we have the following regularity for the solution w: (30)w∈C([0,T];Hγ+1(Ω)), ∂(γ+1) tw∈C([0,T];L2(Ω));∂w ∂ν∈Hγ(Σ). 3.Main Proofs In this section we give the main proofs of the uniqueness and stability results established in the first section. We focus on proving Theorems 1.3 an d 1.4 for the inverse source problem since Theorems 1.1 and 1.2 of the original inve rse problem will then follow from the relation (11) between the two problems and t he regularity theory result recalled in Section 2. Henceforth for convenience we useCto denote a generic positive constant which may depend on Ω, T,c,q1,q0,q,r0,w(i),u(i),R(i), i= 1,···,m+2, but not on the free large parameter τappearing in the Carleman estimate. Proof of Theorem 1.3 . First we consider the case when nis odd, i.e., n= 2m+1, for some m∈N. Then corresponding with the choice of R(i),i= 1,···,m+2, we havem+2 equations of the form (12) with solutions u(i)=u(i)(x,t) that satisfy (31) u(i) tt−c2(x)∆u(i)+q1(x)u(i) t+q0(x)u(i)+q(x)·∇u(i)=S(i)(x,t) inQ u(i)(x,0) =u(i) t(x,0) = 0 in Ω u(i)|Γ×[−T,T]= 0,∂u(i) ∂ν|Γ1×[−T,T]= 0 in Σ ,Σ1, whereS(i)(x,t) is defined in (13) with Rbeing replaced by R(i). Note since c∈C,q1,q0∈L∞(Ω) andq∈(L∞(Ω))n, the equation in (31) can be written as a Riemannian wave equation with respect to the metric g=c−2(x)dx2, modulo lower-order terms2 u(i) tt−∆gu(i)+“lower-order terms” = S(i)(x,t). Moreover, by the regularity assumption (14), we have that S(i)∈L2(Q) and by Cauchy–Schwarz inequality |S(i)(x,t)|2≤C/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig . 2More precisely, we have ∆ gu=c2∆u+cn∇(c2−n)·∇u12 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS Thus we can apply the Carleman estimate (26) for solution u(i)in the class (25) and get the following inequality for sufficiently large τ: τ/integraldisplay Qe2τϕ[(u(i) t)2+|Du(i)|2]dQ+τ3/integraldisplay Q(σ)e2τϕ(u(i))2dxdt ≤C/integraldisplay Qe2τϕ/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig dQ+Ce2τσ.(32) Note here we have dropped the unnecessary terms in the Carleman estimate (26) as well as the boundary terms BT(u(i)) since the homogeneous boundary data u(i)|Γ×[−T,T]=∂u(i) ∂ν|Γ1×[−T,T]= 0 imply BT(u(i))≤0, as suggested in (28). Differentiate the u(i)-system (31) in time t, we get the following u(i) t-problem (33) (u(i) t)tt−c2(x)∆u(i) t+q1(x)(u(i) t)t+q0(x)u(i) t+q(x)·∇u(i) t=S(i) t(x,t) inQ (u(i) t)(x,0) = 0,(u(i) t)t(x,0) =S(i)(x,0) in Ω u(i) t|Γ×[−T,T]= 0,∂u(i) t ∂ν|Γ1×[−T,T]= 0 in Σ ,Σ1. Note again by (14) we have S(i) t∈L2(Q) and by Cauchy–Schwarz inequality |S(i) t(x,t)|2≤C/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig . In addition, BT(u(i) t)≤0 sinceu(i) t|Γ×[−T,T]=∂u(i) t ∂ν|Γ1×[−T,T]= 0. Thus similar to (32) we can apply Carleman estimate (26) for solutions u(i) tin the class (25) and get the following inequality for sufficiently large τ: τ/integraldisplay Qe2τϕ[(u(i) tt)2+|Du(i) t|2]dQ+τ3/integraldisplay Q(σ)e2τϕ(u(i) t)2dxdt ≤C/integraldisplay Qe2τϕ/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig dQ+Ce2τσ.(34) Continue with this process, we differentiate (33) in ttwo more times, and get the corresponding u(i) ttandu(i) ttt-systems (35) (u(i) tt)tt−c2(x)∆u(i) tt+q1(x)(u(i) tt)t+q0(x)u(i) tt+q(x)·∇u(i) tt=S(i) tt(x,t) u(i) tt(x,0) =S(i)(x,0),(u(i) tt)t(x,0) =S(i) t(x,0)−q1(x)S(i)(x,0) u(i) tt|Γ×[−T,T]= 0,∂u(i) tt ∂ν|Γ1×[−T,T]= 0RECOVER ALL COEFFICIENTS 13 and (36) (u(i) ttt)tt−c2(x)∆u(i) ttt+q1(x)(u(i) ttt)t+q0(x)u(i) ttt+q(x)·∇u(i) ttt=S(i) ttt(x,t) (u(i) ttt)(x,0) =S(i) t(x,0)−q1(x)S(i)(x,0) (u(i) ttt)t(x,0) =S(i) tt(x,0)+c2∆S(i)(x,0)−q1S(i) t(x,0)−q0S(i)(x,0)−q·∇S(i)(x,0) u(i) ttt|Γ×[−T,T]= 0,∂u(i) ttt ∂ν|Γ1×[−T,T]= 0. Again by (14), Cauchy–Schwarz inequality and the homogeneous Dir ichlet and Neumann boundary data, we can apply Carleman estimate (26) to th e correspond- ingu(i) tt,u(i) ttt-systems above and get the following inequalities that are similar to (3 2) and (34), for τsufficiently large τ/integraldisplay Qe2τϕ[(u(i) ttt)2+|Du(i) tt|2]dQ+τ3/integraldisplay Q(σ)e2τϕ(u(i) tt)2dxdt ≤C/integraldisplay Qe2τϕ/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig dQ+Ce2τσ.(37) τ/integraldisplay Qe2τϕ[(u(i) tttt)2+|Du(i) ttt|2]dQ+τ3/integraldisplay Q(σ)e2τϕ(u(i) ttt)2dxdt ≤C/integraldisplay Qe2τϕ/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig dQ+Ce2τσ.(38) Add the four inequalities (32), (34), (37), (38) together, we get τ/integraldisplay Qe2τϕ[(u(i) tttt)2+(u(i) ttt)2+(u(i) tt)2+(u(i) t)2+|Du(i) ttt|2+|Du(i) tt|2+|Du(i) t|2+|Du(i)|2]dQ +τ3/integraldisplay Q(σ)e2τϕ[(u(i) ttt)2+(u(i) tt)2+(u(i) t)2+(u(i))2]dxdt ≤C/integraldisplay Qe2τϕ/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig dQ+Ce2τσ.(39) We now analyze the integral term on the right-hand side of (39). Fir st note that by estimating the u(i)-equation in (31) and u(i) t-equation in (33) at time t= 0, we14 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS can get (40) u(i) tt(x,0) =S(i)(x,0) u(i) ttt(x,0) =S(i) t(x,0)−q1(x)S(i)(x,0). Note the above equations hold for any i, 1≤i≤m+2, so putting all of them together we get a ( n+3)×(n+3) linear system (41)/bracketleftBig u(1) tt(x,0),u(1) ttt(x,0),···,u(m+2) tt(x,0),u(m+2) ttt(x,0)/bracketrightBigT =Uq1(x)[f0(x),f1(x),f(x),f2(x)]T where the coefficient matrix Uq1(x) is defined as (42) Uq1(x) = R(1)(x,0)R(1) t(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0) ˜a(1)(x)˜b(1)(x) ˜ m(1) 1(x)··· ˜m(1) n(x) ˜ℓ(1)(x) .................. R(m+2)(x,0)R(m+2) t(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0) ˜a(m+2)(x)˜b(m+2)(x) ˜m(m+2) 1(x)···˜m(m+2) n(x)˜ℓ(m+2)(x) with (43) ˜a(i)(x) =R(i) t(x,0)−q1(x)R(i)(x,0),˜b(i)(x) =R(i) tt(x,0)−q1(x)R(i) t(x,0), ˜m(i) k(x) =∂xkR(i) t(x,0)−q1(x)∂xkR(i)(x),˜ℓ(i)(x,0) = ∆R(i) t(x,0)−q1(x)∆R(i)(x,0). Notice that from doing elementary row operations, specifically, add ingq1multiplied by an odd row to the subsequent even row, the matrix Uq1(x) andU(x) as defined in (16) have the same determinant. Thus the positivity assumption ( 15) implies that we may invert Uq1(x) in (42) to obtain |f0(x)|2+|f1(x)|2+|f2(x)|2+|f(x)|2≤Cm+2/summationdisplay i=1/parenleftBig |u(i) tt(x,0)|2+|u(i) ttt(x,0)|2/parenrightBig =C/parenleftbig |utt(x,0)|2+|uttt(x,0)|2/parenrightbig(44) wherewedenote u(x,t) = (u(1)(x,t),u(2)(x,t),···,u(m+2)(x,t)). Thusbyproperties of the pseudo-convex function ϕand the Cauchy–Schwarz inequality, we can get theRECOVER ALL COEFFICIENTS 15 following estimate /integraldisplay Qe2τϕ(x,t)/parenleftbig |f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig dQ (45) ≤C/integraldisplay Ω/integraldisplayT −Te2τϕ(x,0)/parenleftbig |utt(x,0)|2+|uttt(x,0)|2/parenrightbig dtdΩ ≤C/parenleftbigg/integraldisplay Ω/integraldisplay0 −Td ds[e2τϕ(x,s)/parenleftbig |utt(x,s)|2+|uttt(x,s)|2/parenrightbig ]dsdΩ +/integraldisplay Ωe2τϕ(x,−T)/parenleftbig |utt(x,−T)|2+|uttt(x,−T)|2/parenrightbig dΩ/parenrightbigg ≤C/parenleftbigg τ/integraldisplay Ω/integraldisplay0 −Te2τϕ(x,s)/parenleftbig |utt(x,s)|2+|uttt(x,s)|2/parenrightbig ]dsdΩ + 2/integraldisplay Ω/integraldisplay0 −Te2τϕ(x,s)(utt·uttt+uttt·utttt)]dsdΩ +/integraldisplay Ωe2τϕ(x,−T)/parenleftbig |utt(x,−T)|2+|uttt(x,−T)|2/parenrightbig dΩ/parenrightbigg ≤C/parenleftbigg τ/integraldisplay Qe2τϕ|utt|2dQ+τ/integraldisplay Qe2τϕ|uttt|2dQ+/integraldisplay Qe2τϕ|utttt|2dQ/parenrightbigg . Taking (45) into (39), and note (39) holds for all i= 1,···m+2, thus summing overiin (39) and dropping the non-negative gradient terms on the left-h and side, we get that for τsufficiently large τ/integraldisplay Qe2τϕ/parenleftbig |utttt|2+|uttt|2+|utt|2+|ut|2/parenrightbig dQ (46) +τ3/integraldisplay Q(σ)e2τϕ/parenleftbig |uttt|2+|utt|2+|ut|2+|u|2/parenrightbig dxdt ≤Cτ/integraldisplay Qe2τϕ(|utt|2+|uttt|2)dQ+C/integraldisplay Qe2τϕ|utttt|2dQ+Ce2τσ. Sincee2τϕ< e2τσonQ\Q(σ) from the definition of Q(σ) (23), we have the following /integraldisplay Qe2τϕ/parenleftbig |utt|2+|uttt|2/parenrightbig dQ =/integraldisplay Q(σ)e2τϕ/parenleftbig |utt|2+|uttt|2/parenrightbig dxdt+/integraldisplay Q\Q(σ)e2τϕ/parenleftbig |utt|2+|uttt|2/parenrightbig dxdt ≤/integraldisplay Q(σ)e2τϕ/parenleftbig |utt|2+|uttt|2/parenrightbig dxdt+e2τσ/integraldisplay Q\Q(σ)/parenleftbig |utt|2+|uttt|2/parenrightbig dxdt16 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS and therefore (46) becomes τ/integraldisplay Qe2τϕ/parenleftbig |utttt|2+|uttt|2+|utt|2+|ut|2/parenrightbig dQ (47) +τ3/integraldisplay Q(σ)e2τϕ/parenleftbig |uttt|2+|utt|2+|ut|2+|u|2/parenrightbig dxdt ≤Cτ/integraldisplay Q(σ)e2τϕ/parenleftbig |utt|2+|uttt|2/parenrightbig dxdt+C/integraldisplay Qe2τϕ|utttt|2dQ+Ce2τσ. Note that in (47) the first and second terms on the right-hand side can be ab- sorbed by the corresponding terms on the left-hand side when τis taken large enough. Hence we may get the following estimate for sufficiently large τ: τ3/integraldisplay Q(σ)e2τϕ/parenleftbig |uttt|2+|utt|2+|ut|2+|u|2/parenrightbig dxdt≤Cτe2τσ. Use again the fact that ϕ(x,t)≥σonQ(σ) we hence get τ2/integraldisplay Q(σ)|uttt|2+|utt|2+|ut|2+|u|2dxdt≤C. Sinceτ >0 in a free large parameter and the constants Cdo not depend on τ, the above inequality implies we must have u=0a.e. onQ(σ). Note from (24) the subspace Q(σ) satisfies the property Ω ×[t0,t1]⊂Q(σ)⊂Qwitht0<0< t1, therefore by evaluating the uandut-systems of equations at t= 0, we get the (n+3)×(n+3) linear system (see (41)) Uq1(x)[f0(x),f1(x),f(x),f2(x)]T=0, a.e. x∈Ω. As the coefficient matrix Uq1(x) is invertible from assumption (15), we must have the desired conclusion f0(x) =f1(x) =f2(x) =f(x) = 0, a.e. x∈Ω. For the case when nis even, i.e., n= 2m,m∈N, we can basically repeat the above proof with obvious adjustments. The only difference her e is that since n= 2mis even, the linear system (41) contains an odd number ( n+3) of equations. Therefore we only need m+1 pairs of equations from (40) plus one more equationRECOVER ALL COEFFICIENTS 17 fromu(m+2) tt(x,0). Doing this yields the matrix /tildewideUq1(x), where (48) /tildewideUq1(x) = R(1)(x,0)R(1) t(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0) ˜a(1)(x)˜b(1)(x) ˜ m(1) 1(x)··· ˜m(1) n(x) ˜ℓ(1)(x) .................. R(m+1)(x,0)R(m+1) t(x,0)∂x1R(m+1)(x,0)···∂xnR(m+1)(x,0) ∆R(m+1)(x,0) ˜a(m+1)(x)˜b(m+1)(x) ˜m(m+1) 1(x)···˜m(m+1) n(x)˜ℓ(m+1)(x) R(m+2)(x,0)R(m+2) t(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0) with ˜a(i),˜b(i), ˜m(i) kand˜ℓ(i)defined as in (43). Again since elementary row operations do notchange thedeterminant, /tildewideUq1(x) will have thesamedeterminant asthematrix /tildewideU(x) in the assumption (17). This completes the proof of Theorem 1.3. Proof of Theorem 1.4 . After achieving the uniqueness for the inverse source problem, we now prove the corresponding stability estimate (21). T he proof below works essentially for both of the cases whether nis odd or even, the only difference is in the choices of the functions R(i),i= 1,···,m+2, as indicated in the Theorem 1.3. First we go back to the inequality (44), integrate over Ω gives (49) /⌊a∇d⌊lf0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2 L2(Ω)≤Cm+2/summationdisplay i=1/parenleftBig /⌊a∇d⌊lu(i) tt(·,0)/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lu(i) ttt(·,0)/⌊a∇d⌊l2 L2(Ω)/parenrightBig . For each i, 1≤i≤m+2, we return to the u(i) tt-system: (50) (u(i) tt)tt−c2(x)∆u(i) tt+q1(x)(u(i) tt)t+q0(x)u(i) tt+q(x)·∇u(i) tt=S(i) tt(x,t) u(i) tt(x,0) =S(i)(x,0),(u(i) tt)t(x,0) =S(i) t(x,0)−q1(x)S(i)(x,0) u(i) tt|Γ×[−T,T]= 0 withS(i)(x,t) =f0(x)R(i)+f1(x)R(i) t+f(x)·∇R(i)+f2(x)∆R(i). Here we assume (51)c∈C, q0,q1,q2∈L∞(Ω),q∈(L∞(Ω))n,f0,f1,f2∈H1 0(Ω),f∈/parenleftbig H1 0(Ω)/parenrightbign andR(i)satisfies (14) and (15) (or (17) if nis even). By linearity, we split u(i) tt into two systems, u(i) tt=y(i)+z(i), wherey(i)=y(i)(x,t) satisfies the homogeneous18 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS forcing term and nonhomogeneous initial conditions (52) y(i) tt−c2(x)∆y(i)+q1(x)y(i) t+q0(x)y(i)+q(x)·∇y(i)= 0 in Q y(i)(x,0) =u(i) tt(x,0) =S(i)(x,0) in Ω y(i) t(x,0) = (u(i) tt)t(x,0) =S(i) t(x,0)−q1(x)S(i)(x,0) in Ω y(i)|Γ×[−T,T]= 0 in Σ andz(i)=z(i)(x,t) has the nonhomogeneous forcing term and homogeneous initial conditions (53) z(i) tt−c2(x)∆z(i)+q1(x)z(i) t+q0(x)z(i)+q(x)·∇z(i)=S(i) tt(x,t) inQ z(i)(x,0) =z(i) t(x,0) = 0 in Ω z(i)|Γ×[−T,T]= 0 in Σ. For they(i)-system, note by assumptions (51) and (14) we have S(i)(·,0)∈H1 0(Ω) and S(i) t(·,0)−q1(·)S(i)(·,0)∈L2(Ω). Thuswemayapplythecontinuousobservabilityinequality(29)(with g=c−2(x)dx2) to get /⌊a∇d⌊ly(i)(·,0)/⌊a∇d⌊l2 H1 0(Ω)+/⌊a∇d⌊ly(i) t(·,0)/⌊a∇d⌊l2 L2(Ω)=/⌊a∇d⌊lu(i) tt(·,0)/⌊a∇d⌊l2 H1 0(Ω)+/⌊a∇d⌊lu(i) ttt(·,0)/⌊a∇d⌊l2 L2(Ω)≤C/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂y(i) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1). Sum the above inequality over i, use (49) and the decomposition u(i) tt=y(i)+z(i), as well as Poincar´ e’s inequality, we have /⌊a∇d⌊lf0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2 L2(Ω) (54) ≤Cm+2/summationdisplay i=1/parenleftBig /⌊a∇d⌊lu(i) ttt(·,0)/⌊a∇d⌊l2 H1 0(Ω)+/⌊a∇d⌊lu(i) ttt(·,0)/⌊a∇d⌊l2 L2(Ω)/parenrightBig ≤Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂y(i) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1) =Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt ∂ν−∂z(i) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1) ≤Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1)+Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂z(i) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1).RECOVER ALL COEFFICIENTS 19 Note this is the desired stability estimate (21) polluted by the z(i)terms. Next we show those terms can be absorbed through a compactness–uniqu eness argument, where the uniqueness relies on Theorem 1.3. To start, note for the z(i)-system (53), we have the following proposition. Proposition 3.1. For each i= 1,···,m+2, the operator define by Ki:L2(Ω)×L2(Ω)×L2(Ω)×/parenleftbig L2(Ω)/parenrightbign→L2(Σ1) (55) (f0,f1,f2,f)/mapsto→∂z(i) ∂ν|Σ1, is a compact operator. Proof.Note assumptions (51) and (14) imply S(i) tt∈H1(Q), thus by the regularity result (30) we have S(i) tt∈H1(Q)⇒∂z(i) ∂ν∈H1(Σ1) continuously . This then implies the map ( f0,f1,f2,f)/mapsto→Ki(f0,f1,f2,f)∈H1(Σ1) is continuous and hence ( f0,f1,f2,f)/mapsto→Ki(f0,f1,f2,f)∈L2(Σ1) is compact. /square Combine K(i),i= 1,···,m+ 2, being compact, together with the uniqueness result in Theorem 1.3, we may drop the z(i)terms in (54) to get the desired stability estimate (21). To carry this out, suppose by contradiction the st ability estimate (21) does not hold, then there exist sequences {fk 0},{fk 1},{fk 2}and{fk}, with fk 0,fk 1,fk 2∈H1 0(Ω) andfk∈(H1 0(Ω))n,∀k∈N, such that (56)/vextenddouble/vextenddoublefk 0/vextenddouble/vextenddouble2 L2(Ω)+/vextenddouble/vextenddoublefk 1/vextenddouble/vextenddouble2 L2(Ω)+/vextenddouble/vextenddoublefk 2/vextenddouble/vextenddouble2 L2(Ω)+/vextenddouble/vextenddoublefk/vextenddouble/vextenddouble2 L2(Ω)= 1 and (57) lim k→∞m+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(fk 0,fk 1,fk 2,fk) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(Σ1)= 0 whereu(i)(fk 0,fk 1,fk 2,fk) solves the system (31) with f0=fk 0,f1=fk 1,f2=fk 2and f=fk. From (56), there exist subsequences, still denoted as {fk 0},{fk 1},{fk 2}and {fk}, such that (58) fik⇀ f∗ iandfk⇀f∗weakly for some f∗ i∈L2(Ω) andf∗∈/parenleftbig L2(Ω)/parenrightbign,i= 0,1,2. Moreover, in view of the compactness of Ki,i= 1,···,m+ 2, we also have the strong convergence (59) lim k,l→∞/vextenddouble/vextenddoubleKi(fk 0,fk 1,fk 2,fk)−Ki(fl 0,fl 1,fl 2,fl)/vextenddouble/vextenddouble L2(Σ1)= 0,∀i= 1,···,m+2.20 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS On the other hand, since the map ( f0,f1,f2,f)/mapsto→u(i)(f0,f1,f2,f) is linear, we have from (54) that /vextenddouble/vextenddoublefk 0−fl 0/vextenddouble/vextenddouble2 L2(Ω)+/vextenddouble/vextenddoublefk 1−fl 1/vextenddouble/vextenddouble2 L2(Ω)+/vextenddouble/vextenddoublefk 2−fl 2/vextenddouble/vextenddouble2 L2(Ω)+/vextenddouble/vextenddoublefk−fl/vextenddouble/vextenddouble2 L2(Ω) ≤Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(fk 0,fk 1,fk 2,fk) ∂ν−∂u(i) tt(fl 0,fl 1,fl 2,fl) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1) +Cm+2/summationdisplay i=1/vextenddouble/vextenddoubleKi(fk 0,fk 1,fk 2,fk)−Ki(fl 0,fl 1,fl 2,fl)/vextenddouble/vextenddouble2 L2(Σ1) ≤Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(fk 0,fk 1,fk 2,fk) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1)+Cm+2/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(fl 0,fl 1,fl 2,fl) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1) +Cm+2/summationdisplay i=1/vextenddouble/vextenddoubleKi(fk 0,fk 1,fk 2,fk)−Ki(fl 0,fl 1,fl 2,fl)/vextenddouble/vextenddouble2 L2(Σ1) and therefore by (57) and (59) we get lim k,l→∞/vextenddouble/vextenddoublefk i−fl i/vextenddouble/vextenddouble L2(Ω)= lim k,l→∞/vextenddouble/vextenddoublefk−fl/vextenddouble/vextenddouble L2(Ω)= 0, i= 0,1,2. Namely, {fk 0},{fk 1},{fk 2}are Cauchy sequences in L2(Ω) and {fk}is a Cauchy sequence in ( L2(Ω))n. By uniqueness of limit and in view of (58), we must have {fk i}converges to f∗ istrongly, i= 0,1,2, and{fk}converges to f∗strongly. Hence we have from (56) (60) /⌊a∇d⌊lf∗ 0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf∗ 1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf∗ 2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf∗/⌊a∇d⌊l2 L2(Ω)= 1. Now again for the u(i) tt-system (50), by the regularity theory (30) we have that the map ( f0,f1,f2,f)/mapsto→∂u(i) tt(f0,f1,f2,f) ∂ν∈L2(Σ) is continuous and hence /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(f0,f1,f2,f) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ)≤C/parenleftBig /⌊a∇d⌊lf0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2 L2(Ω)/parenrightBig . Since the map ( f0,f1,f2,f)/mapsto→u(i) tt(f0,f1,f2,f)|Σis linear, we thus have /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(fk 0,fk 1,fk 2,fk) ∂ν−∂u(i) tt(f∗ 0,f∗ 1,f∗ 2,f∗) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1)(61) ≤C/parenleftBig /⌊a∇d⌊lfk 0−f∗ 0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk 1−f∗ 1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk 2−f∗ 2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk−f∗/⌊a∇d⌊l2 L2(Ω)/parenrightBig .RECOVER ALL COEFFICIENTS 21 This then implies, by virtue of fk i→f∗ i,i= 0,1,2 andfk→f∗strongly, that lim k→∞/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i) tt(fk 0,fk 1,fk 2,fk) ∂ν−∂u(i) tt(f∗ 0,f∗ 1,f∗ 2,f∗) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(Σ1)= 0 and hence∂u(i) tt(f∗ 0,f∗ 1,f∗ 2,f∗) ∂ν= 0 inL2(Σ1) in view of (57). In other words, ∂u(i) t(f∗ 0,f∗ 1,f∗ 2,f∗) ∂νisaconstantin t∈[−T,T]. Weclaimthat∂u(i) t(f∗ 0,f∗ 1,f∗ 2,f∗) ∂ν= 0 on Σ 1. To see this, we consider the u(i) t(fk 0,fk 1,fk 2,fk)-system (62) (u(i) t)tt−c2(x)∆u(i) t+q1(x)(u(i) t)t+q0(x)u(i) t+q(x)·∇u(i) t= (S(i) k)t(x,t) inQ (u(i) t)(x,0) = 0,(u(i) t)t(x,0) =S(i) k(x,0) in Ω u(i) t|Γ×[−T,T]= 0 in Σ where for i= 1,···,m+2,R(i)=R(i)(x,t) and S(i) k(x,t) =fk 0(x)R(i)+fk 1(x)R(i) t+fk(x)·∇R(i)+fk 2(x)∆R(i). The standard regularity theory (30) and trace theory implies /vextenddouble/vextenddouble/vextenddoubleu(i) t(fk 0,fk 1,fk 2,fk)−u(i) t(f∗ 0,f∗ 1,f∗ 2,f∗)/vextenddouble/vextenddouble/vextenddouble2 C([0,T];H1 0(Ω)) ≤C/parenleftBig /⌊a∇d⌊lfk 0−f∗ 0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk 1−f∗ 1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk 2−f∗ 2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk−f∗/⌊a∇d⌊l2 L2(Ω)/parenrightBig and /vextenddouble/vextenddouble/vextenddoubleu(i) t(fk 0,fk 1,fk 2,fk)−u(i) t(f∗ 0,f∗ 1,f∗ 2,f∗)/vextenddouble/vextenddouble/vextenddouble2 C([0,T];H1 2(Σ) ≤C/parenleftBig /⌊a∇d⌊lfk 0−f∗ 0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk 1−f∗ 1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk 2−f∗ 2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lfk−f∗/⌊a∇d⌊l2 L2(Ω)/parenrightBig . Noteu(i) t(fk 0,fk 1,fk 2,fk)(x,0) = 0 as well as the strong convergence of fk i→f∗ i, i= 0,1,2 andfk→f∗. Thus letting k→ ∞we getu(i) t(f∗ 0,f∗ 1,f∗ 2,f∗)(x,0) = 0 in Ω andu(i) t(f∗ 0,f∗ 1,f∗ 2,f∗)|Σ= 0. Hence∂u(i) t(f∗ 0,f∗ 1,f∗ 2,f∗) ∂ν(x,0) = 0 on Σ. Since we know∂u(i) t(f∗ 0,f∗ 1,f∗ 2,f∗) ∂νis a constant in t, we must have∂u(i) t(f∗ 0,f∗ 1,f∗ 2,f∗) ∂ν= 0 on Σ1, as desired.22 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS The above then implies∂u(i)(f∗ 0,f∗ 1,f∗ 2,f∗) ∂νis also a constant in t. By repeating the same argument, this time using the regularity theory for the u(i)(fk 0,fk 1,fk 2,fk)- system and taking limit k→ ∞, we finally get∂u(i)(f∗ 0,f∗ 1,f∗ 2,f∗) ∂ν= 0 on Σ 1. Hence we have that u(i)(f∗ 0,f∗ 1,f∗ 2,f∗) satisfies the following (63) u(i) tt−c2(x)∆u(i)+q1(x)u(i) t+q0(x)u(i)+q(x)·∇u(i)=S(i) ∗(x,t) inQ u(i)(x,0) =u(i) t(x,0) = 0 in Ω u(i)|Γ×[−T,T]= 0,∂u(i) ∂ν|Γ1×[−T,T]= 0 in Σ ,Σ1 with S(i) ∗(x,t) =f∗ 0(x)R(i)+f∗ 1(x)R(i) t+f∗(x)·∇R(i)+f∗ 2(x)∆R(i),i= 1,···,m+2. By the uniqueness result we proved in Theorem 1.3, this must imply f∗ 0=f∗ 1= f∗ 2=f∗= 0, which contradicts with (60). Hence we must be able to drop the z(i) terms in (54). This completes the proof of Theorem 1.4. Proof of Theorems 1.1 and 1.2 . Finally, we provide the proofs of uniqueness and stability of the original inverse problem. These results are pret ty much direct consequences of Theorems 1.3 and 1.4 given the relationship (11) be tween the orig- inal inverse problem and the inverse source problem. More precisely , we have the positivity conditions (4) and (6) imply (15) and (17). In addition, by t he regularity theory (30) the assumption (3) on the initial and boundary conditio ns{w(i) 0,w(i) 1,h} implies the solutions w(i),i= 1,···,m+2, satisfy {w(i),w(i) t,w(i) tt,w(i) ttt} ∈C/parenleftbig [−T,T];Hγ+1(Ω)×Hγ(Ω)×Hγ−1(Ω)×Hγ−2(Ω)/parenrightbig . Asγ >n 2+ 4, we have the following embedding Hγ−2(Ω)֒→W2,∞(Ω) and hence the regularity assumption (3) implies the corresponding regularity a ssumption (14) for the inverse source problem. This completes the proof of all the theorems. 4.Some Examples and Concluding Remarks In this last section we first provide some concrete examples such th at the key positivity conditions (4), (6), (15), (17) are satisfied, and then g ive some general remarks.RECOVER ALL COEFFICIENTS 23 Example 1 . Consider the following functions R(i)(x,t),x= (x1,···,xn)∈Ω, t∈[−T,T],i= 1,···,m+2, defined by R(1)(x,t) =t, R(i)(x,t) =x2i−3+tx2i−2,2≤i≤m+1, R(m+2)(x,t) =/braceleftBigg x2m+1+1 2tx2 1ifn= 2m+1 is odd 1 2x2 1 ifn= 2mis even. Then we may easily see that the matrices U(x) and/tildewideU(x) are lower triangular matrices with all 1s at the diagonal after swapping the first two colu mns. Thus the determinants of the matrices U(x) and/tildewideU(x) are both −1 and hence conditions (15), (17) are satisfied. Correspondingly, we may choose the m+2 pairs of initial conditions {w(i) 0(x),w(i) 1(x)}as w(1) 0(x) = 0, w(1) 1(x) = 1, w(i) 0(x) =x2i−3, w(i) 1(x) =x2i−2,2≤i≤m+1, w(m+2) 0(x) =/braceleftBigg x2m+1ifn= 2m+1 is odd 1 2x2 1ifn= 2mis even w(m+2) 1(x) =/braceleftBigg1 2x2 1ifn= 2m+1 is odd 0 ifn= 2mis even. Then the matrices W(x) and/tildewiderW(x) are also lower triangular matrices with all 1s at the diagonal after swapping the first two columns and hence condit ions are (4), (6) are satisfied. Example 2 . Considering the following functions R(i)(x,t),x= (x1,···,xn)∈Ω, t∈[−T,T],i= 1,···,m+2, defined by R(1)(x,t) = sin t, R(i)(x,t) = costex2i−3+sintex2i−2,2≤i≤m+1, R(m+2)(x,t) =/braceleftBigg costex2m+1+sinte−x1ifn= 2m+1 is odd coste−x1 ifn= 2mis even.24 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS Then the matrix U(x) becomes U(x) = 0 1 0 0 ···0 0 0 1 0 0 0 ···0 0 0 ex1ex2ex10 0 ···0ex1 ex2−ex10ex20···0ex2 ........................ exn−1−exn−20···0exn−10exn−1 exne−x10···0 0 exnexn e−x1−exn−e−x10···0 0 e−x1 (64) Notice that U(x) is not a lower triangular matrix. However, we can easily transform theitinto alower triangularmatrixbyswapping thefirst twocolumns a ndsubtract- ing the 3rd, 4th, ..., (n+2)th column from the last column. As a conseq uence we get detU(x) =−2/producttextn i=2exi. Inasimilar fashionwecanalsogetdet ˜U(x) =−2/producttextn i=2exi. As Ω is a bounded domain, we hence have the conditions (15) and (17) are satisfied. Correspondingly we may choose the m+2 pairs of initial conditions {w(i) 0,w(i) 1} as w(1) 0(x) = 0, w(1) 1(x) = 1, w(i) 0(x) =ex2i−3, w(i) 1(x) =ex2i−2,2≤i≤m+1, w(m+2) 0(x) =/braceleftBigg exnifn= 2m+1 is odd e−x1ifn= 2mis even w(m+2) 1(x) =/braceleftBigg e−x1ifn= 2m+1 is odd 0 if n= 2mis even. Then the determinants of both W(x) and/tildewiderW(x) are also −2/producttextn i=2exi, calculated in the same manner as in the case of U(x) and/tildewideU(x). Hence the conditions (4) and (6) are satisfied. Example 3 . In general if we have f(j)∈C2(Ω) with f(j)(x) =f(j)(x1,···,xj), 1≤j≤nandg,h∈C2[−T,T] that satisfies /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂f(j) ∂xj/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥rj>0,1≤j≤n,/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2f(1) ∂x2 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥˜r1>0 g(0) =h′(0) = 1, g′(0) =h(0) = 0.RECOVER ALL COEFFICIENTS 25 for some positive rj, 1≤j≤n, and ˜r1. Then we may consider the functions R(i)(x,t),x= (x1,···,xn)∈Ω,t∈[−T,T],i= 1,···,m+ 2, of the following form: R(1)(x,t) =h(t), R(i)(x,t) =f(2i−3)(x)g(t)+f(2i−2)(x)h(t),2≤i≤m+1, R(m+2)(x,t) =/braceleftBigg f(n)(x)g(t)+f(1)(ax)h(t) ifn= 2m+1 is odd f(1)(ax)g(t) if n= 2mis even. wherea <0 so that ax∈Ω. Correspondingly we may choose the m+2 pairs of initial conditions {w(i) 0,w(i) 1} as w(1) 0(x) = 0, w(1) 1(x) = 1, w(i) 0(x) =f(2i−3)(x), w(i) 1(x) =f(2i−2)(x),2≤i≤m+1, w(m+2) 0(x) =/braceleftBigg f(n)(x) ifn= 2m+1 is odd f(1)(ax) ifn= 2mis even w(m+2) 1(x) =/braceleftBigg f(1)(ax) ifn= 2m+1 is odd 0 if n= 2mis even. In this case, after swapping the first and second column, the last c olumn with the preceding ( n+2)th, (n+1)th,···, and finally the 3rd column, as well as swapping the last row with the preceding ( n+2)th, (n+1)th,···, and finally the 3rd row. We may get the determinants of the matrices U(x),/tildewideU(x),W(x) and/tildewiderW(x) are equal to /parenleftbig a∂x1f(1)(ax)∂2 x1f(1)(x)−a2∂2 x1f(1)(ax)∂x1f(1)(x)/parenrightbign/productdisplay j=2∂xjf(j)(x). Sincef(1)∈C2(Ω),|∂x1f(1)| ≥r1>0 and|∂2 x1f(1)| ≥˜r1>0,∂x1f(1)and∂2 x1f(1) do not change sign. Hence we have |/parenleftbig a∂x1f(1)(ax)∂2 x1f(1)(x)−a2∂2 x1f(1)(ax)∂x1f(1)(x)/parenrightbign/productdisplay j=2∂xjf(j)(x)| ≥(a2+|a|)˜r1n/productdisplay j=1rj. Hence the positivity conditions (4), (6), (15) and (17) are satisfie d. Finally, we end the paper with some comments and remarks. (1) We have shown in this paper that in order to recover all the coeffi cients, we need to appropriately choose ⌊n+4 2⌋pairs of initial conditions {w0,w1}and a boundary condition h, and then use their corresponding boundary measurements. As mentioned earlier, since in total there are n+3 unknown functions, it is natural26 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS to expect to recover them from n+3boundary measurements. Indeed, following the approach of this paper, we can also achieve the recovery by appro priately choosing n+ 3 initial positions w0with an initial velocity w1and a boundary condition h, and then use their corresponding boundary measurements. In pa rticular, in this case the positivity condition becomes det w(1) 0(x)w1(x)∂x1w(1) 0(x)···∂xnw(1) 0(x) ∆w(1) 0(x) w(2) 0(x)w1(x)∂x1w(2) 0(x)···∂xnw(2) 0(x) ∆w(2) 0(x) .................. w(n+3) 0(x)w1(x)∂x1w(n+3) 0(x)···∂xnw(n+3) 0(x) ∆w(n+3) 0(x) ≥r0>0. Note although in this case we need more measurements, an advanta ge is that we only need to differentiate the u-equation with respect to ttwice, rather than three times. We may also get a better stability estimate of the form /⌊a∇d⌊lc2−˜c2/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lq1−p1/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lq0−p0/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊lq−p/⌊a∇d⌊l2 L2(Ω) ≤Cn+3/summationdisplay i=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂w(i) t(c,q1,q0,q) ∂ν−∂w(i) t(˜c,p1,p0,p) ∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(Σ1) in the sense that we only need to differentiate the measurements in t ime once. (2) In our problem formulation we use the time interval [ −T,T] and regard the middlet= 0 as initial time. This is not essential since a simple change of variable t→t−Ttransforms t= 0 tot=−T. However, this present choice allows the recovery of all coefficients with fewer choices of initial condition s and hence fewer boundary measurements. This is because we may use both eq uations in (40), comparetojustoneifweassumethetimeintervalas[0 ,T]andthenextendsolutions to [−T,0]. (3) It is also possible to set up the inverse problem by assuming Neuma nn bound- ary condition∂w ∂νon Σ = Γ ×[−T,T] and making measurements of Dirichlet bound- ary traces of the solution wover Σ 1= Γ1×[−T,T], such as in [18]. This, however, would require more demanding geometrical assumption on the unobs erved portion of the boundaryΓ 0. Forexample, we may need to assume∂d ∂ν=/a\}⌊∇a⌋ketle{tDd,ν/a\}⌊∇a⌋ket∇i}ht= 0 onΓ 0in the geometrical assumption to account for the Neumann boundar y condition [22]. In addition, the more delicate regularity theory of second-order h yperbolic equation with nonhomogeneous Neumann boundary condition will also need to b e invoked [14], [15]. Nevertheless, the main ideas of solving the inverse problem r emain the same. AcknowledgementsRECOVER ALL COEFFICIENTS 27 The first author would like to thank Professor Yang Yang for many v ery useful discussions. References [1] M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR 297(1987), no.3, 524–527. [2] M. Belishev, Boundary control in reconstruction of manifolds an d metrics (BC method), Inverse Problems, 13(1997), R1–R45. [3] M. Bellassoued, Uniqueness and stability in determining the speed o f propagation of second- order hyperbolic equation with variable coefficients, Appl. Anal., 83(2004), 983–1014. [4] M. Bellassoued and M. Yamamoto, “Carleman Estimates and Applica tions to Inverse Prob- lems for Hyperbolic Systems,” Springer Monographs in Mathematics S eries, 2017. [5] A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problems , Dokl. Akad. Nauk SSSR, 260(1981), 269–272. [6] V. Isakov, “Inverse Source Problems,” American Mathematical Society, 2000. [7] V.Isakov,“InverseProblemsforPartialDifferentialEquation s,”2ndedition, Springer-Verlag, New York, 2006. [8] D. Jellali, An inverse problem for the acoustic wave equation with finite sets of boundary data , Journal of Inverse and Ill-posed Problems, 14(2006), 665–684. [9] M. Klibanov, “Carleman Estimates for Coefficient Inverse Problem s and Numerical Applica- tions,” VSP, Utrecht, 2004. [10] M. Klibanov and J. Li, “Inverse Problems and Carleman Estimates : Global Uniqueness, Global Convergence and Experimental Data,” Inverse and Ill-pose d Problems Series 63, De Gruyter, 2021. [11] A. Katchalov, Y. Kurylev and M. Lassas, “Inverse Boundary S pectral Problems,” Chapman & Hall/CRC, 2001. [12] Y. Kurylev and M. Lassas, Hyperbolic inverse boundary-value problem and time-conti nuation of the non-stationary Dirichlet-to-Neumann map , Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), no. 4, 931–949. [13] I. Lasiecka, J.L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators , J. Math. Pures Appl., 65(1986), 149–192. [14] I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equati ons of Neumann type. Part I. L2Nonhomogeneous data , Ann. Mat. Pura. Appl. (IV), CLVII (1990), 285–367. [15] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homoge neous Neumann boundary conditions. II. General boundary data , Journal of Differential Equations, 94(1991), 112–164. [16] I. Lasiecka, R. Triggianiand X. Zhang, Nonconservative wave equations with unobserved Neu- mann B.C.: Global uniqueness and observability in one shot , Differential geometric methods in the control of partial differential equations, 227–325, Amer. M ath. Soc., Providence, RI, 2000. [17] M.M. Lavrentev, V.G. Romanov and S.P. Shishataskii, “Ill-Posed P roblems of Mathematical Physics and Analysis,” Vol. 64, Amer. Math. Soc., Providence, RI, 1986. [18] S.Liu andR.Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Neum ann B.C. through an additional Dirichlet boundary trace , SIAM J. Math. Anal., 43(2011), 1631–1666.28 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS [19] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem , Nonlinear Anal. Real World Appl., 91 (2011), 1562–1590. [20] S. Liu, Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement , Evolution Equations and Control Theory, 2(2013), 355–364. [21] S. Liu and R. Triggiani, Boundary control and boundary inverse theory for non-homog eneous second order hyperbolic equations: a common Carleman estim ates approach , HCDTE Lecture Notes, AIMS on Applied Mathematics Vol 6 (2012), 227–343. [22] R. Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Ri e- mannian wave equations. Global uniqueness and observabili ty in one shot , Appl. Math. Op- tim.,46(2002), 331–375. School of Mathematical and Statistical Sciences, Clemson U niversity, Clemson, SC 29634 Email address :liul@clemson.edu Email address :srscrug@g.clemson.edu Email address :ampierr@g.clemson.edu

2109.03684v2.Room_Temperature_Intrinsic_and_Extrinsic_Damping_in_Polycrystalline_Fe_Thin_Films.pdf

Room-Temperature Intrinsic and Extrinsic Damping in Polycrystalline Fe Thin Films Shuang Wu,1David A. Smith,1Prabandha Nakarmi,2Anish Rai,2Michael Clavel,3Mantu K. Hudait,3Jing Zhao,4F. Marc Michel,4Claudia Mewes,2Tim Mewes,2and Satoru Emori1 1Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA 2Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL 35487 USA 3Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA 4Department of Geosciences, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Abstract We examine room-temperature magnetic relaxation in polycrystalline Fe lms. Out-of-plane fer- romagnetic resonance (FMR) measurements reveal Gilbert damping parameters of 0.0024 for Fe lms with thicknesses of 4-25 nm, regardless of their microstructural properties. The remarkable invariance with lm microstructure strongly suggests that intrinsic Gilbert damping in polycrys- talline metals at room temperature is a local property of nanoscale crystal grains, with limited impact from grain boundaries and lm roughness. By contrast, the in-plane FMR linewidths of the Fe lms exhibit distinct nonlinear frequency dependences, indicating the presence of strong extrinsic damping. To t our in-plane FMR data, we have used a grain-to-grain two-magnon scat- tering model with two types of correlation functions aimed at describing the spatial distribution of inhomogeneities in the lm. However, neither of the two correlation functions is able to reproduce the experimental data quantitatively with physically reasonable parameters. Our ndings advance the fundamental understanding of intrinsic Gilbert damping in structurally disordered lms, while demonstrating the need for a deeper examination of how microstructural disorder governs extrinsic damping. 1arXiv:2109.03684v2 [cond-mat.mtrl-sci] 24 Feb 2022I. INTRODUCTION In all magnetic materials, magnetization has the tendency to relax toward an eective magnetic eld. How fast the magnetization relaxes governs the performance of a variety of magnetic devices. For example, magnetization relaxation hinders ecient precessional dynamics and should be minimized in devices such as precessional magnetic random access memories, spin-torque oscillators, and magnonic circuits1{4. From the technological perspec- tive, it is important to understand the mechanisms behind magnetic relaxation in thin-lm materials that comprise various nanomagnetic device applications. Among these materials, bcc Fe is a prototypical elemental ferromagnet with attractive properties, including high sat- uration magnetization, soft magnetism5, and large tunnel magnetoresistance6,7. Our present study is therefore motivated by the need to uncover magnetic relaxation mechanisms in Fe thin lms { particularly polycrystalline lms that can be easily grown on arbitrary substrates for diverse applications. To gain insights into the contributions to magnetic relaxation, a common approach is to examine the frequency dependence of the ferromagnetic resonance (FMR) linewidth. The most often studied contribution is viscous Gilbert damping8{13, which yields a linear increase in FMR linewidth with increasing precessional frequency. In ferromagnetic metals, Gilbert damping arises predominately from \intrinsic" mechanisms14{16governed by the electronic band structure17. Indeed, a recent experimental study by Khodadadi et al.18has shown that intrinsic, band-structure-based Gilbert damping dominates magnetic relaxation in high- quality crystalline thin lms of Fe, epitaxially grown on lattice-matched substrates. However, it is yet unclear how intrinsic damping is impacted by the microstructure of polycrystalline Fe lms. Microstructural disorder in polycrystalline Fe lms can also introduce extrinsic magnetic relaxation. A well-known extrinsic relaxation mechanism is two-magnon scattering, where the uniform precession mode with zero wave vector scatters into a degenerate magnon mode with a nite wave vector19{22. Two-magnon scattering generally leads to a nonlinear fre- quency dependence of the FMR linewidth, governed by the nature of magnon scattering centers at the surfaces23,24or in the bulk of the lm25{28. While some prior experiments point to the prominent roles of extrinsic magnetic relaxation in polycrystalline ferromag- netic lms29{31, systematic studies of extrinsic relaxation (e.g., two-magnon scattering) on 2polycrystalline Fe thin lms are still lacking. Here, we investigate both the intrinsic and extrinsic contributions to magnetic relax- ation at room temperature in polycrystalline Fe lms. We have measured the frequency dependence of the FMR linewidth with (1) the lm magnetized out-of-plane (OOP), where two-magnon scattering is suppressed25such that intrinsic Gilbert damping is quantied re- liably, and (2) the lm magnetized in-plane (IP), where two-magnon scattering is generally expected to coexist with intrinsic Gilbert damping. From OOP FMR results, we nd that the intrinsic Gilbert damping of polycrystalline Fe lms at room temperature is independent of their structural properties and almost identical to that of epitaxial lms. Such insensitivity to microstructure is in contrast to disorder- sensitive Gilbert damping recently shown in epitaxial Fe at cryogenic temperature18. Our present work implies that Gilbert damping at a suciently high temperature becomes a local property of the metal, primarily governed by the structure within nanoscale crystal grains rather than grain boundaries or interfacial disorder. This implication refutes the intuitive expectation that intrinsic Gilbert damping should depend on structural disorder in polycrystalline lms. In IP FMR results, the frequency dependence of the FMR linewidth exhibits strong nonlinear trends that vary signicantly with lm microstructure. To analyze the nonlin- ear trends, we have employed the grain-to-grain two-magnon scattering model developed by McMichael and Krivosik25with two types of correlation functions for capturing inho- mogeneities in the lm. However, neither of the correlation functions yields quantitative agreement with the experimental results or physically consistent, reasonable parameters. This nding implies that a physical, quantitative understanding of extrinsic magnetic re- laxation requires further corrections of the existing two-magnon scattering model, along with much more detailed characterization of the nanoscale inhomogeneities of the magnetic lm. Our study stimulates opportunities for a deeper examination of fundamental magnetic relaxation mechanisms in structurally disordered ferromagnetic metal lms. II. FILM DEPOSITION AND STRUCTURAL PROPERTIES Polycrystalline Fe thin lms were deposited using DC magnetron sputtering at room temperature on Si substrates with a native oxide layer of SiO 2. The base pressure of the 3chamber was below 1 107Torr and all lms were deposited with 3 mTorr Ar pressure. Two sample series with dierent seed layers were prepared in our study: subs./Ti(3 nm)/Cu(3 nm)/Fe(2-25 nm)/Ti(3 nm) and subs./Ti(3 nm)/Ag(3 nm)/Fe(2-25 nm)/Ti(3 nm). In this paper we refer to these two sample series as Cu/Fe and Ag/Fe, respectively. The layer thicknesses are based on deposition rates derived from x-ray re ectivity (XRR) of thick calibration lms. The Ti layer grown directly on the substrate ensures good adhesion of the lm, whereas the Cu and Ag layers yield distinct microstructural properties for Fe as described below. We note that Cu is often used as a seed layer for growing textured polycrystalline ferromagnetic metal lms32,33. Our initial motivation for selecting Ag as an alternative seed layer was that it might promote qualitatively dierent Fe lm growth34, owing to a better match in bulk lattice parameter 𝑎between Fe ( 𝑎286A) and Ag (𝑎p 2288A) compared to Fe and Cu ( 𝑎p 2255A). We performed x-ray diraction (XRD) measurements to compare the structural properties of the Cu/Fe and Ag/Fe lms. Figure 1(a,b) shows symmetric 𝜃-2𝜃XRD scan curves for several lms from both the Cu/Fe and Ag/Fe sample series. For all Cu/Fe lms, the (110) body-center-cubic (bcc) peak can be observed around 2 𝜃=44°45°(Fig. 1(a)). This observation conrms that the Fe lms grown on Cu are polycrystalline and textured, where the crystal grains predominantly possess (110)-oriented planes that are parallel to the sample surface. For Ag/Fe (Fig. 1(b)), the (110) bcc peak is absent or extremely weak, from which one might surmise that the Fe lms grown on Ag are amorphous or only possess weak crystallographic texture. However, we nd that the Ag/Fe lms are, in fact, also polycrystalline with evidence of (110) texturing. In the following, we elaborate on our XRD results, rst for Cu/Fe and then Ag/Fe. We observe evidence for a peculiar, non-monotonic trend in the microstructural properties of the Cu/Fe lms. Specically, the height of the 𝜃-2𝜃diraction peak (Fig. 1(a)) increases with Fe lm thickness up to 10 nm but then decreases at higher Fe lm thicknesses. While we do not have a complete explanation for this peculiar nonmonotonic trend with lm thickness, a closer inspection of the XRD results (Fig. 1) provides useful insights. First, the Fe lm diraction peak shifts toward a higher 2 𝜃value with increasing lm thickness. This signies that thinner Fe lms on Cu are strained (with the Fe crystal lattice tetragonally distorted), whereas thicker Fe lms undergo structural relaxation such that the out-of-plane lattice parameter converges toward the bulk value of 2.86 A, as summarized in Fig. 1(e). 4354 04 55 05 5Ag/Fe2 nm6 nmIntensity [arb. unit] Cu/Febulk bcc Fe (110)1 0 nm15 nm25 nm8 nm( a) 4045502 θ [deg]10 nm2 5 nm 2 θ [deg]10 nm15 nm6 nm2 nm8 nm( b) 16182022242628Ag/Fe2 nm6 nmIntensity [arb. unit] Cu/Febulk bcc Fe (110)1 0 nm15 nm25 nm8 nm( c)2 5 nm θ [deg]10 nm15 nm6 nm2 nm8 nm( d) 2.842.862.882.902.922.940 5 10152025051015Bulk value 2.86 Cu/Fe Ag/FeOut-of-planel attice parameter [Å]( e)Crystallite size [nm]T hickness [nm](f)FIG. 1. (Color online) 𝜃-2𝜃X-ray diraction scan curves for (a) Cu/Fe (blue lines) and (b) Ag/Fe (red lines) sample series. The inset in (b) is the grazing-incidence XRD scan curve for 10 nm thick Ag/Fe lm. Rocking curves for (c) Cu/Fe (blue lines) and (d) Ag/Fe (red lines) sample series. (e) Out-of-plane lattice parameter estimated via Bragg's law using the 2 𝜃value at the maximum of the tallest lm diraction peak. (f) Crystallite size estimated via the Scherrer equation using the full-width-at-half-maximum of the tallest lm diraction peak. In (e) and (f), the data for the Ag/Fe lm series at a few thickness values are missing because of the absence of the bcc (110) peak in𝜃-2𝜃XRD scans. Second, as the Fe lm thickness approaches 10 nm, additional diraction peaks appear to the left of the tall primary peak. We speculate that these additional peaks may originate from Fe crystals that remain relatively strained (i.e., with an out-of-plane lattice parameter larger than the bulk value), while the primary peak arises from more relaxed Fe crystals (i.e., with a lattice parameter closer to the bulk value). The coexistence of such dierent Fe crystals appears to be consistent with the rocking curve measurements (Fig. 1(c)), which exhibit a large broad background peak in addition to a small sharp peak for Cu/Fe lms with thicknesses near 10 nm. As we describe in Sec. IV, these 10 nm thick Cu/Fe samples also show distinct behaviors in extrinsic damping (highly nonlinear frequency dependence of 5the FMR linewidth) and static magnetization reversal (enhanced coercivity), which appear to be correlated with the peculiar microstructural properties evidenced by our XRD results. On the other hand, it is worth noting that the estimated crystal grain size (Fig. 1(f)) { derived from the width of the 𝜃-2𝜃diraction peak { does not exhibit any anomaly near the lm thickness of10 nm, but rather increases monotonically with lm thickness. Unlike the Cu/Fe lms discussed above, the Ag/Fe lms do not show a strong (110) bcc peak in the 𝜃-2𝜃XRD results. However, the lack of pronounced peaks in the symmetric 𝜃-2𝜃 scans does not necessarily signify that Ag/Fe is amorphous. This is because symmetric 𝜃-2𝜃 XRD is sensitive to crystal planes that are nearly parallel to the sample surface, such that the diraction peaks capture only the crystal planes with out-of-plane orientation with a rather small range of misalignment (within 1°, dictated by incident X-ray beam divergence). In fact, from asymmetric grazing-incidence XRD scans that are sensitive to other planes, we are able to observe a clear bcc Fe (110) diraction peak even for Ag/Fe samples that lack an obvious diraction peak in 𝜃-2𝜃scans (see e.g. inset of Fig. 1(b)). Furthermore, rocking curve scans (conducted with 2 𝜃xed to the expected position of the (110) Fe lm diraction peak) provide orientation information over an angular range much wider than 1°. As shown in Fig. 1(d), a clear rocking curve peak is observed for each Ag/Fe sample, suggesting that Fe lms grown on Ag are polycrystalline and (110)-textured { albeit with the (110) crystal planes more misaligned from the sample surface compared to the Cu/Fe samples. The out- of-plane lattice parameters of Ag/Fe lms (with discernible 𝜃-2𝜃diraction lm peaks) show the trend of relaxation towards the bulk value with increasing Fe thickness, similar to the Cu/Fe series. Yet, the lattice parameters for Ag/Fe at small thicknesses are systematically closer to the bulk value, possibly because Fe is less strained (i.e., better lattice matched) on Ag than on Cu. We also nd that the estimation of the crystal grain size for Ag/Fe { although made dicult by the smallness of the diraction peak { yields a trend comparable to Cu/Fe, as shown in Fig. 1(f). We also observe a notable dierence between Cu/Fe and Ag/Fe in the properties of lm interfaces, as revealed by XRR scans in Fig. 2. The oscillation period depends inversely on the lm thickness. The faster decay of the oscillatory re ectivity signal at high angles for the Ag/Fe lms suggests that the Ag/Fe lms may have rougher interfaces compared to the Cu/Fe lms. Another interpretation of the XRR results is that the Ag/Fe interface is more diuse than the Cu/Fe interface { i.e., due to interfacial intermixing of Ag and Fe. By 60.000.050.100.150.200.250.3010 nm2 5 nmReflectivity [a.u.] ( a) Cu/Fe Ag/Fe q z [Å-1](b)FIG. 2. (Color online) X-ray re ectivity scans of 10 nm and 25 nm thick lms from (a) Cu/Fe (blue circles) and (b) Ag/Fe (red squares) sample series. Black solid curves are ts to the data. tting the XRR results35, we estimate an average roughness (or the thickness of the diuse interfacial layer) of .1 nm for the Fe layer in Cu/Fe, while it is much greater at 2-3 nm for Ag/Fe36. Our structural characterization described above thus reveals key attributes of the Cu/Fe and Ag/Fe sample series. Both lm series are polycrystalline, exhibit (110) texture, and have grain sizes of order lm thickness. Nevertheless, there are also crucial dierences between Cu/Fe and Ag/Fe. The Cu/Fe series overall exhibits stronger 𝜃-2𝜃diraction peaks than the Ag/Fe series, suggesting that the (110) bcc crystal planes of Fe grown on Cu are aligned within a tighter angular range than those grown on Ag. Moreover, Fe grown on Cu has relatively smooth or sharp interfaces compared to Fe grown on Ag. Although identifying the origin of such structural dierences is beyond the scope of this work, Cu/Fe 7and Ag/Fe constitute two qualitatively distinct series of polycrystalline Fe lms for exploring the in uence of microstructure on magnetic relaxation. III. INTRINSIC GILBERT DAMPING PROBED BY OUT-OF-PLANE FMR Having established the dierence in structural properties between Cu/Fe and Ag/Fe, we characterize room-temperature intrinsic damping for these samples with OOP FMR mea- surements. The OOP geometry suppresses two-magnon scattering25such that the Gilbert damping parameter can be quantied in a straightforward manner. We use a W-band shorted waveguide in a superconducting magnet, which permits FMR measurements at high elds ( &4 T) that completely magnetize the Fe lms out of plane. The details of the mea- surement method are found in Refs.18,37. Figure 3(a) shows the frequency dependence of half-width-at-half-maximum (HWHM) linewidth Δ𝐻OOP for selected thicknesses from both sample series. The linewidth data of 25 nm thick epitaxial Fe lm from a previous study18 is plotted in Fig. 3 (a) as well. The intrinsic damping parameter can be extracted from the linewidth plot using Δ𝐻OOP=Δ𝐻0¸2𝜋 𝛾𝛼OOP𝑓 (1) whereΔ𝐻0is the inhomogeneous broadening38,𝛾=𝑔𝜇𝐵 ℏis the gyromagnetic ratio ( 𝛾2𝜋 2.9 MHz/Oe [Ref.39], obtained from the frequency dependence of resonance eld37), and 𝛼OOP is the measured viscous damping parameter. In general, 𝛼OOP can include not only intrinsic Gilbert damping, parameterized by 𝛼int, but also eddy-current, radiative damping, and spin pumping contributions40, which all yield a linear frequency dependence of the linewidth. Damping due to eddy current is estimated to make up less than 10% of the total measured damping parameter37and is ignored here. Since we used a shorted waveguide in our setup, the radiative damping does not apply here. Spin pumping is also negligible for most of the samples here because the materials in the seed and capping layers (i.e., Ti, Cu, and Ag) possess weak spin-orbit coupling and are hence poor spin sinks31,41,42. We therefore proceed by assuming that the measured OOP damping parameter 𝛼OOP is equivalent to the intrinsic Gilbert damping parameter. The extracted damping parameter is plotted as a function of Fe lm thickness in Fig. 3(b). The room-temperature damping parameters of all Fe lms with thicknesses of 4-25 80204060801001200306090120150180 25nm epitaxial Fe 10nm Cu/Fe 25nm Cu/Fe 10nm Ag/Fe 25nm Ag/FeΔHOOP [Oe]f [GHz](a) 05101520250.0000.0010.0020.0030.004 epitaxial Fe Cu/Fe Ag/FeαOOPT hickness [nm](b)FIG. 3. (Color online) (a) OOP FMR half-width-at-half-maximum linewidth Δ𝐻OOPas a function of resonance frequency 𝑓. Lines correspond to ts to the data. (b) Gilbert damping parameter 𝛼𝑚𝑎𝑡ℎ𝑟𝑚𝑂𝑂𝑃 extracted from OOP FMR as a function of lm thickness. The red shaded area highlights the damping value range that contains data points of all lms thicker than 4 nm. The data for the epitaxial Fe sample (25 nm thick Fe grown on MgAl 2O4) are adapted from Ref.18. nm fall in the range of 0.0024 0.0004, which is shaded in red in Fig. 3(b). This damping parameter range is quantitatively in line with the value reported for epitaxial Fe (black symbol in Fig. 3(b))18. For 2 nm thick samples, the damping parameter is larger likely due to an additional interfacial contribution43{45{ e.g., spin relaxation through interfacial Rashba spin-orbit coupling46that becomes evident only for ultrathin Fe. The results in Fig. 3(b) therefore indicate that the structural properties of the &4 nm thick polycrystalline bcc Fe lms have little in uence on their intrinsic damping. It is remarkable that these polycrystalline Cu/Fe and Ag/Fe lms { with dierent thick- 9nesses and microstructural properties (as revealed in Sec. II) { exhibit essentially the same room-temperature intrinsic Gilbert damping parameter as single-crystalline bcc Fe. This nding is qualitatively distinct from a prior report18on intrinsic Gilbert damping in single- crystalline Fe lms at cryogenic temperature, which is sensitive to microstructural disorder. In the following, we discuss the possible dierences in the mechanisms of intrinsic damping between these temperature regimes. Intrinsic Gilbert damping in ferromagnetic metals is predominantly governed by transi- tions of spin-polarized electrons between electronic states, within a given electronic band (intraband scattering) or in dierent electronic bands (interband scattering) near the Fermi level15. For Fe, previous studies15,18,47indicate that intraband scattering tends to dominate at low temperature where the electronic scattering rate is low (e.g., 1013s1); by contrast, interband scattering likely dominates at room temperature where the electronic scattering rate is higher (e.g., 1014s1). According to our results (Fig. 3(b)), intrinsic damping at room temperature is evidently unaected by the variation in the structural properties of the Fe lms. Hence, the observed intrinsic damping is mostly governed by the electronic band structure within the Fe grains , such that disorder in grain boundaries or lm interfaces has minimal impact. The question remains as to why interband scattering at room temperature leads to Gilbert damping that is insensitive to microstructural disorder, in contrast to intraband scattering at low temperature yielding damping that is quite sensitive to microstructure18. This dis- tinction may be governed by what predominantly drives electronic scattering { specically, defects (e.g., grain boundaries, rough or diuse interfaces) at low temperature, as opposed to phonons at high temperature. That is, the dominance of phonon-driven scattering at room temperature may eectively diminish the roles of microstructural defects in Gilbert damping. Future experimental studies of temperature-dependent damping in polycrystalline Fe lms may provide deeper insights. Regardless of the underlying mechanisms, the robust consistency of 𝛼OOP (Fig. 3(b)) could be an indication that the intrinsic Gilbert damp- ing parameter at a suciently high temperature is a local property of the ferromagnetic metal, possibly averaged over the ferromagnetic exchange length of just a few nm48that is comparable or smaller than the grain size. In this scenario, the impact on damping from grain boundaries would be limited in comparison to the contributions to damping within the grains. 10Moreover, the misalignment of Fe grains evidently does not have much in uence on the intrinsic damping. This is reasonable considering that intrinsic Gilbert damping is predicted to be nearly isotropic in Fe at suciently high electronic scattering rates49{ e.g.,1014s1 at room temperature where interband scattering is expected to be dominant15,18,47. It is also worth emphasizing that 𝛼OOP remains unchanged for Fe lms of various thicknesses with dierent magnitudes of strain (tetragonal distortion, as evidenced by the variation in the out-of-plane lattice parameter in Fig. 1(e)). Strain in Fe grains is not expected to impact the intrinsic damping, as Ref.18suggests that strain in bcc Fe does not signicantly alter the band structure near the Fermi level. Thus, polycrystalline Fe lms exhibit essentially the same magnitude of room-temperature intrinsic Gilbert damping as epitaxial Fe, as long as the grains retain the bcc crystal structure. The observed invariance of intrinsic damping here is quite dierent from the recent study of polycrystalline Co 25Fe75alloy lms31, reporting a decrease in intrinsic damping with in- creasing structural disorder. This inverse correlation between intrinsic damping and disorder in Ref.31is attributed to the dominance of intraband scattering, which is inversely propor- tional to the electronic scattering rate. It remains an open challenge to understand why the room-temperature intrinsic Gilbert damping of some ferromagnetic metals might be more sensitive to structural disorder than others. IV. EXTRINSIC MAGNETIC RELAXATION PROBED BY IN-PLANE FMR Although we have shown via OOP FMR in Sec. III that intrinsic Gilbert damping is essentially independent of the structural properties of the Fe lms, it might be expected that microstructure has a pronounced impact on extrinsic magnetic relaxation driven by two-magnon scattering, which is generally present in IP FMR. IP magnetized lms are more common in device applications than OOP magnetized lms, since the shape anisotropy of thin lms tends to keep the magnetization in the lm plane. What governs the performance of such magnetic devices (e.g., quality factor50,51) may not be the intrinsic Gilbert damping parameter but the total FMR linewidth. Thus, for many magnetic device applications, it is essential to understand the contributions to the IP FMR linewidth. IP FMR measurements have been performed using a coplanar-waveguide-based spectrom- eter, as detailed in Refs.18,37. Examples of the frequency dependence of IP FMR linewidth 110501001502002500 10203040506070050100150200250Cu/FeA g/Fe 2 nm 6 nm 8 nm 10 nm 15 nm 25 nmΔHIP [Oe] 12( a) f [GHz](b)FIG. 4. (Color online) IP FMR half-width-at-half-maximum linewidth Δ𝐻IPas a function of resonance frequency 𝑓for (a) Cu/Fe and (b) Ag/Fe. The vertical dashed line at 12 GHz highlights the hump in linewidth vs frequency seen for many of the samples. are shown in Fig. 4. In contrast to the linear frequency dependence that arises from in- trinsic Gilbert damping in Fig. 3(a), a nonlinear hump is observed for most of the lms in the vicinity of 12 GHz. In some lms, e.g., 10 nm thick Cu/Fe lm, the hump is so large that its peak even exceeds the linewidth at the highest measured frequency. Similar nonlinear IP FMR linewidth behavior has been observed in Fe alloy lms52and epitaxial Heusler lms53in previous studies, where two-magnon scattering has been identied as a signicant contributor to the FMR linewidth. Therefore, in the following, we attribute the nonlinear behavior to two-magnon scattering. To gain insight into the origin of two-magnon scattering, we plot the linewidth at 12 122550751001251500 5101520250255075100125150 Cu/Fe Ag/Fe Cu/Fe Ag/FeΔHIP @ 12 GHz [Oe](a)HC [Oe]T hickness [nm](b)FIG. 5. (Color online) (a) IP FMR half-width-at-half-maximum linewidth at 12 GHz { approxi- mately where the maximum (\hump") in linewidth vs frequency is seen (see Fig. 4) { as a function of lm thickness for both Cu/Fe and Ag/Fe. (b) Coercivity 𝐻𝑐as a function of lm thickness for both Cu/Fe and Ag/Fe. The red shaded area highlights thickness region where the Cu/Fe sample series show a peak behavior in both plots. GHz { approximately where the hump is seen in Fig. 4 { against the Fe lm thickness in Fig. 5(a). We do not observe a monotonic decay in the linewidth with increasing thickness that would result from two-magnon scattering of interfacial origin54. Rather, we observe a non-monotonic thickness dependence in Fig. 5(a), which indicates that the observed two-magnon scattering originates within the bulk of the lms. We note that Ag/Fe with greater interfacial disorder (see Sec. II) exhibits weaker two-magnon scattering than Cu/Fe, particularly in the lower thickness regime ( .10 nm). This observation further corroborates 13that the two-magnon scattering here is not governed by the interfacial roughness of Fe lms. The contrast between Cu/Fe and Ag/Fe also might appear counterintuitive, since two-magnon scattering is induced by defects and hence might be expected to be stronger for more \defective" lms (i.e., Ag/Fe in this case). The counterintuitive nature of the two-magnon scattering here points to more subtle mechanisms at work. To search for a possible correlation between static magnetic properties and two-magnon scattering, we have performed vibrating sample magnetometry (VSM) measurements with a Microsense EZ9 VSM. Coercivity extracted from VSM measurements is plotted as a function of lm thickness in Fig. 5(b), which shows a remarkably close correspondence with linewidth vs thickness (Fig. 5(a)). In particular, a pronounced peak in coercivity is observed for Cu/Fe around 10 nm, corresponding to the same thickness regime where the 12 GHz FMR linewidth for Cu/Fe is maximized. Moreover, the 10 nm Cu/Fe sample (see Sec. II) exhibits a tall, narrow bcc (110) diraction peak, which suggests that its peculiar microstructure plays a possible role in the large two-magnon scattering and coercivity (e.g., via stronger domain wall pinning). While the trends shown in Fig. 5 provide some qualitative insights, we now attempt to quantitatively analyze the frequency dependence of FMR linewidth for the Cu/Fe and Ag/Fe lms. We assume that the Gilbert damping parameter for IP FMR is equal to that for OOP FMR, i.e.,𝛼IP=𝛼OOP. This assumption is physically reasonable, considering that Gilbert damping is theoretically expected to be isotropic in Fe lms near room temperature49. While a recent study has reported anisotropic Gilbert damping that scales quadratically with magnetostriction55, this eect is likely negligible in elemental Fe whose magnetostriction is several times smaller56,57than that of the Fe 07Ga03alloy in Ref.55. Thus, from the measured IP linewidth Δ𝐻IP, the extrinsic two-magnon scattering linewidthΔ𝐻TMS can be obtained by Δ𝐻TMS=Δ𝐻IP2𝜋 𝛾𝛼IP (2) where2𝜋 𝛾𝛼IPis the Gilbert damping contribution. Figure 6 shows the obtained Δ𝐻TMSand t attempts using the \grain-to-grain" two-magnon scattering model developed by McMicheal and Krivosik25. This model captures the inhomogeneity of the eective internal magnetic eld in a lm consisting of many magnetic grains. The magnetic inhomogeneity can arise from the distribution of magnetocrystalline anisotropy eld directions associated with the 14randomly oriented crystal grains52. In this model the two-magnon scattering linewidth Δ𝐻TMS is a function of the Gilbert damping parameter 𝛼IP, the eective anisotropy eld 𝐻𝑎of the randomly oriented grain, and the correlation length 𝜉within which the eective internal magnetic eld is correlated. Further details for computing Δ𝐻TMS are provided in the Appendix and Refs.25,52,53. As we have specied above, 𝛼IPis set to the value derived from OOP FMR results (i.e., 𝛼OOP in Fig. 3(b)). This leaves 𝜉and𝐻𝑎as the only free parameters in the tting process. The modeling results are dependent on the choice of the correlation function 𝐶¹Rº, which captures how the eective internal magnetic eld is correlated as a function of lateral distance Rin the lm plane. We rst show results obtained with a simple exponentially decaying correlation function, as done in prior studies of two-magnon scattering25,52,53, i.e., 𝐶¹Rº=exp jRj 𝜉 (3) Equation 3 has the same form as the simplest correlation function used to model rough topographical surfaces (when they are assumed to be \self-ane")58. Fit results with Eq. (3) are shown in dashed blue curves in Fig. 6. For most samples, the tted curve does not reproduce the experimental data quantitatively. Moreover, the tted values of 𝜉and𝐻𝑎 often reach physically unrealistic values, e.g., with 𝐻𝑎¡104Oe and𝜉 1 nm (see Table I). These results suggest that the model does not properly capture the underlying physics of two-magnon scattering in our samples. A possible cause for the failure to t the data is that the simple correlation function (Eq. 3) is inadequate. We therefore consider an alternative correlation function by again invoking an analogy between the spatially varying height of a rough surface58and the spa- tially varying eective internal magnetic eld in a lm. Specically, we apply a correlation function (i.e., a special case of Eq. (4.3) in Ref.58where short-range roughness 𝛼=1) for the so-called \mounded surface," which incorporates the average distance 𝜆between peaks in topographical height (or, analogously, eective internal magnetic eld): 𝐶¹Rº=p 2jRj 𝜉𝐾1 p 2jRj 𝜉! 𝐽02𝜋jRj 𝜆 (4) where𝐽0and𝐾1are the Bessel function of the rst kind of order zero and the modied Bessel function of the second kind of order one, respectively. This oscillatory decaying function is chosen because its Fourier transform (see Appendix) does not contain any transcendental 15020406080100120 Experimental Self-affine MoundedΔHTMS [Oe]Cu/FeA g/Fe6 nm8 nm1 0 nm1 5 nm2 5 nm(a)( f)0 50100150ΔHTMS [Oe]( b)( g)0 50100150ΔHTMS [Oe]( c)( h)0 255075100125ΔHTMS [Oe]( d)( i)0 2 04 06 0050100150200ΔHTMS [Oe]f [GHz](e)0 2 04 06 0f [GHz](j)FIG. 6. (Color online) Extrinsic two-magnon scattering linewidth Δ𝐻TMSvs frequency 𝑓and tted curves for 6, 8, 10, 15, and 25 nm Cu/Fe and Ag/Fe lms. Black squares represent experimental FMR linewidth data. Dashed blue and solid red curves represent the tted curves using correlation functions proposed for modeling self-ane and mounded surfaces, respectively. In (d), (e), (h), (i), dashed blue curves overlap with solid red curves. 16functions, which simplies the numerical calculation. We also stress that while Eq. (4) in the original context (Ref.58) was used to model topographical roughness, we are applying Eq. (4) in an attempt to model the spatial uctuations (\roughness") of the eective internal magnetic eld { rather than the roughness of the lm topography. The tted curves using the model with Eq. (4) are shown in solid red curves in Fig. 6. Fit results for some samples show visible improvement, although this is perhaps not surprising with the introduction of 𝜆as an additional free parameter. Nevertheless, the tted values of𝐻𝑎or𝜆still diverge to unrealistic values of ¡104Oe or¡104nm in some cases (see Table I), which means that the new correlation function (Eq. (4)) does not fully re ect the meaningful underlying physics of our samples either. More detailed characterization of the microstructure and inhomogeneities, e.g., via synchrotron x-ray and neutron scattering, could help determine the appropriate correlation function. It is also worth pointing out that for some samples (e.g. 15 nm Cu/Fe and Ag/Fe lms), essentially identical t curves are obtained regardless of the correlation function. This is because when 𝜆𝜉, the Fourier transform of Eq. (4) has a very similar form as the Fourier transform of Eq. (3), as shown in the Appendix. In such cases, the choice of the correlation function has almost no in uence on the behavior of the two-magnon scattering model in the tting process. V. SUMMARY We have examined room-temperature intrinsic and extrinsic damping in two series of polycrystalline Fe thin lms with distinct structural properties. Out-of-plane FMR mea- surements conrm constant intrinsic Gilbert damping of 0.0024, essentially independent of lm thickness and structural properties. This nding implies that intrinsic damping in Fe at room temperature is predominantly governed by the crystalline and electronic band structures within the grains, rather than scattering at grain boundaries or lm surfaces. The results from in-plane FMR, where extrinsic damping (i.e., two-magnon scattering) plays a signicant role, are far more nuanced. The conventional grain-to-grain two-magnon scatter- ing model fails to reproduce the in-plane FMR linewidth data with physically reasonable parameters { pointing to the need to modify the model, along with more detailed character- ization of the lm microstructure. Our experimental ndings advance the understanding of intrinsic Gilbert damping in polycrystalline Fe, while motivating further studies to uncover 17TABLE I. Summary of IP FMR linewidth t results. Note the divergence to physically unreasonable values in many of the results. Standard error is calculated using equation√︁ SSRDOFdiag¹COVº, where SSR stands for the sum of squared residuals, DOF stands for degrees of freedom, and COV stands for the covariance matrix. Self-ane Mounded Sample SeriesThickness (nm)𝜉 (nm)𝐻𝑎 (Oe)𝜉 (nm)𝐻𝑎 (Oe)𝜆 (nm) Cu/Fe6 7010 17010 8090 243 >1104 8 200100 15020 7001000 252 900100 10 14040 20020 16050 331 800200 15 92 800100 1020 10080 >1104 25 05 >11046030 >110410.410.01 Ag/Fe6 040 >110415040 >110411.70.7 8 030 >110417050 >1104124 10 61 1500300 840 200500 >1104 15 22 40003000 39 500900 >6103 25 06 >110414050 >1104156 the mechanisms of extrinsic damping in structurally disordered thin lms. ACKNOWLEDGMENTS S.W. acknowledges support by the ICTAS Junior Faculty Program. D.A.S. and S.E. acknowledge support by the National Science Foundation, Grant No. DMR-2003914. P. N. would like to acknowledge support through NASA Grant NASA CAN80NSSC18M0023. A. R. would like to acknowledge support through the Defense Advanced Research Project Agency (DARPA) program on Topological Excitations in Electronics (TEE) under Grant No. D18AP00011. This work was supported by NanoEarth, a member of National Nan- otechnology Coordinated Infrastructure (NNCI), supported by NSF (ECCS 1542100). 18Appendix A: Details of the Two-Magnon Scattering Model In the model developed by McMichael and Krivosik, the two-magnon scattering contri- butionΔ𝐻TMS to the FMR linewidth is given by25,52,53 Δ𝐻TMS=𝛾2𝐻2 𝑎 2𝜋𝑃𝐴¹𝜔º∫ Λ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔𝜔𝑘ºd2𝑘 (A1) where𝜉is correlation length, 𝐻𝑎is the eective anisotropy eld of the randomly oriented grain.𝑃𝐴¹𝜔º=𝜕𝜔 𝜕𝐻 𝐻=𝐻FMR=√︃ 1¸¹4𝜋𝑀𝑠 2𝜔𝛾º2accounts for the conversion between the fre- quency and eld swept linewidth. Λ0𝑘represents the averaging of the anisotropy axis uc- tuations over the sample. It also takes into account the ellipticity of the precession for both the uniform FMR mode and the spin wave mode52. The detailed expression of Λ0𝑘can be found in the Appendix of Ref.52. The coecients in the expression of Λ0𝑘depend on the type of anisotropy of the system. Here, we used rst-order cubic anisotropy for bcc Fe. 𝛿𝛼¹𝜔𝜔𝑘ºselects all the degenerate modes, where 𝜔represents the FMR mode frequency and𝜔𝑘represents the spin wave mode frequency. The detailed expression of 𝜔𝑘can be found in Ref.25. In the ideal case where Gilbert damping is 0, 𝛿𝛼is the Dirac delta function. For a nite damping, 𝛿𝛼¹𝜔0𝜔𝑘ºis replaced by a Lorentzian function1 𝜋¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻 ¹𝜔𝑘𝜔º2¸»¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻¼2, which is centered at 𝜔and has the width of ¹2𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻. Finally,𝐶𝑘¹𝜉º(or𝐶𝑘¹𝜉𝜆º) is the Fourier transform of the grain-to-grain internal eld correlation function, Eq. (3) (or Eq. (4)). For the description of magnetic inhomogeneity analogous to the simple self-ane topographical surface58, the Fourier transform of the correlation function, Eq. (3), is 𝐶𝑘¹𝜉º=2𝜋𝜉2 »1¸¹𝑘𝜉º2¼3 2 (A2) as also used in Refs.25,52,53. For the description analogous to the mounded surface, the Fourier transform of the correlation function, Eq. (4), is58 𝐶𝑘¹𝜉𝜆º=8𝜋3𝜉2 1¸2𝜋2𝜉2 𝜆2¸𝜉2 2𝑘2  1¸2𝜋2𝜉2 𝜆2¸𝜉2 2𝑘22  2𝜋𝜉2 𝜆𝑘232 (A3) When𝜆𝜉, Eq. (A3) becomes 𝐶𝑘¹𝜉º8𝜋3𝜉2  1¸𝜉2 2𝑘22 (A4) 19100102104106108101010-2410-2210-2010-1810-1610-1410-1210-10 Self-affine Mounded λ = 10 nm Mounded λ = 100 nm Mounded λ = 1000 nmCk [m2]k [m-1]ξ = 100 nmFIG. 7. Fourier transform of correlation function for mounded surfaces as a function of wavenumber 𝑘for three dierent 𝜆values. Fourier transform of correlation function for self-ane surfaces as a function of 𝑘is also included for comparison purpose. 𝜉is set as 100 nm for all curves. which has a similar form as Eq. (A2). This similarity can also be demonstrated graphically. Figure 7 plots a self-ane 𝐶𝑘curve (Eq. (A2)) at 𝜉=100 nm and three mounded 𝐶𝑘curves (Eq. (A3)) at 𝜆=10, 100, 1000 nm. 𝜉in mounded 𝐶𝑘curves is set as 100 nm as well. It is clearly shown in Fig. 7 that when 𝜆=1000 nm, the peak appearing in 𝜆=10 and 100 nm mounded 𝐶𝑘curves disappears and the curve shape of mounded 𝐶𝑘resembles that of self-ane𝐶𝑘. The hump feature in Fig. 4 is governed by both 𝛿𝛼and𝐶𝑘(see Eq. A1). 𝛿𝛼has the shape of1in reciprocal space ( 𝑘space), as shown in our videos in the Supplemental Material as well as Fig. 5(b) of Ref.53and Fig 2 (b) of Ref.25. The size of the contour of the degenerated spin wave modes in 𝑘space increases as the microwave frequency 𝑓increases, which means the number of available degenerate spin wave modes increases as 𝑓increases. As shown in Fig. 7, self-ane 𝐶𝑘is nearly constant with the wavenumber 𝑘until𝑘reaches1𝜉. This suggests that the system becomes eectively more uniform (i.e. weaker inhomogeneous perturbation) when the length scale falls below the characteristic correlation length 𝜉(i.e., 𝑘 ¡1𝜉). Because inhomogeneities serve as the scattering centers of two-magnon scattering 20process, degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be scattered into. Now we consider the 𝑓dependence of the two-magnon scattering rate. When 𝑓is small, the two-magnon scattering rate increases as 𝑓increases because more degenerate spin wave modes become available as 𝑓increases. When 𝑓further increases, the wavenumber 𝑘of some degenerate spin wave modes exceeds 1 𝜉. This will decrease the overall two-magnon scattering rate because the degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be scattered into, as discussed above. Furthermore, the portion of degenerate spin wave modes with𝑘 ¡ 1𝜉increases as 𝑓continues to increase. When the impact of decreasing two- magnon scattering rate for degenerate spin wave modes with high 𝑘surpasses the impact of increasing available degenerate spin wave modes, the overall two-magnon scattering rate will start to decrease as 𝑓increases. Consequently, the nonlinear trend { i.e., a \hump" { in FMR linewidth Δ𝐻TMS vs𝑓appears in Fig. 4. However, the scenario discussed above can only happen when 𝜉is large enough, because the wavenumber 𝑘of degenerate spin wave modes saturates (i.e., reaches a limit) as 𝑓 approaches innity. If the limit value of 𝑘is smaller than 1𝜉, the two-magnon scattering rate will increase monotonically as 𝑓increases. In that case the hump feature will not appear. See our videos in the Supplemental Material that display the 𝑓dependence of Λ0𝑘, 𝛿𝛼¹𝜔𝜔𝑘º,𝐶𝑘¹𝜉º 2𝜋𝜉2,Λ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔𝜔𝑘º 2𝜋𝜉2 , andΔ𝐻TMS for various𝜉values. Previous discussions of the hump feature are all based on the self-ane correlation func- tion (Eq. 3). The main dierence between the mounded correlation function (Eq. 4) and the self-ane correlation function (Eq. 3) is that the mounded correlation function has a peak when𝜆is not much larger than 𝜉as shown in Fig. 7. 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0808.1373v1.Gilbert_Damping_in_Conducting_Ferromagnets_I__Kohn_Sham_Theory_and_Atomic_Scale_Inhomogeneity.pdf

arXiv:0808.1373v1 [cond-mat.mes-hall] 9 Aug 2008Gilbert Damping in Conducting Ferromagnets I: Kohn-Sham Theory and Atomic-Scale Inhomogeneity Ion Garate and Allan MacDonald Department of Physics, The University of Texas at Austin, Au stin TX 78712 (Dated: October 27, 2018) We derive an approximate expression for the Gilbert damping coefficient αGof itinerant electron ferromagnets which is based on their description in terms of spin-density-functional-theory (SDFT) and Kohn-Sham quasiparticle orbitals. We argue for an expre ssion in which the coupling of mag- netization fluctuations to particle-hole transitions is we ighted by the spin-dependent part of the theory’s exchange-correlation potential, a quantity whic h has large spatial variations on an atomic length scale. Our SDFT result for αGis closely related to the previously proposed spin-torque correlation-function expression. PACS numbers: I. INTRODUCTION The Gilbert parameter αGcharacterizes the damping of collective magnetization dynamics1. The key role of αGin current-driven2and precessional3magnetization reversal has renewed interest in the microscopic physics of this important material parameter. It is generally accepted that in metals the damping of magnetization dynamics is dominated3by particle-hole pair excitation processes. The main ideas which arise in the theory of Gilbert damping have been in place for some time4,5. It has however been difficult to apply them to real materi- als with the precision required for confident predictions which would allow theory to play a larger role in design- ing materials with desired damping strengths. Progress has recently been achieved in various directions, both through studies6of simple models for which the damp- ing can be evaluated exactly and through analyses7of transition metal ferromagnets that are based on realis- tic electronic structure calculations. Evaluation of the torquecorrelationformula5forαGusedinthelatercalcu- lations requires knowledge only of a ferromagnet’s mean- field electronic structure and of its Bloch state lifetime, which makes this approach practical. Realistic ab initio theories normally employ spin- density-functional theory9which has a mean-field theory structure. In this article we use time-dependent spin- density functional theory to derive an explicit expression for the Gilbert damping coefficient in terms of Kohn- Sham theory eigenvalues and eigenvectors. Our final result is essentially equivalent to the torque-correlation formula5forαG, but has the advantages that its deriva- tion is fully consistent with density functional theory, that it allows for a consistent microscopic treatments of both dissipative and reactive coefficients in the Landau- Liftshitz Gilbert (LLG) equations, and that it helps establish relationships between different theoretical ap- proaches to the microscopic theory of magnetization damping. Our paper is organized as follows. In Section II we relate the Gilbert damping parameter αGof a fer- romagnet to the low-frequency limit of its transversespin response function. Since ferromagnetism is due to electron-electron interactions, theories of magnetism are always many-electron theories, and it is necessary to evaluate the many-electron response function. In time- dependent spin-density functional theory the transverse response function is calculated using a time-dependent self- consistent-field calculation in which quasiparticles respond both to external potentials and to changesin the interaction-induced effective potential. In Section III we use perturbation theory and time-dependent mean-field theory to express the coefficients which appear in the LLG equations in terms of the Kohn-Sham eigenstates and eigenvaluesof the ferromagnet’sground state. These formal expressions are valid for arbitrary spin-orbit cou- pling, arbitrary atomic length scale spin-dependent and scalarpotentials, and arbitrarydisorder. By treating dis- order approximately, in Section IV we derive and com- pare two commonly used formulas for Gilbert damping. Finally, in Section V we summarize our results. II. MANY-BODY TRANSVERSE RESPONSE FUNCTION AND THE GILBERT DAMPING PARAMETER The Gilbert damping parameter αGappears in the Landau-Liftshitz-Gilbert expression for the collective magnetization dynamics of a ferromagnet: ∂ˆΩ ∂t=ˆΩ×Heff−αGˆΩ×∂ˆΩ ∂t. (1) In Eq.( 1) Heffis an effective magnetic field which we comment on further below and ˆΩ = (Ω x,Ωy,Ωz) is the direction of the magnetization. This equation de- scribes the slow dynamics of smooth magnetization tex- tures and is formally the first term in an expansion in time-derivatives. The damping parameter αGcan be measured by per- forming ferromagnetic resonance (FMR) experiments in which the magnetization direction is driven weakly away from an easy direction (which we take to be the ˆ z- direction.). To relate this phenomenological expression2 formally to microscopic theory we consider a system in which external magnetic fields couple only11to the elec- tronic spin degree of freedom and associate the magneti- zation direction ˆΩ with the direction of the total electron spin. Forsmalldeviationsfromtheeasydirection,Eq.(1) reads Heff,x= +∂ˆΩy ∂t+αG∂ˆΩx ∂t Heff,y=−∂ˆΩx ∂t+αG∂ˆΩy ∂t. (2) The gyromagnetic ratio has been absorbed into the unitsof the field Heffso that this quantity has energy units and we set /planckover2pi1= 1 throughout. The corresponding formal linear response theory expression is an expansion of the long wavelength transverse total spin response function to first order12in frequency ω: S0ˆΩα=/summationdisplay β[χst α,β+ωχ′ α,β]Hext,β (3) whereα,β∈ {x,y},ω≡i∂tis the frequency, S0is the to- tal spin ofthe ferromagnet, Hextis the external magnetic field and χis the transverse spin-spin response function: χα,β(ω) =i/integraldisplay∞ 0dtexp(iωt)/an}bracketle{t[Sα(t),Sβ(t)]/an}bracketri}ht=/summationdisplay n/bracketleftbigg/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht ωn,0−ω−iη+/an}bracketle{tΨ0|Sβ|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sα|Ψ0/an}bracketri}ht ωn,0+ω+iη/bracketrightbigg (4) Here|Ψn/an}bracketri}htis an exact eigenstate of the many-body Hamiltonian and ωn,0is the excitation energy for state n. We use this formal expression below to make some general comments abou t the microscopic theory of αG. In Eq.( 3) χst α,βis the static ( ω= 0) limit of the response function, and χ′ α,βis the first derivative with respect to ωevaluated at ω= 0. Notice that we have chosen the normalization in which χis the total spin response to a transverse field; χis therefore extensive. The keystep in obtainingthe Landau-Liftshitz-Gilbert form for the magnetization dynamics is to recognize that in the static limit the transverse magnetization responds to an external magnetic field by adjusting orientation to minimize the total energy including the internal energy Eintand the energy due to coupling with the external magnetic field, Eext=−S0ˆΩ·Hext. (5) It follows that χst α,β=S2 0/bracketleftBigg ∂2Eint ∂ˆΩαˆΩβ/bracketrightBigg−1 . (6) We obtain a formal equation for Heffcorresponding to Eq.( 2) by multiplying Eq.( 3) on the left by [ χst α,β]−1and recognizing Hint,α=−1 S0/summationdisplay β∂2Eint ∂ˆΩα∂ˆΩβˆΩβ=−1 S0∂Eint ∂ˆΩα(7) as the internal energy contribution to the effective mag- netic field Heff=Hint+Hext. With this identification Eq.( 3) can be written in the form Heff,α=/summationdisplay βLα,β∂tˆΩβ (8) where Lα,β=−S0[i(χst)−1χ′(χst)−1]α,β=iS0∂ωχ−1 α,β.(9)According to the Landau-Liftshitz Gilbert equation then Lx,y=−Ly,x= 1 and Lx,x=Ly,y=αG. (10) Explicit evaluation of the off-diagonal components of L will in general yield very small deviation from the unit result assumed by the Landau-Liftshitz-Gilbert formula. The deviation reflects mainly the fact that the magneti- zation magnitude varies slightly with orientation. We do not comment further on this point because it is of little consequence. Similarly Lx,xis not in general identical toLy,y, although the difference is rarely large or impor- tant. Eq.( 10) is the starting point we use later to derive approximate expressions for αG. In Eq.( 9) χα,β(ω) is the correlation function for an interacting electron system with arbitrary disorder and arbitrary spin-orbit coupling. In the absence of spin- orbit coupling, but still with arbitrary spin-independent periodic and disorder potentials, the ground state of a ferromagnet is coupled by the total spin-operator only to states in the same total spin multiplet. In this case it follows from Eq.( 4) that χst α,β= 2/summationdisplay nRe/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht] ωn,0=δα,βS0 H0 (11) whereH0is a static external field, which is necessary in the absence of spin-orbit coupling to pin the magne- tization to the ˆ zdirection and splits the ferromagnet’s3 ground state many-body spin multiplet. Similarly χ′ α,β= 2i/summationdisplay nIm[/an}bracketle{tΨ0|Sα|Ψn/an}bracketri}ht/an}bracketle{tΨn|Sβ|Ψ0/an}bracketri}ht] ω2 n,0=iǫα,βS0 H2 0. (12) whereǫx,x=ǫy,y= 0 and ǫx,y=−ǫy,x= 1, yielding Lx,y=−Ly,x= 1 and Lx,x=Ly,y= 0. Spin-orbit coupling is required for magnetization damping8. III. SDF-STONER THEORY EXPRESSION FOR GILBERT DAMPING Approximate formulas for αGin metals are inevitably based on on a self-consistent mean-field theory (Stoner) descriptionofthemagneticstate. Ourgoalistoderivean approximate expression for αGwhen the adiabatic local spin-densityapproximation9isused forthe exchangecor- relation potential in spin-density-functional theory. The effective Hamiltonian which describes the Kohn-Sham quasiparticle dynamics therefore has the form HKS=HP−∆(n(/vector r),|/vector s(/vector r)|)ˆΩ(/vector r)·/vector s,(13) whereHPis the Kohn-Sham Hamiltonian of a paramag- netic state in which |/vector s(/vector r)|(the local spin density) is set to zero,/vector sis the spin-operator, and ∆(n,s) =−d[nǫxc(n,s)] ds(14) is the magnitude of the spin-dependent part of the exchange-correlation potential. In Eq.( 14) ǫxc(n,s) is the exchange-correlation energy per particle in a uni- form electron gas with density nand spin-density s. We assume that the ferromagnet is described using some semi-relativistic approximation to the Dirac equa- tion like those commonly used13to describe magnetic anisotropy or XMCD, even though these approximations are not strictly consistent with spin-density-functional theory. Within this framework electrons carry only a two-componentspin-1/2degreeoffreedomandspin-orbit coupling terms are included in HP. Sincenǫxc(n,s)∼ [(n/2 +s)4/3+ (n/2−s)4/3], ∆0(n,s)∼n1/3is larger closertoatomic centersand farfrom spatiallyuniform on atomic length scales. This property figures prominently in the considerations explained below. In SDFT the transverse spin-response function is ex- pressed in terms of Kohn-Sham quasiparticle response to both external and induced magnetic fields: s0(/vector r)Ωα(/vector r) =/integraldisplayd/vectorr′ VχQP α,β(/vector r,/vectorr′) [Hext,β(/vectorr′)+∆0(/vectorr′)Ωβ(/vectorr′)]. (15) In Eq.( 15) Vis the system volume, s0(/vector r) is the magni- tude of the ground state spin density, ∆ 0(/vector r) is the mag- nitude of the spin-dependent part of the ground stateexchange-correlation potential and χQP α,β(/vector r,/vectorr′) =/summationdisplay i,jfj−fi ωi,j−ω−iη/an}bracketle{ti|/vector r/an}bracketri}htsα/an}bracketle{t/vector r|j/an}bracketri}ht/an}bracketle{tj|/vectorr′/an}bracketri}htsβ/an}bracketle{t/vectorr′|i/an}bracketri}ht, (16) wherefiis the ground state Kohn-Sham occupation fac- tor for eigenspinor |i/an}bracketri}htandωij≡ǫi−ǫjis a Kohn- Sham eigenvalue difference. χQP(/vector r,/vectorr′) has been normal- ized so that it returns the spin-density rather than total spin. Like the Landau-Liftshitz-Gilbert equation itself, Eq.( 15) assumes that only the direction of the mag- netization, and not the magnitudes of the charge and spin-densities, varies in the course of smooth collective magnetization dynamics14. This property should hold accurately as long as magnetic anisotropies and exter- nal fields are weak compared to ∆ 0. We are able to use this property to avoid solving the position-space integral equation implied by Eq.( 15). Multiplying by ∆ 0(/vector r) on both sides and integrating over position we find15that S0Ωα=/summationdisplay β1 ¯∆0˜χQP α,β(ω)/bracketleftbig Ωβ+Hext,β ¯∆0/bracketrightbig (17) where we have taken advantage of the fact that in FMR experiments Hext,βandˆΩ are uniform. ¯∆0is a spin- density weighted average of ∆ 0(/vector r), ¯∆0=/integraltext d/vector r∆0(/vector r)s0(/vector r)/integraltext d/vector rs0(/vector r), (18) and ˜χQP α,β(ω) =/summationdisplay ijfj−fi ωij−ω−iη/an}bracketle{tj|sα∆0(/vector r)|i/an}bracketri}ht/an}bracketle{ti|sβ∆0(/vector r)|j/an}bracketri}ht (19) is the response function of the transverse-part of the quasiparticleexchange-correlationeffective field response function, notthe transverse-part of the quasiparticle spin response function. In Eq.( 19), /an}bracketle{ti|O(/vector r)|j/an}bracketri}ht=/integraltext d/vector rO(/vector r)/an}bracketle{ti|/vector r/an}bracketri}ht/an}bracketle{t/vector r|j/an}bracketri}htdenotes a single-particle matrix ele- ment. Solving Eq.( 17) for the many-particle transverse susceptibility (the ratio of S0ˆΩαtoHext,β) and inserting the result in Eq.( 9) yields Lα,β=iS0∂ωχ−1 α,β=−S0¯∆2 0∂ωIm[˜χQP−1 α,β].(20) Our derivation of the LLG equation has the advantage that the equation’s reactive and dissipative components are considered simultaneously. Comparing Eq.( 15) and Eq.( 7) we find that the internal anisotropy field can also be expressed in terms of ˜ χQP: Hint,α=−¯∆2 0S0/summationdisplay β/bracketleftbig ˜χQP−1 α,β(ω= 0)−δα,β S0¯∆0/bracketrightbig Ωβ.(21) Eq.( 20) and Eq.( 21) provide microscopic expressions for all ingredients that appear in the LLG equations4 linearized for small transverse excursions. It is gener- ally assumed that the damping coefficient αGis inde- pendent of orientation; if so, the present derivation is sufficient. The anisotropy-field at large transverse ex- cursions normally requires additional information about magnetic anisotropy. We remark that if the Hamiltonian does not include a spin-dependent mean-field dipole in- teraction term, as is usually the case, the above quantity will return only the magnetocrystalline anisotropy field. Since the magnetostatic contribution to anisotropy is al- ways well described by mean-field-theory it can be added separately. We conclude this section by demonstrating that the Stoner theory equations proposed here recover the exact results mentioned at the end of the previous section for the limit in which spin-orbit coupling is neglected. We consider a SDF theory ferromagnet with arbitrary scalar and spin-dependent effective potentials. Since the spin- dependent part of the exchange correlation potential is then the only spin-dependent term in the Hamiltonian it follows that [HKS,sα] =−iǫα,β∆0(/vector r)sβ (22) and hence that /an}bracketle{ti|sα∆0(/vector r)|j/an}bracketri}ht=−iǫα,βωij/an}bracketle{ti|sβ|j/an}bracketri}ht.(23) Inserting Eq.( 23) in one of the matrix elements of Eq.( 19) yields for the no-spin-orbit-scattering case ˜χQP α,β(ω= 0) =δα,βS0¯∆0. (24)The internal magnetic field Hint,αis therefore identically zero in the absence of spin-orbit coupling and only exter- nal magnetic fields will yield a finite collective precession frequency. Inserting Eq.( 23) in both matrix elements of Eq.( 19) yields ∂ωIm[˜χQP α,β] =ǫα,βS0. (25) Using both Eq.( 24) and Eq.( 25) to invert ˜ χQPwe re- cover the results proved previously for the no-spin-orbit case using a many-body argument: Lx,y=−Ly,x= 1 andLx,x=Ly,y= 0. The Stoner-theory equations de- rived here allow spin-orbit interactions, and hence mag- netic anisotropy and Gilbert damping, to be calculated consistently from the same quasiparticle response func- tion ˜χQP. IV. DISCUSSION As long as magnetic anisotropy and external magnetic fields are weak compared to the exchange-correlation splitting in the ferromagnet we can use Eq.( 24) to ap- proximate ˜ χQP α,β(ω= 0). Using this approximation and assuming that damping is isotropic we obtain the follow- ing explicit expression for temperature T→0: αG=Lx,x=−S0¯∆2 0∂ωIm[˜χQP−1 x,x] =π S0/summationdisplay ijδ(ǫj−ǫF)δ(ǫi−ǫF)/an}bracketle{tj|sx∆0(/vector r)|i/an}bracketri}ht/an}bracketle{ti|sx∆0(/vector r)|j/an}bracketri}ht =π S0/summationdisplay ijδ(ǫj−ǫF)δ(ǫi−ǫF)/an}bracketle{tj|[HP,sy]|i/an}bracketri}ht/an}bracketle{ti|[HP,sy]|j/an}bracketri}ht.(26) The second form for αGis equivalent to the first and follows from the observation that for m atrix elements between states that have the same energy /an}bracketle{ti|[HKS,sα]|j/an}bracketri}ht=−iǫα,β/an}bracketle{ti|∆0(/vector r)sβ|j/an}bracketri}ht+/an}bracketle{ti|[HP,sα]|j/an}bracketri}ht= 0 (for ωij= 0). (27) Eq. ( 26) is valid for any scalar and any spin-dependent potential. It is clear however that the numerical value of αG in a metal is very sensitive to the degree of disorder in its lattice. To s ee this we observe that for a perfect crystal the Kohn-Sham eigenstates are Bloch states. Since the operator ∆0(/vector r)sαhas the periodicity of the crystal its matrix elements are non-zero only between states with the same Bloch wav evector label /vectork. For the case of a perfect crystal then αG=π s0/integraldisplay BZd/vectork (2π)3/summationdisplay nn′δ(ǫ/vectorkn′−ǫF)δ(ǫ/vectorkn−ǫF)/an}bracketle{t/vectorkn′|sx∆0(/vector r)|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn′/an}bracketri}ht =π s0/integraldisplay BZd/vectork (2π)3/summationdisplay nn′δ(ǫ/vectorkn′−ǫF)δ(ǫ/vectorkn−ǫF)/an}bracketle{t/vectorkn′|[HP,sy]|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|[HP,sy]|/vectorkn′/an}bracketri}ht. (28) wherenn′are band labels and s0is the ground state spin per unit volume and the integral over /vectorkis over theBrillouin-zone (BZ).5 Clearly αGdiverges16in a perfect crystal since /an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn/an}bracketri}htis generically non-zero. A theory of αGmust therefore always account for disorder in a crys- tal. The easiest way to account for disorder is to replace theδ(ǫ/vectorkn−ǫF) spectral function of a Bloch state by a broadened spectral function evaluated at the Fermi en- ergyA/vectorkn(ǫF). If disorder is treated perturbatively this simpleansatzcan be augmented17by introducing impu- rity vertex corrections in Eq. ( 28). Provided that the quasiparticlelifetimeiscomputedviaFermi’sgoldenrule,these vertex corrections restore Ward identities and yield an exact treatment of disorder in the limit of dilute im- purities. Nevertheless, this approach is rarely practical outside the realm of toy models, because the sources of disorder are rarely known with sufficient precision. Although appealing in its simplicity, the δ(ǫ/vectorkn−ǫF)→ A/vectorkn(ǫF) substitution is prone to ambiguity because it gives rise to qualitatively different outcomes depending on whether it is applied to the first or second line of Eq. ( 28): α(TC) G=π s0/integraldisplay BZd/vectork (2π)3/summationdisplay nn′A/vectork,n(ǫF)A/vectork,n′(ǫF)/an}bracketle{t/vectorkn′|[HP,sy]|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|[HP,sy]|/vectorkn′/an}bracketri}ht, α(SF) G=π s0/integraldisplay BZd/vectork (2π)3/summationdisplay nn′A/vectork,n(ǫF)A/vectork,n′(ǫF)/an}bracketle{t/vectorkn′|sx∆0(/vector r)|/vectorkn/an}bracketri}ht/an}bracketle{t/vectorkn|sx∆0(/vector r)|/vectorkn′/an}bracketri}ht. (29) α(TC) Gis the torque-correlation (TC) formula used in realistic electronic structure calculations7andα(SF) Gis the spin-flip (SF) formula used in certain toy model calculations18. The discrepancy between TC and SF ex- pressions stems from inter-band ( n/ne}ationslash=n′) contributions to damping, which may now connect states with dif- ferentband energies due to the disorder broadening of the spectral functions. Therefore, /an}bracketle{t/vectorkn|[HKS,sα]|/vectorkn′/an}bracketri}htno longer vanishes for n/ne}ationslash=n′and Eq. ( 27) indicates that α(TC) G≃α(SF) Gonly if the Gilbert damping is dominated by intra-band contributions and/or if the energy differ- ence between the states connected by inter-band transi- tions is small compared to ∆ 0. When α(TC) G/ne}ationslash=α(SF) G, it isa priori unclear which approach is the most accu- rate. One obvious flaw of the SF formula is that it pro- ducesaspuriousdampinginabsenceofspin-orbitinterac- tions; this unphysical contribution originates from inter- band transitions and may be cancelled out by adding the leading order impurity vertex correction19. In con- trast, [HP,sy] = 0 in absence of spin-orbit interaction andhencetheTCformulavanishesidentically, evenwith- out vertex corrections. From this analysis, TC appears to have a pragmatic edge over SF in materials with weak spin-orbitinteraction. However, insofarasit allowsinter- band transitions that connect states with ωi,j>∆0, TC is not quantitatively reliable. Furthermore, it canbe shown17that when the intrinsic spin-orbit coupling is significant (e.g. in ferromagnetic semiconductors), the advantage of TC over SF (or vice versa) is marginal, and impurity vertex corrections play a significant role. V. CONCLUSIONS Using spin-density functional theory we have derived a Stoner model expression for the Gilbert damping co- efficient in itinerant ferromagnets. This expression ac- counts for atomic scale variations of the exchange self energy, as well as for arbitrary disorder and spin-orbit interaction. By treating disorder approximately, we have derived the spin-flip and torque-correlationformulas pre- viously used in toy-model and ab-initio calculations, re- spectively. Wehavetracedthediscrepancybetweenthese equations to the treatment of inter-band transitions that connect states which are not close in energy. A better treatment of disorder, which requires the inclusion of im- purity vertex corrections, will be the ultimate judge on the relativereliabilityofeitherapproach. Whendamping is dominated by intra-band transitions, a circumstance which we believe is common, the two formulas are identi- cal and both arelikely to provide reliable estimates. This work was suported by the National Science Foundation under grant DMR-0547875. 1For a historical perspective see T.L. Gilbert, IEEE Trans. Magn.40, 3443 (2004). 2Foranintroductoryreviewsee D.C. RalphandM.D.Stiles, J. Magn. Mag. Mater. 320, 1190 (2008).3J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag- netic Structures III: Fundamentals of Nanomagnetism (Springer-Verlag, New York, 2005). 4V. Korenman and R. E. Prange, Phys. Rev. B 6, 27696 (1972). 5V. Kambersky, Czech J. Phys. B 26, 1366 (1976). 6Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl. Phys. Lett. 84, 5234 (2004); E.M. Hankiewicz, G. Vig- nale and Y. Tserkovnyak, Phys. Rev. B 75, 174434 (2007); Y. Tserkovnyak et al., Phys. Rev. B 74, 144405 (2006) ; H.J. Skadsem, Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, Phys. Rev. B 75, 094416 (2007); H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan 75, 113706 (2006); R.A. Duine et al., Phys. Rev. B 75, 214420 (2007). Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, J. Magn. Mag. Mater. 320, 1282 (2008). 7K. Gilmore, Y.U.IdzerdaandM.D. Stiles, Phys.Rev.Lett. 99, 27204 (2007); V. Kambersky, Phys. Rev. B 76, 134416 (2007). 8For zero spin-orbit coupling αGvanishes even in presence of magnetic impurities, provided that their spins follow th e dynamics of the magnetization adiabatically. 9O. Gunnarsson, J. Phys. F 6, 587 (1976). 10Z. Qian, G. Vignale, Phys. Rev. Lett. 88, 056404 (2002). 11In doing so we dodge the subtle difficulties which compli- cate theories of orbital magnetism in bulk metals. See for example J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett. 99, 197202 (2007); I. Souza and D. Vanderbilt, Phys. Rev. B 77, 054438 (2008) and work cited therein. This simplification should have little influence on the the- ory of damping because the orbital contribution to the magnetization is relatively small in systems of interest an dbecause it in any event tends to be collinear with the spin magnetization. 12For most materials the FMR frequency is by far the small- est energy scale in the problem. Expansion to linear order is almost always appropriate. 13See for example A.C. Jenkins and W.M. Temmerman, Phys. Rev. B 60, 10233 (1999) and work cited therein. 14This approximation does not preclude strong spatial varia- tions of|s0(/vector r)|and|∆0(/vector r)|at atomic lenghtscales; rather it is assumed that such spatial profiles will remain unchanged in the course of the magnetization dynamics. 15For notational simplicity we assume that all magnetic atoms are identical. Generalizations to magnetic com- pounds are straight forward. 16Eq. ( 26) is valid provided that ωτ <<1. While this con- dition is normally satisfied in cases of practical interest, it invariably breaks down as τ→ ∞. Hence the divergence of Eq. ( 26) in perfectcrystals is spurious. 17I. Garate and A.H. MacDonald (in preparation). 18J. Sinova et al., Phys. Rev. B 69, 85209 (2004). In order to get the equivalence, trade hzby ∆0and use ∆ 0=JpdS0, whereJpdis the p-d exchange coupling between GaAs va- lence band holes and Mn d-orbitals. In addition, note that our spectral function differs from theirs by a factor 2 π. 19H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan 75, 113706 (2006).

1901.01941v1.Giant_anisotropy_of_Gilbert_damping_in_epitaxial_CoFe_films.pdf

Giant anisotropy of Gilbert damping in epitaxial CoFe lms Yi Li,1, 2Fanlong Zeng,3Steven S.-L. Zhang,2Hyeondeok Shin,4Hilal Saglam,2, 5Vedat Karakas,2, 6Ozhan Ozatay,2, 6John E. Pearson,2Olle G. Heinonen,2Yizheng Wu,3, 7,Axel Homann,2,yand Wei Zhang1, 2,z 1Department of Physics, Oakland University, Rochester, MI 48309, USA 2Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA 3State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China 4Computational Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USA 5Department of Physics, Illinois Institute of Technology, Chicago IL 60616, USA 6Department of Physics, Bogazici University, Bebek 34342, Istanbul, Turkey 7Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China (Dated: January 8, 2019) Tailoring Gilbert damping of metallic ferromagnetic thin lms is one of the central interests in spintronics applications. Here we report a giant Gilbert damping anisotropy in epitaxial Co 50Fe50 thin lm with a maximum-minimum damping ratio of 400 %, determined by broadband spin-torque as well as inductive ferromagnetic resonance. We conclude that the origin of this damping anisotropy is the variation of the spin orbit coupling for dierent magnetization orientations in the cubic lattice, which is further corroborate from the magnitude of the anisotropic magnetoresistance in Co 50Fe50. In magnetization dynamics the energy relaxation rate is quantied by the phenomenological Gilbert damping in the Landau-Lifshits-Gilbert equation [1], which is a key parameter for emerging spintronics applications [2{ 6]. Being able to design and control the Gilbert damp- ing on demand is crucial for versatile spintronic device engineering and optimization. For example, lower damp- ing enables more energy-ecient excitations, while larger damping allows faster relaxation to equilibrium and more favorable latency. Nevertheless, despite abundant ap- proaches including interfacial damping enhancement [7{ 9], size eect [10, 11] and materials engineering [12{14], there hasn't been much progress on how to manipulate damping within the same magnetic device. The only well-studied damping manipulation is by spin torque [15{ 18], which can even fully compensate the intrinsic damp- ing [19, 20]. However the requirement of large current density narrows its applied potential. An alternative approach is to explore the intrinsic Gilbert damping anisotropy associated with the crys- talline symmetry, where the damping can be continu- ously tuned via rotating the magnetization orientation. Although there are many theoretical predictions [21{25], most early studies of damping anisotropy are disguised by two-magnon scattering and linewidth broadening due to eld-magnetization misalignment [26{29]. In addition, those reported eects are usually too weak to be consid- ered in practical applications [30, 31]. In this work, we show that a metallic ferromagnet can exhibit a giant Gilbert damping variation by a factor of four along with low minimum damping. We inves- tigated epitaxial cobalt-iron alloys, which have demon- strated new potentials in spintronics due to their ultralow dampings [32, 33]. Using spin-torque-driven and induc- tive ferromagnetic resonance (FMR), we obtain a four- fold (cubic) damping anisotropy of 400% in Co 50Fe50thin lms between their easy and hard axes. For each angle,the full-range frequency dependence of FMR linewidths can be well reproduced by a single damping parame- ter. Furthermore, from rst-principle calculations and temperature-dependent measurements, we argue that this giant damping anisotropy in Co 50Fe50is due to the variation of the spin-orbit coupling (SOC) in the cu- bic lattice, which diers from the anisotropic density of state found in ultrathin Fe lm [30]. We support our conclusion by comparing the Gilbert damping with the anisotropic magnetoresistance (AMR) signals. Our re- sults reveal the key mechanism to engineer the Gilbert damping and may open a new pathway to develop novel functionality in spintronic devices. Co50Fe50(CoFe) lms were deposited on MgO(100) substrates by molecular beam epitaxy at room temper- ature, under a base pressure of 2 1010Torr [34]. For spin-torque FMR measurements, i) CoFe(10 nm)/Pt(6 nm) and ii) CoFe(10 nm) samples were prepared. They were fabricated into 10 m40m bars by photolithog- raphy and ion milling. Coplanar waveguides with 100- nm thick Au were subsequently fabricated [18, 35]. For each layer structure, 14 devices with dierent orienta- tions were fabricated, as shown in Fig. 1(a). The geom- etry denes the orientation of the microwave current, I, and the orientation of the biasing eld, H, with respect to the MgO [100] axis (CoFe [1 10]).Iranges from 0 to 180with a step of 15(D1 to D14, with D7 and D8 pointing to the same direction). For each device we x H=I+ 45for maximal rectication signals. In addi- tion, we also prepared iii) CoFe(20 nm) 40 m200m bars along dierent orientations with transmission copla- nar waveguides fabricated on top for vector network an- alyzer (VNA) measurements. See the Supplemental Ma- terials for details [36]. Fig. 1(b) shows the angular-dependent spin-torque FMR lineshapes of CoFe(10 nm)/Pt devices from dif- ferent samples (D1 to D4, hard axis to easy axis) atarXiv:1901.01941v1 [cond-mat.mtrl-sci] 7 Jan 20192 FIG. 1. (a) Upper: crystalline structure, axes of bcc Co 50Fe50 lm on MgO(100) substrate and denition of HandI. Lower: device orientation with respect to the CoFe crystal axis. (b) Spin-torque FMR lineshapes of i) CoFe(10 nm)/Pt devices D1 to D4 measured. (c) Resonances of D1 and D4 from (b) for 0Hres<0. (d) Resonances of iii) CoFe(20 nm) forH= 45and 90measured by VNA FMR. In (b-d) !=2= 20 GHz and oset applies. !=2= 20 GHz. A strong magnetocrystalline anisotropy as well as a variation of resonance signals are observed. Moreover, the linewidth increases signicantly from easy axis to hard axis, which is shown in Fig. 1(c). We have also conducted rotating-eld measurements on a sec- ond CoFe(10 nm)/Pt device from a dierent deposition and the observations can be reproduced. This linewidth anisotropy is even more pronounced for the CoFe(20 nm) devices without Pt, measured by VNA FMR (Fig. 1d). For the CoFe(10 nm) devices, due to the absence of the Pt spin injector the spin-torque FMR signals are much weaker than CoFe/Pt and completely vanish when the microwave current is along the easy axes. Figs. 2(a-b) show the angular and frequency de- pendence of the resonance eld Hres. In Fig. 2(a), the Hresfor all four sample series match with each other, which demonstrates that the magnetocrystalline proper- ties of CoFe(10 nm) samples are reproducible. A slightly smallerHresfor CoFe(20 nm) is caused by a greater eec- tive magnetization when the thickness increases. A clear fourfold symmetry is observed, which is indicative of the cubic lattice due to the body-center-cubic (bcc) texture of Co 50Fe50on MgO. We note that the directions of the hard axes has switched from [100] and [010] in iron-rich alloys [33] to [110] and [1 10] in Co 50Fe50, which is con- ω/2πμ0Hres (T) μ0Hres (T) [110] [110][100][010](a) (b) CoFe(10 nm)/Pt ω/2π=2045o90 o135o 135o180o 225oCoFe(10 nm)/Pt CoFe(10 nm) CoFe(20 nm) θH: [100] [110][010]FIG. 2. (a) Resonance eld 0Hresas a function of Hat !=2= 20 GHz for dierent samples. Diamonds denote the rotating-eld measurement from the second CoFe(10 nm)/Pt device. The black curve denotes the theoretical prediction. (b)0Hresas a function of frequency for the CoFe(10 nm)/Pt devices. Solid curves denote the ts to the Kittel equation. sistent with previous reports [37, 38]. The magnetocrystalline anisotropy can be quanti- ed from the frequency dependence of 0Hres. Fig. 2(b) shows the results of CoFe(10 nm)/Pt when HB is aligned to the easy and hard axes. A small uniax- ial anisotropy is found between [1 10] (0and 180) and [110] (90) axes. By tting the data to the Kittel equa- tion!2= 2=2 0(HresHk)(HresHk+Ms), where = 2(geff=2)
28 GHz/T, we obtain geff= 2:16, 0Ms= 2:47 T,0H[100] k= 40 mT,0H[010] k= 65 mT and0H[110] k=0H[110] k=43 mT. Taking the disper- sion functions from cubic magnetocrystalline anisotropy [39, 40], we obtain an in-plane cubic anisotropy eld 0H4jj= 48 mT and a uniaxial anisotropy eld 0H2jj= 12 mT. Fig. 2(a) shows the theoretical predictions from H4jjandH2jjin black curve, which aligns well with all 10-nm CoFe samples. With good magnetocrystalline properties, we now turn to the energy relaxation rate. Fig. 3(a) shows the full- width-half-maximum linewidths 0
H1=2of the spin- torque FMR signals at !=2= 20 GHz. Again, a fourfold symmetry is observed for CoFe(10 nm)/Pt and CoFe(10 nm), with the minimal (maximal) linewidth measured when the eld lies along the easy (hard) axes. For CoFe(10 nm) devices, we did not measure any spin-torque FMR signal for HBalong the hard axes ( H= 45, 135 and 225). This is due to the absence of the Pt spin injector as well as the near-zero AMR ratio when the rf current ows along the easy axes, which will be discussed later. For all other measurements, the linewidths of CoFe devices are smaller than for CoFe/Pt by the same con- stant, independent of orientation (upper diagram of Fig. 3a). This constant linewidth dierence is due to the spin pumping contribution to damping from the additional Pt layer [41, 42]. Thus we can deduce the intrinsic damp- ing anisotropy from CoFe(10 nm)/Pt devices, with the3 ω/2π 105, 195 deg 75, 165 deg 120, 210 deg 135, 225 deg(HA) 45, 135 deg (HA) 60, 150 deg 90, 180 deg(EA) θHCoFe(10 nm)/Pt (b) = - =- [100] [110] [110] [010](a) ω/2π=20 ω/2π θH 0, 90 deg 15, 75 deg 22.5, 67.5 deg 30, 60 deg 42.5, 50 deg 40, 52.5 deg 37.5, 55 deg CoFe(20 nm) (VNA) (c)CoFe(10 nm)/Pt CoFe(10 nm) 90 deg (EA) for CoFe FIG. 3. (a) 0
H1=2as a function of Hat!=2= 20 GHz for the CoFe(10 nm) series in Fig. 2(a). Top: Addtional linewidth due to spin pumping of Pt. The green region de- notes the additional linewidth as 4 :50:7 mT. (b-c) 0
H1=2 as a function of frequency for (b) CoFe(10 nm)/Pt and (c) CoFe(20 nm) samples. Solid lines and curves are the ts to the data. damping shifted from CoFe(10 nm) devices by a constant and is much easier to measure. In Fig. 3(b-c) we show the frequency dependence of 0
H1=2of CoFe(10 nm)/Pt devices from spin-torque FMR and CoFe(20 nm) devices from VNA FMR. For both the easy and hard axes, linear relations are ob- tained, and the Gilbert damping can be extracted from0
H1=2=0
H0+ 2!= with the ts shown as solid lines. Here 0
H0is the inhomogeneous broad- ening due to the disorders in lattice structures. In Fig. 3(b) we also show the linewidths of the CoFe(10 nm) device along the easy axis ( H= 90), which has a signicant lower linewidth slope than the easy axis of CoFe(10 nm)/Pt. Their dierences yield a spin pump- ing damping contribution of
sp= 0:0024. By using
sp= hg"#=(4MstM), we obtain a spin mixing con- ductance of g"#(CoFe/Pt) = 25 nm2, which is compa- rable to similar interfaces such as NiFe/Pt [43, 44]. For Hbetween the easy and hard axes, the low-frequency linewidth broadenings are caused by the deviation of magnetization from the biasing eld direction, whereas at high frequencies the eld is sucient to saturate the magnetization. In order to nd the damping anisotropy, we t the linewidths to the angular model developed bySuhl [45, 46], using a single t parameter of and the extractedH2jjandH4jjfrom Fig. 2. The solid tting curves in Fig. 3(b) nicely reproduce the experimental points. The obtained damping anisotropy for all the samples are summarized in Fig. 4, which is the main result of the paper. For CoFe(10 nm)/Pt samples, varies from 0.0056 along the easy axis to 0.0146 along the hard axis. By subtracting the spin pumping
spfrom both values, we derive a damping anisotropy of 380%. For CoFe(20 nm) samples measured by VNA FMR, varies from 0.0054 to 0.0240, which yields an anisotropy of 440% and reproduces the large anisotropy from spin-torque FMR. This giant damping anisotropy implies, technologically, nearly four times smaller critical current to switch the magnetization in a spin-torque magnetic random access memory, or to excite auto-oscillation in a spin-torque os- cillator, by simply changing the magnetization orienta- tion from the hard axis to the easy axis within the same device. In addition, we emphasize that our reported damping anisotropy is not subject to a dominant two- magnon scattering contribution, which would be mani- fested as a nonlinear linewidth softening at high frequen- cies [28, 31]. For this purpose we have extended the fre- quency of spin-torque FMR on CoFe(10 nm)/Pt up to 39 GHz, see the Supplemental Materials for details [36]. We choose CoFe(10 nm)/Pt samples because they provide the best signals at high frequencies and the additional Pt layer signicantly helps to excite the dynamics. Linear frequency dependence of linewidth persists throughout the frequency range and
H0is unchanged for the two axes, with which we can exclude extrinsic eects to the linewidths. We also note that our result is substantially dierent from the recent report on damping anisotropy in Fe/GaAs [30], which is due to the interfacial SOC and disappears quickly as Fe becomes thicker. In compari- son, the Gilbert damping anisotropy in Co 50Fe50is the intrinsic property of the material, is bonded to its bulk crystalline structure, and thus holds for dierent thick- nesses in our experiments. In order to investigate the dominant mechanism for such a large Gilbert damping anisotropy, we perform temperature-dependent measurements of and the re- sistivity. Fig. 5(a) plots as a function of 1 =for the CoFe(10 nm)/Pt and CoFe(20 nm) samples and for HBalong the easy and hard axes. The dominant lin- ear dependence reveals a major role of conductivitylike damping behavior. This is described by the breathing Fermi surface model for transition-metal ferromagnets, in whichcan be expressed as [23, 24, 47{49]: N(EF)jj2 (1) whereN(EF) is the density of state at the Fermi level,  is the electron relaxation time and =h[;Hso]iE=EF is the matrix for spin- ip scatterings induced by the SOC Hamiltonian Hsonear the Fermi surface [48, 49]. Here 4 (b) CoFe(10 nm) CoFe(20 nm) CoFe(20 nm) CoFe(10 nm)/Pt - ∆α sp CoFe(10 nm ) 400 % 100 % FIG. 4. Renormalized damping and its anisotropy for CoFe(10 nm) and CoFe(20 nm), measured from spin-torque FMR and VNA FMR, respectively. For CoFe(20 nm)/Pt sam- ples,
sphas been subtracted from the measured damping. is proportional to the conductivity (1 =) from the Drude model, with which Eq. (1) gives rise to the behaviors shown in Fig. 5(a). For the origin of damping anisotropy, we rst check the role of N(EF) by ab-initio calculations for dierent ordered cubic supercells, which is shown in the Supple- mental Materials [36]. However, a negligible anisotropy inN(EF) is found for dierent magnetization orienta- tions. This is consistent with the calculated anisotropy in Ref. [30], where less than 0.4% change of N(EF) was obtained in ultrathin Fe lms. The role of can also be excluded from the fact that the resistivity dierence be- tween the easy and hard axes is less than 2% [36]. Thus we deduce that the giant damping anisotropy of 400% is due to the change of jj2, or the SOC, at dierent crys- talline directions. In particular, unlike the single element Fe, disordered bcc Fe-Co alloy can possess atomic short- range order, which gives rise to local tetragonal crystal distortions due to the dierent lattice constants of Fe and Co [2{4]. Such local tetragonal distortions will preserve global cubic symmetry but can have large eects on the SOC. We emphasize that our CoFe samples, which did not experience annealing, preserve the random disorder. Our rst principle calculations also conrm the role of lo- cal tetragonal distortions and its enhancement on SOC, see the Supplemental Materials for details [36]. The anisotropy of the SOC in Co 50Fe50can be re ected by its AMR variation along dierent crystalline orienta- tions. The AMR ratio can be dened as: AMR(I) =k(I) ?(I)1 (2) wherek(I) and?(I) are measured for the biasing eld parallel and perpendicular to the current direction, respectively. The main contribution of AMR is the asym- metrics-delectron scatterings where the s-orbitals are mixed with magnetization-containing d-orbitals due toSOC [53, 54]. Since both the damping and AMR origi- nate from SOC and, more precisely, are proportional to the second order of SOC, a large damping anisotropy is expected to be accompanied by a large AMR anisotropy and vice versa. Furthermore, due to the fourfold sym- metry, the AMR should be invariant when the current direction is rotated by 90 degrees, as the AMR is a func- tion ofIas dened in Eq. (1). Thus the damping and AMR should exhibit similar angular dependence on H andI, respectively. In Fig. 5(b) we compare renormalized (H) with CoFe(20 nm) CoFe(10 nm)/Pt : (a) 300 K 8 K F(θI)/F max (b) ,10 nm 20 nm 10 nm 20 nm FIG. 5. (a) (T) as a function of 1 =(T).T= 8 K, 30 K, 70 K, 150 K and 300 K for CoFe(10 nm)/Pt and T= 8 K and 300 K for CoFe(20 nm). Dashed and dotted lines are guides to eyes. (b) Renormalized (H) and AMR( I) andF(I) for CoFe(10 nm)/Pt and CoFe(20 nm). Circles, crosses and + denote, AMR and F, respectively. AMR(I) for 10-nm and 20-nm CoFe samples, where the AMR values are measured from Hall bars with dierent I. The AMR ratio is maximized along h100iaxes and minimized alongh110iaxes, with a large anisotropy by a factor of 10. This anisotropy is also shown by the inte- grated spin-torque FMR intensity for CoFe(10 nm)/Pt, dened asF(I) =
H1=2Vmax dc [17, 18] and plotted in Fig. 5(b). The large AMR anisotropy and its symme- try clearly coincide with the damping anisotropy mea- sured in the same samples, which conrms our hypoth- esis of strong SOC anisotropy in CoFe. Thus we con- clude that the damping anisotropy is dominated by the variation of SOC term in Eq. (1). In parallel, we also compare(H) and AMR( I) for epitaxial Fe(10 nm) samples grown on GaAs substrates [36]. Experimentally we nd the anisotropy less is than 30% for both damping and AMR, which helps to explain the presence of weak damping anisotropy in epitaxial Fe [30].5 We compare our results with prior theoretical works on damping anisotropy [23, 24]. First, despite their propor- tional relationship in Fig. 5(a), the giant anisotropy in is not re ected in 1 =. This is because the s-dscatter- ing, which dominates in the anisotropic AMR, only con- tributes a small portion to the total resistivity. Second, neither the anisotropy of damping nor AMR are sensitive to temperature. This is likely because the thermal excita- tions at room temperature ( 0:025 eV) are much smaller than the spin-orbit coupling ( 0:1 eV [47]). Third, the damping tensor has been expressed as a function of M anddM=dt[24]. However in a fourfold-symmetry lat- tice and considering the large precession ellipticity, these two vectors are mostly perpendicular to each other, point towards equivalent crystalline directions, and contribute equivalently to the symmetry of damping anisotropy. In summary, we have experimentally demonstrated very large Gilbert damping anisotropy up to 400% in epitaxial Co 50Fe50thin lms which is due to their bulk, cubic crystalline anisotropy. We show that the damping anisotropy can be explained by the change of spin-orbit coupling within the breathing Fermi surface model, which can be probed by the corresponding AMR change. Our results provide new insights to the damping mechanism in metallic ferromagnets, which are important for opti- mizing dynamic properties of future magnetic devices. We are grateful for fruitful discussions with Bret Hein- rich. W.Z. acknowledges supports from the U.S. Na- tional Science Foundation under Grants DMR-1808892, Michigan Space Grant Consortium and DOE Visit- ing Faculty Program. Work at Argonne, including transport measurements and theoretical modeling, was supported by the U.S. Department of Energy, Of- ce of Science, Materials Science and Engineering Di- vision. Work at Fudan, including thin lm growth and fabrication, was supported by the Nat'l Key Ba- sic Research Program (2015CB921401), Nat'l Key Re- search and Development Program (2016YFA0300703), NSFC (11734006,11474066,11434003), and the Program of Shanghai Academic Research Leader (17XD1400400) of China. 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B 10, 4626 (1974).7 Supplemental Materials:Giant anisotropy of Gilbert damping in epi- taxial CoFe lms byYi Li, Fanlong Zeng, Steven S.-L. Zhang, Hyeondeok Shin, Hilal Saglam, Vedat Karakas, Ozhan Ozatay, John E. Pearson, Olle G. Heinonen, Yizheng Wu, Axel Homann and Wei Zhang Crystallographic quality of Co 50Fe50lms FIG. S-1. Crystallographic characterization results of CoFe lms. (a) RHEED pattern of the CoFe(10 nm) lm. (b) XRD of the CoFe(10 nm) and (20 nm) lms. (c) X-ray re ectometry measured for the CoFe(20 nm) lm. (d) AFM scans of the CoFe(20 nm) lm. (e) Rocking curves of the CoFe(20 nm) lm for [100] and [110] rotating axes. Fig. S-1 shows the crystallographic characterization for the epitaxial CoFe samples. Re ection high-energy electron diraction (RHEED) shows very clear and sharp patterns which shows high quality of the epitaxal lms. X-ray diraction (XRD) yields clear CoFe(002) peaks at 2 = 66:5. X-ray re ectometry scan of the CoFe (20 nm) lm shows a good periodic pattern and the t gives a total thickness of 19.84 nm. Atomic-force microscopy (AFM) scans for 10m10m and 100 nm100 nm scales show smooth surface with a roughness of 0.1 nm. Lastly XRD rocking curves for [100] and [110] rotating axes show a consistent linewidth of 1.45, which indicates isotropic mosaicity of the CoFe lms. As a result of the crystallographic characterizations, we believe our MBE-grown CoFe samples are epitaxial, have smooth surfaces and exhibit excellent crystalline quality. Moreover, we can exclude the source of inhomogeneity from misorientation of crystallities (mosaicity) due to isotropic rocking curves. This means the inhomogeneous FMR linewidth broadening is isotropic, as is consistent with the experiments. Device geometries for Spin-torque FMR and VNA FMR measurements. Fig. S-2 shows the device geometry for Spin-torque FMR and VNA FMR measurements. For spin-torque FMR, we have prepared CoFe(10 nm)/Pt, CoFe(10 nm) and Fe(10 nm) devices. A second CoFe(10 nm)/Pt sample is also prepared for rotating-eld measurements. For VNA FMR, we have prepared CoFe(20 nm) samples. All the CoFe lms are grown on MgO(100) substrates; the Fe lm is grown on a GaAs(100) substrate. Au (100 nm) coplanar waveguides are subsequently fabricated on top of all devices. For VNA FMR samples, an additional SiO 2(100 nm) is8 FIG. S-2. (a) Spin-torque FMR devices of CoFe(10 nm)/Pt samples. (b) Illustration of the Spin-torque FMR circuit. (c) Front and (d) back view of the VNA FMR devices for CoFe(20 nm) samples. deposited between CoFe and Au for electric isolation. The CoFe(20 nm) bars is only visible from the back view in Fig. S-2(d). Spin-torque FMR lineshapes Figure S-3 shows the full lineshapes of (a) CoFe(10 nm)/Pt(6 nm), (b) CoFe(10 nm) and (c) Fe(10 nm) devices measured at !=2= 20 GHz. The Fe lms were deposited on GaAs substrates by MBE growth. (a) and (b) are used to extract the resonance elds and linewidths in Figs. 2(a) and 3(a) of the main text. (c) is used to examine the correlation between damping anisotropy and AMR anisotropy. Spin-torque FMR linewidths as a function of frequency for CoFe(10 nm) devices. Figure S-4(a) shows the spin-torque FMR linewidths for CoFe(10 nm) devices. Because there is no spin torque injection from Pt layer, the FMR signals are much weaker than CoFe(10 nm)/Pt and the extracted linewidths are more noisy. The excitation of the dynamics is due to the magnon charge pumping eect [1] or inhomogeneities of the Oersted elds. No signal is measured for the rf current owing along the easy axis (magnetic eld along the hard axis, see Fig. S-3b), because of the negligible AMR ratio. Figure S-4(b) shows the angular dependence of the extracted Gilbert damping for CoFe(10 nm)/Pt and CoFe(10 nm). The former is extracted from Fig. 3(b) of the main text. The latter is extracted from Fig. S-4(a). The blue data points for CoFe(10 nm)/Pt are obtained from the resonances at negative biasing elds. Those data are used in Fig. 4 of the main text.9 FIG. S-3. Spin-torque FMR lineshapes of (a) CoFe(10 nm)/Pt, (b) CoFe(10 nm) and (c) Fe(10 nm) devices measured at !=2= 20 GHz. HIis xed to 45. FIG. S-4. (a) 0
H1=2as a function of frequency for CoFe(10 nm) devices. Solid lines and curves are the ts to the experiments. HIis xed to 45. (b)as a function of Hfor CoFe(10 nm)/Pt and CoFe(10 nm) devices. Spin-torque FMR for CoFe(10 nm)/Pt up to 39 GHz. Fig. S-5 shows the spin-torque FMR lineshapes and linewidths up to 39 GHz for CoFe(10 nm)/Pt devices along the easy and hard axes ( H= 90and45). At!=2= 32:1 GHz (Fig. S-5a), the spin-torque FMR amplitude is 0.1V for the easy axis and 0.02 V for the hard axis. 10 seconds of time constant is used to obtained the signals. Throughout the frequency range, linewidths demonstrate good linear dependence on frequency as shown Fig. S-5(b). For the hard axis the signal has reached the noise bottom limit at 32.1 GHz. For the easy axis the noise bottom limit is reached at 39 GHz. The two linear ts yield = 0:0063 and0
H0= 1:8 mT for the easy axis and = 0:00153 and0
H0= 1:5 mT for the hard axis. The two damping parameters are close to the values obtained below 20 GHz in the main text. Also the inhomogeneous linewidth 0
H0nicely match between easy and hard axes.10 FIG. S-5. High-frequency ST-FMR measurement of i) CoFe(10 nm)/Pt for the biasing eld along the easy axis ( H= 90) and hard axis (H= 45). Left: lineshapes of ST-FMR at !=2= 32:1 GHz. Right: linewidth as a function of frequency. Lines are linear ts to the data by setting both and
H0as free parameters. Low-temperature FMR linewidths and dampings for CoFe(10 nm)/Pt and CoFe(20 nm). FIG. S-6. (a-b) 0
H1=2as a function of frequency for CoFe(10 nm)/Pt devices at dierent temperatures. (c) Extracted damping at dierent temperatures, same as in Fig. 4 of the main text. Figure S-6 shows the frequency dependence of linewidths for extracting temperature-dependent Gilbert damping in Fig. 5(a) of the main text. For CoFe(10 nm)/Pt samples, we plot both and resistivity measured at dierent temperatures in Fig. S-6(c). The measurements of were conducted with a biasing magnetic eld of 1 Tesla parallel to the current direction, so that the AMR in uence is excluded. Also the resistivity variation between the easy and hard axes is very small, about 1%, which is much smaller than the damping anisotropy. We have also conducted the low-temperature VNA FMR of the new CoFe(20 nm) samples at 8 K, in addition to the room-temperature measurements. The linewidths data are shown in Fig. S-6(d) for both easy and hard axes. The extracted damping are: = 0:0054 (EA, 300 K), 0.0061 (EA, 8 K), 0.0240 (HA, 300 K) and 0.0329 (HA, 8 K). Those values are used in Fig. 4(b) and Fig. 5(a) of the main text. For CoFe(10 nm) the damping anisotropy decreases from 380 % at 300 K to 273 % at 30 K by taking out the spin pumping damping enhancement (an unexpected reduction of alpha happens at 8 K for the hard axis). For CoFe(2011 nm) the damping anisotropy increases from 440 % at 300 K to 540 % at 8 K. Thus a clear variation trend of damping anisotropy in CoFe lms remains to be explored. First-principle calculation of N(EF)anisotropy for Co 50Fe50 FIG. S-7. Density of states as a function of energy. EFis the Fermi level. First-principle calculations were done using QUANTUM ESPRESSO for a cubic lattice of Co 50Fe50of CsCl, Zintl and random alloy structures. Supercells consisting of 4 44 unit cells were considered with a total of 128 atoms (64 cobalt and 64 iron atoms). The calculations were done using plane-wave basis set with a 180 Ry kinetic energy cut-o and 1440 Ry density cut-o. For both Co and Fe atoms, fully relativistic PAW pseudopotentials were used. Figure S-7 shows the density of states (DOS) of the CsCl form for dierent magnetization orientations in thexy-plane. Clearly, DOS exhibits no anisotropy ( <0:1% variation at E=EF). No anisotropy was found in the Zintl form, either. Thus, we conclude that the Gilbert damping anisotropy in Co 50Fe50cannot be caused by a variation of N(EF) with respect to magnetization direction in ideal ordered structures. SOC induced by atomic short-range order (ASRO) In our experiment, because the Co 50Fe50lms were grown by MBE at low temperatures, they do not form the ordered bcc B2 structure but instead exhibit compositional disorder. Transition metal alloys such as CoPt, NiFe, and CoFe tend to exhibit ASRO [2{4]. The ASRO in CoFe is likely to give rise to local tetragonal distortions because of the dierent lattice constants of bcc Fe and (metastable) bcc Co at 2.856 A and 2.82 A, respectively. Such local tetragonal distortions will preserve global cubic (or four-fold in-plane) symmetry, but can have large eects on the SOC, with concomitant eect on spin-orbit induced magnetization damping. For example, rst-principle calculations using the coherent-potential approximation for the substitutionally disordered system shows that a tetragonal distortion of 10% in the ratio of the tetragonal axes aandcgives rise to an magnetocrystalline anisotropy energy (MAE) density [2, 3] of about 1 MJ/m3. These results are consistent with our observed MAE in Co 50Fe50. To conrm this mechanism, we performed DFT-LDA calculations on 50:50 CoFe supercells consisting of a total of 16 atoms for CsCl, zintl, and random alloy structures; in the random alloy supercell, Co or Fe atoms randomly occupied the atomic positions in the supercell. Note that all CoFe geometries are fully relaxed, including supercell lattice vectors. 1. Structural relaxation including spin-orbit coupling (SOC) shows local tetragonal distortions for random alloy supercell. Among the three dierent CoFe phases, tetragonal c/a ratio for the supercell in optimized geometry is largest (1.003) in the random alloy supercell with SOC, which means local tetragonal distortions are more12 FIG. S-8. Density of states (DOS) for (a) CsCl, (b) Zintl, and (c) alloy form of CoFe with SOC (black solid) and without SOC (red solid). TABLE I. Relaxed atomic positions (including SOC) of the alloy structure. In the ideal CsCl or Zintl structures, the atomic positions are all multiples of 0.25 in units of the lattice vector components. Atom x-position y-position z-position Co 0.003783083 0.000000000 0.000000000 Fe -0.001339230 0.000000000 0.500000000 Fe -0.002327721 0.500000000 0.000000000 Fe 0.002079922 0.500000000 0.500000000 Fe 0.502327721 0.000000000 0.000000000 Fe 0.497920078 0.000000000 0.500000000 Co 0.496216917 0.500000000 0.000000000 Fe 0.501339230 0.500000000 0.500000000 Co 0.250000000 0.250000000 0.254117992 Fe 0.250000000 0.250000000 0.752628048 Fe 0.250000000 0.750000000 0.247371952 Co 0.250000000 0.750000000 0.745882008 Co 0.750000000 0.250000000 0.250415490 Co 0.750000000 0.250000000 0.746688258 Co 0.750000000 0.750000000 0.253311742 Co 0.750000000 0.750000000 0.749584510 dominant in random alloy compared to CsCl and Zintl structures. [c/a values : CsCl (0.999), Zintl (0.999), Alloy (1.003)]. In addition, the alloy system exhibited local distortions of Co and Fe position relative to their ideal positions. In contrast, in CsCl and Zintl structures the Co and Fe atoms exhibited almost imperceptible distortions. Table shows the relaxed atomic positions in the alloys structure in units of the lattice vectors. In the ideal (unrelaxed) system, the positions are all at multiples of 0.25; the relaxed CsCl and Zintl structures no deviations from these positions larger than 1 part in 106 2. SOC changes the density of states (DOS) at the Fermi energy, notably for the random alloy but notfor the CsCl and Zintl structures. Figure S-8 shows DOS for (a) CsCl, (b) Zintl, and (c) random alloy structure with SOC (black lines) and without it red lines). We can see signicant DOS dierence for the random alloy supercell with SOC where tetragonal distortions occurred, while almost no changes are observed in the CsCl and Zintl structures. 3. The local distortions in the alloy structure furthermore gave rise to an energy anisotropy with respect to the magnetization direction. The energy (including SOC) of the relaxed alloy structure for dierent directions of the magnetization is shown in Fig. S-9. While the supercell was rather small, because of the computational expense in relaxing the structure with SOC, so that no self-averaging can be inferred, the gure demonstrates an induced magnetic anisotropy that arises from the SOC and local distortions. No magnetic anisotropy was discernible in the CsCl and Zintl structures. As a result from the DFT calculation, we attribute the large SOC eect in damping anisotropy of Co 50Fe50to local tetragonal distortions in disordered Co and Fe alloys. These distortions give rise to SOC-induced changes of DOS at the Fermi level, as well as magnetic anisotropy energy with respect to the crystallographic axes.13 FIG. S-9. Change in total energy (per supercell) of the alloy structure as function of the magnetization direction. wuyizheng@fudan.edu.cn yhomann@anl.gov zweizhang@oakland.edu [1] C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas and A. Ferguson, Nature Nano. 10, 50 (2015) [2] S. Razee, J. Staunton, B. Ginatempo, E. Bruno, and F. Pinski, Phys. Rev. B 64, 014411 (2001). [3] Y. Kota and A. Sakuma, Appl. Phys. Express 5, 113002 (2012). [4] I. Turek, J. Kudrnovsk y, and K. Carva, Phys. Rev. B 86, 174430 (2012).

1909.08004v1.Microwave_induced_tunable_subharmonic_steps_in_superconductor_ferromagnet_superconductor_Josephson_junction.pdf

arXiv:1909.08004v1 [cond-mat.supr-con] 17 Sep 2019Microwave induced tunable subharmonic steps in superconductor-ferromagnet-superconductor Josephson j unction M. Nashaat,1,2,∗Yu. M. Shukrinov,2,3,†A. Irie,4A.Y. Ellithi,1and Th. M. El Sherbini1 1Department of Physics, Cairo University, Cairo, 12613, Egy pt 2BLTP, JINR, Dubna, Moscow Region, 141980, Russian Federati on 3Dubna State University, Dubna, 141982, Russian Federation 4Department of Electrical and Electronic Systems Engineeri ng, Utsunomiya University, Utsunomiya, Japan. We investigate the coupling between ferromagnet and superc onducting phase dynamics in superconductor-ferromagnet-superconductor Josephson j unction. The current-voltage character- istics of the junction demonstrate a pattern of subharmonic current steps which forms a devil’s staircase structure. We show that a width of the steps become s maximal at ferromagnetic reso- nance. Moreover, we demonstrate that the structure of the st eps and their widths can be tuned by changing the frequency of the external magnetic field, rat io of Josephson to magnetic energy, Gilbert damping and the junction size. This paper is submitted to LTP Journal. I. INTRODUCTION Josephson junction with ferromagnet layer (F) is widely considered to be the place where spintronics and superconductivity fields interact1. In these junctions the supercurrent induces magnetization dynamics due to the coupling between the Josephson and magnetic subsystems. The possibility of achieving electric con- trol over the magnetic properties of the magnet via Josephson current and its counterpart, i.e., achieving magnetic control over Josephson current, recently at- tracted a lot of attention1–7. The current-phase rela- tion in the superconductor-ferromagnet-superconductor junction (SFS) junctions is very sensitive to the mutual orientation of the magnetizations in the F-layer8,9. In Ref.[10] the authors demonstrate a unique magnetization dynamics with a series of specific phase trajectories. The origin of these trajectories is related to a direct coupling between the magnetic moment and the Josephson oscil- lations in these junctions. External electromagnetic field can also provide a cou- pling between spin wave and Josephson phase in SFS junctions11–17. Spin waves are elementary spin excita- tions which considered to be as both spatial and time dependent variations in the magnetization18,19. The fer- romagnetic resonance(FMR) correspondsto the uniform precession of the magnetization around an external ap- plied magnetic field18. This mode can be resonantly ex- cited by ac magnetic field that couples directly to the magnetization dynamics as described by the Landau- Lifshitz-Gilbert (LLG) equation18,19. In Ref.[18] the authors show that spin wave resonance at frequency ωrin SFS implies a dissipation that is mani- fested as adepressionin the IV-characteristicofthe junc- tion when /planckover2pi1ωr= 2eV, where/planckover2pi1is the Planck’s constant, e is the electron charge and Vis the voltage across the junction. The ac Josephson current produces an oscil- lating magnetic field and when the Josephson frequencymatches the spin wave frequency, this resonantly excites the magnetization dynamics M(t)18. Due to the non- linearity of the Josephson effect, there is a rectification of current across the junction, resulting in a dip in the average dc component of the suppercurrent18. In Ref.[13] the authors neglect the effective field due to Josephson energy in LLG equation and the results re- veal that even steps appear in the IV-characteristic of SFS junction under external magnetic field. The ori- gin of these steps is due to the interaction of Cooper pairs with even number of magnons. Inside the ferro- magnet, if the Cooper pairs scattered by odd number of magnons, no Josephson current flows due to the forma- tion of spin triplet state13. However, if the Cooper pairs interact with even number of magnons, the Josephson coupling between the s-wave superconductor is achieved and the spin singlet state is formed, resulting in flows of Josephsoncurrent13. In Ref.[20]weshowthat takinginto account the effective field due to Josepshon energy and at FMR, additional subharmonic current steps appear in the IV-characteristic for overdamped SFS junction with spin wave excitations (magnons). It is found that the po- sition of the current steps in the IV-characteristics form devil’s staircase structure which follows continued frac- tion formula20. The positions of those fractional steps are given by V= N±1 n±1 m±1 p±.. Ω, (1) where Ω = ω/ωc,ωis the frequency of the external ra- diation, ωcis the is the characteristic frequency of the Josephson junction and N,n,m,pare positive integers. In this paper, we present a detailed analysis for the IV-characteristics of SFS junction under external mag- netic field, and show how we can control the position of the subharmonic steps and alter their widths. The coupling between spin wave and Josephson phase in SFS junction is achieved through the Josephson energy and gauge invariant phase difference between the S-layers. In the framework of our approach, the dynamics of the SFS2 junction isfully describedbytheresistivelyshuntedjunc- tion (RSJ) model and LLG equation. These equations are solved numerically by the 4thorder Runge-Kutta method. The appearance and position of the observed current steps depend directly on the magnetic field and junction parameters. II. MODEL AND METHODS F ss Hacxyz H0I I FIG. 1. SFS Josephson junction. The bias current is applied in x-direction, an external magnetic field with amplitude Hac and frequency ωis applied in xy-plane and an uniaxial con- stant magnetic field H0is applied in z-direction. In Fig 1 we consider a current biased SFS junction where the two superconductors are separated by ferro- magnet layer with thickness d. The area of the junction isLyLz. An uniaxial constant magnetic field H0is ap- plied in z-direction, while the magnetic field is applied in xy-plane Hac= (Haccosωt,Hacsinωt,0)withamplitude Hacand frequency ω. The magnetic field is induced in the F-layer through B(t) = 4πM(t), and the magnetic fluxes in z- and y-direction are Φ z(t) = 4πdLyMz(t), Φy(t) = 4πdLzMy(t), respectively. The gauge-invariant phase difference in the junction is given by21: ∇y,zθ(y,z,t) =−2πd Φ0B(t)×n, (2) whereθis the phase difference between superconducting electrodes, and Φ 0=h/2eis the magnetic flux quantum andnis a unit vector normal to yz-plane. The gauge- invariantphasedifference in terms ofmagnetizationcom- ponents reads as θ(y,z,t) =θ(t)−8π2dMz(t) Φ0y+8π2dMy(t) Φ0z,(3) where Φ 0=h/(2e) is the magnetic flux quantum. AccordingtoRSJ model, the currentthroughthe junc- tion is given by13: I I0c= sinθ(y,z,t)+Φ0 2πI0cRdθ(y,z,t) dt,(4) whereI0 cis the critical current, and R is the resistance in the Josephson junction. After taking into account thegaugeinvarianceincludingthemagnetizationoftheferro- magnetandintegratingoverthejunction areatheelectric current reads13: I I0c=Φ2 osin(θ(t))sin/parenleftBig 4π2dMz(t)Ly Φo/parenrightBig sin/parenleftBig 4π2dMy(t)Lz Φo/parenrightBig 16π4d2LzLyMz(t)My(t) +Φ0 2πRI0cdθ(y,z,t) dt. (5) The applied magnetic field in the xy-plane causes pre- cessionalmotionofthemagnetizationinthe F-layer. The dynamics of magnetization Min the F-layer is described by LLG equation (1+α2)dM dt=−γM×Heff−γ α |M|[M×(M×Heff)](6) The total energy of junction in the proposed model is givenby E=Es+EM+EacwhereEsistheenergystored in Josephson junction, EMis the energy of uniaxial dc magnetic field (Zeeman energy) and Eacis the energy of ac magnetic field: Es=−Φ0 2πθ(y,z,t)I+EJ[1−cos(y,z,t)], EM=−VFH0Mz(t), Eac=−VFMx(t)Haccos(ωt)−VFMy(t)Hacsin(ωt)(7) Here,EJ= Φ0I0 c/2πis the the Josephson energy, H0= ω0/γ,ω0is the FMR frequency, and VFis the volume of the ferromagnet. We neglect the anisotropy energy due to demagnetizing effect for simplicity. The effective field in LLG equation is calculated by Heff=−1 VF∇ME (8) Thus, the effective field Hmdue to microwave radiation Hacand uniaxial magnetic field H0is given by Hm=Haccos(ωt)ˆex+Hacsin(ωt)ˆey+H0ˆez.(9) while the effective field ( Hs) due to superconducting part is found from Hs=−EJ VFsin(θ(y,z,t))∇Mθ(y,z,t).(10) One should take the integration of LLG on coordinates, however, the superconducting part is the only part which depends on the coordinate so, we can integrate the ef- fective field due to the Josephson energy and insert the result into LLG equation. Then, the y- and z-component are given by Hsy=EJcos(θ(t))sin(πΦz(t)/Φ0) VFπMy(t)Φz(t)/bracketleftbigg Φ0cos(πΦy(t)/Φ0) −Φ2 0sin(πΦy(t)/Φ0) πΦy(t)/bracketrightbigg ˆey, (11) Hsz=EJcos(θ(t))sin(πΦy(t)/Φ0) VFπMz(t)Φy(t)/bracketleftbigg Φ0cos(πΦz(t)/Φ0) −Φ2 0sin(πΦz(t)/Φ0) πΦz(t)/bracketrightbigg ˆez. (12)3 As a result, the total effective field is Heff=Hm+ Hs. In the dimensionless form we use t→tωc,ωc= 2πI0 cR/Φ0is the characteristic frequency, m=M/M0, M0=∝ba∇dblM∝ba∇dbl,heff=Heff/H0,ǫJ=EJ/VFM0H0,hac= Hac/H0, Ω =ω/ωc, Ω0=ω0/ωc,φsy=4π2LydM0/Φo, φsz=4π2lzdM0/Φo. Finally, the voltage V(t) =dθ/dtis normalized to /planckover2pi1ωc/(2e). The LLG and the effective field equations take the form dm dt=−Ω0 (1+α2)/parenleftbigg m×heff+α[m×(m×heff)]/parenrightbigg (13) with heff=haccos(Ωt)ˆex+(hacsin(Ωt)+ΓijǫJcosθ)ˆey + (1+Γ jiǫJcosθ)ˆez, (14) Γij=sin(φsimj) mi(φsimj)/bracketleftbigg cos(φsjmi)−sin(φsjmi) (φsjmi)/bracketrightbigg ,(15) wherei=y,j=z. The RSJ in the dimensionless form is given by I/I0 c=sin(φsymz)sin(φszmy) (φsymz)(φszmy)sinθ+dθ dt.(16) The magnetization and phase dynamics of the SFS junction can be described by solving Eq.(16) together with Eq.(13). To solve this system of equations, we em- ploy the fourth-order Runge-Kutta scheme. At each cur- rent step, we find the temporal dependence of the volt- ageV(t), phase θ(t), andmi(i=x,y,z) in the (0 ,Tmax) interval. Then the time-average voltage Vis given by V=1 Tf−Ti/integraltext V(t)dt, whereTiandTfdetermine the in- terval for the temporal averaging. The current value is increased or decreased by a small amount of δI (the bias current step) to calculate the voltage at the next point of the IV-characteristics. The phase, voltage and mag- netization components achieved at the previous current step are used as the initial conditions for the next cur- rent step. The one-loop IV-characteristic is obtained by sweeping the bias current from I= 0 toI= 3 and back down to I= 0. The initial conditions for the magnetiza- tion components are assumed to be mx= 0,my= 0.01 andmz=/radicalBig 1−m2x−m2y, while for the voltage and phase we have Vini= 0,θini=0. The numerical param- eters (if not mentioned) are taken as α= 0.1,hac= 1, φsy=φsz= 4,ǫJ= 0.2 and Ω 0= 0.5. III. RESULTS AND DISCUSSIONS Itiswell-knownthatJosephsonoscillationscanbesyn- chronized by external microwave radiation which leads to Shapiro steps in the IV-characteristic22. The position of the Shapiro step is determined by relation V=n mΩ, wheren,mare integers. The steps at m= 1 are calledharmonics, otherwise we deal with synchronized subhar- monic (fractional) steps. We show below the appearance of subharmonics in our case. First we present the simulated IV-characteristics at different frequencies of the magnetic field. The IV- characteristics at three different values of Ω are shown in Fig 2(a). FIG. 2. (a) IV-characteristic at three different values of Ω. For clarity, the IV-characteristics for Ω = 0 .5 and Ω = 0 .7 have been shifted to the right, by ∆ I= 0.5 and ∆ I= 1, respectively with respect to Ω = 0 .2; (b) An enlarged part of the IV-characteristic with Ω = 0 .7. To get step voltage multiply the corresponding fraction with Ω = 0 .7. As we see, the second harmonic has the largest step width at the ferromagnetic resonance frequency Ω = Ω 0, i.e., the FMR is manifested itself by the step’s width. There are also many subharmonic current steps in the IV-characteristic. We have analyzed the steps position between V= 0 and V= 0.7 for Ω = 0 .7 and found dif- ferent level continued fractions, which follow the formula given by Eq.(1) and demonstrated in Fig.2(b). We see4 the reflection of the second level continued fractions 1 /n and 1−1/nwithN= 1. In addition to this, steps with third level continued fractions 1 /(n−1/m) withN= 1 is manifested. In the inset we demonstrate part of the fourth level continued fraction 1 −1/(n+ 1/(m+1/p)) withn= 2 and m= 2. In case of external electromagnetic field which leads to the additional electric current Iac=AsinΩt, the width of the Shapiro step is proportional to ∝Jn(A/Ω), where Jnis the Bessel function of first kind. The preliminary results (not presented here) show that the width of the Shapiro-like steps under external magnetic field has a more complex frequency dependence20. This question will be discussed in detail somewhere else. The coupling between Josephson phase and magneti- zation manifests itself in the appearance of the Shapiro steps in the IV-characteristics at fractional and odd mul- tiplies of Ω20. In Fig.3 we demonstrate the effect of the ratio of the Josephson to magnetic energy ǫJon appear- ance of the steps and their width for Ω = 0 .5 where the enlarged parts of the IV-characteristics at three differ- ent values of ǫJare shown. As it is demonstrated in the figures, at ǫJ= 0.05 only two subharmonic steps appear between V= 1 and V= 1.5 (see hollow ar- rows). An enhanced staircase structure appears by in- creasing the value of ǫJ, which can be see at ǫJ= 0.3 and 0.5. Moreover, an intense subharmonic steps appear between V= 1.75 andV= 2 forǫJ= 0.5. The posi- tions for these steps reflect third level continued fraction (N−1)+1/(n+1/m)withN=4 andn=1 [see Fig.3(b)]. Let us now demonstrate the effect of Gilbert damping on the devil’s staircase structure. The Gilbert damping αis introduced into LLG equation23?to describe the relaxation of magnetization dynamics. To reflect effect of Gilbert damping, we show an enlarged part of the IV- characteristic at three different values of αin Fig.4. Thewidthofcurrentstepat V= 2Ωisalmostthesame at different values of α(e.g., see upward inset V= 2Ω). The subharmonic current step width for V= (n/m)Ω (n is odd,mis integer) is decreasing with increasing α. In addition a horizontal shift for the current steps occurs. We see the intense current steps in the IV-characteristic for small value of α= 0.03 (see black solid arrows). With increase in Gilbert damping (see α= 0.1, 0.16 and 0 .3) the higher level subharmonic steps disappear. It is well- knownthatatlargevalueof αtheFMRlinewidthbecome more broadening and the resonance frequency is shifted from Ω 0. Accordingly, the subharmonic steps disappear at large value of α. Furthermore, using the formula pre- sented in Ref.[20] the width at Ω = Ω 0for the fractional and odd current steps is proportional to (4 α2+α4)−q/2 ×(12+3α2)−k/2, whereqandkare integers. Finally, we demonstrate the effect of the junction size on the devil’s staircase in the IV-characteristic under ex- ternalmagneticfield. Thejunction sizechangesthe value ofφsyandφsz. In Fig.5(a) we demonstrate the effect of the junction thickness by changing φsz(φsyis qualita- FIG. 3. (a) An enlarged part of the IV-characteristic at different values of ǫJin the interval between V= 1 and V= 1.5; (b)Thesameintheintervalbetween V= 1.75andV= 2. For clarity, the IV-characteristics for ǫJ= 0.3, and 0 .5 have been shifted to right, by ∆ I= 0.07, and 0 .14, respectively with respect to the case with ǫJ= 0.05. tively the same). We observe an enhanced subharmonic structure with increase of junction size or the thickness of the ferro- magnet. In Ref.[13] the authors demonstrated that the critical current and the width of the step at V= 2Ω as a function of Lz/Lyfollow Bessel function of first kind. In Fig.5(b), we can see the parts of continued fraction se- quences for subharmonic steps between V= 1 andV= 2 atφsz=φsy= 6. Current steps between V= 1 and V= 1.5 reflect the two second level continued fractions (N−1)+ 1/nandN−1/nwithN= 3 in both cases, while for the steps between V= 1.5 andV= 2 follow the second level continued fraction ( N−1) + 1/nwith N= 4. Finally, wediscussthepossibilityofexperimentallyob-5 FIG. 4. An enlarged part of IV-characteristic for four differ - ent values of Gilbert damping for Ω = 0 .5. The inset shows an enlargedpartofcurrentstepwithconstantvoltage at V= 2Ω. serving the effects presented in this paper. For junction sized= 5nm, Ly=Lz= 80nm, critical current I0 c≈ 200µA, saturation magnetization M0≈5×105A/m, H0≈40mT and gyromagnetic ratio γ= 3πMHz/T, we find the value of φsy(z)=4π2Ly(z)dM0/Φ0= 4.8 and ǫJ= 0.1. With the same junction parameters one can control the appearance of the subharmonic steps by tun- ing the strength of the constant magnetic field H0. Esti- mations showthat for H0= 10mT, the value of ǫJ= 0.4, and the fractional subharmonic steps are enhanced. In general, the subharmonic steps are sensitive to junction parameters, Gilbert damping and the frequency of the external magnetic field. IV. CONCLUSIONS In this work, we have studied the IV-characteristics of superconductor-ferromagnet-superconductor Joseph- son junction under external magnetic field. We used a modified RSJ model which hosts magnetization dynam- ics in F-layer. Due to the external magnetic field, the couplingbetweenmagneticmomentandJosephsonphase is achieved through the effective field taking into account the Josephson energy and gauge invariant phase differ- ence between the superconducting electrodes. We have solvedasystemofequationswhichdescribethe dynamics of the Josephson phase by the RSJ equation and magne- tization dynamics by Landau-Lifshitz-Gilbert equation. The IV-characteristic demonstrates subharmonic current steps. The pattern of the subharmonic steps can be con- trolled by tuning the frequency of the ac magnetic field. We show that by increasing the ratio of the Josephson to magneticenergyanenhancedstaircasestructureappears. Finally, we demonstrate that Gilbert damping and junc- FIG. 5. (a) IV-characteristic at three different values of φsz= 0.7,3,6 andφsy=φsz. (b) An enlarged part of the IV- characteristic at φsz=φsy=6. The hollow arrows represent the starting point of the sequences. To get step voltage we multiply the corresponding fraction by Ω = 0 .5. tion parameters can change the subharmonic step struc- ture. The observed features might find an application in superconducting spintronics. V. ACKNOWLEDGMENT We thank Dr. D. V. Kamanin and Egypt JINR col- laboration for support this work. The reported study was partially funded by the RFBR research Projects No. 18-02-00318 and No. 18-52-45011-IND. Numerical cal- culations have been made in the framework of the RSF Project No. 18-71-10095.6 REFERENCES ∗majed@sci.cu.edu.eg †shukrinv@theor.jinr.ru 1J. Linder and K. Halterman, Phys. Rev. B 90, 104502 (2014). 2Yu. M. Shukrinov, A. Mazanik, I. Rahmonov, A. Botha, and A. Buzdin, EPL122, 37001 (2018). 3Yu. M. Shukrinov, I. Rahmonov, K. Sengupta, and A. Buzdin, Appl. Phys. Lett. 110, 182407 (2017). 4A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008). 5A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). 6F. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. 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1001.4576v1.Effect_of_spin_conserving_scattering_on_Gilbert_damping_in_ferromagnetic_semiconductors.pdf

arXiv:1001.4576v1 [cond-mat.mtrl-sci] 26 Jan 2010Effect of spin-conserving scattering on Gilbert damping in f erromagnetic semiconductors K. Shen,1G. Tatara,2and M. W. Wu1,∗ 1Hefei National Laboratory for Physical Sciences at Microsc ale and Department of Physics, University of Science and Technology of China, Hefei, Anhui , 230026, China 2Department of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan (Dated: November 12, 2018) The Gilbert damping in ferromagnetic semiconductors is the oretically investigated based on the s-dmodel. In contrast to the situation in metals, all the spin-c onserving scattering in ferromagnetic semiconductors supplies an additional spin relaxation cha nnel due to the momentum dependent effective magnetic field of the spin-orbit coupling, thereby modifies the Gilbert damping. In the presence of a pure spin current, we predict a new contributio n due to the interplay of the anisotropic spin-orbit coupling and a pure spin current. PACS numbers: 72.25.Dc, 75.60.Ch, 72.25.Rb, 71.10.-w The ferromagnetic systems have attracted much at- tention both for the abundant fundamental physics and promising applications in the past decade.1,2The study on the collective magnetization dynamics in such sys- tems has been an active field with the aim to control the magnetization. In the literature, the magnetization dynamics is usually described by the phenomenological Landau-Lifshitz-Gilbert (LLG) equation,3 ˙n=γHeff×n+αn×˙n, (1) withndenoting the direction of the magnetization. The first and second terms on the right hand side of the equa- tion represent the precession and relaxation of the mag- netization under the effective magnetic field Heff, respec- tively. The relaxation term is conventionally named as the Gilbert damping term with the damping coefficient α. The time scale of the magnetization relaxation then can be estimated by 1 /(αγHeff),4which is an important parameter for dynamic manipulations. The coefficient α is essential in determining the efficiency of the current- induced magnetizationswiching, andexperimentaldeter- mination of αhas been carried out intensively in metals5 and magnetic semiconductors.6 To date, many efforts have been made to clarify the microscopic origin of the Gilbert damping.7–12Kohno et al.8employed the standard diagrammatic pertur- bation approach to calculate the spin torque in the small-amplitude magnetization dynamics and obtained a Gilberttorquewiththedampingcoefficientinverselypro- portional to the electron spin lifetime. They showed that the electron-non-magnetic impurity scattering, a spin- conserving process, does not affect the Gilbert damping. Later, they extended the theory into the finite-amplitude dynamics by introducing an SU(2) gauge field2and ob- tained a Gilbert torque identical to that in the case of small-amplitude dynamics.9In those calculations, the electron-phonon and electron-electron scatterings were discarded. One may infer that both of them should be irrelevant to the Gilbert damping in ferromagnetic met- als, since they are independent of the electron spin re-laxation somewhat like the electron-non-magnetic impu- rity scattering. However, the situation is quite different in ferromagnetic semiconductors, where the spin-orbit coupling (SOC) due to the bulk inversion asymmetry13 and/or the structure inversion asymmetry14presents a momentum-dependent effective magnetic field (inhomo- geneous broadening15). As a result, any spin-conserving scattering, including the electron-electron Coulomb scat- tering,canresultinaspinrelaxationchanneltoaffectthe Gilbert damping. In this case, many-body effects on the Gilbert damping due to the electron-electron Coulomb scatteringshould be expected. Sinova et al.16studied the Gilbert damping in GaMnAs ferromagnetic semiconduc- tors by including the SOC to the energy band structure. In that work, the dynamics of the carrier spin coherence was missed.17The issue of the present work is to study the Gilbert damping in a coherent frame. In this Report, we apply the gauge field approach to investigate the Gilbert damping in ferromagnetic semi- conductors. In our frame, all the relevant scattering pro- cesses, even the electron-electron scattering which gives rise to many-body effects, can be included. The goal of this work is to illustrate the role of the SOC and spin-conserving scattering on Gilbert damping. We show that the spin-conserving scattering can affect the Gilbert damping due to the contribution on spin relaxation pro- cess. We also discuss the case with a pure spin current, from which we predict a new Gilbert torque due to the interplay of the SOC and the spin current. Our calculation is based on the s-dmodel with itiner- antsand localized delectrons. The collectivemagnetiza- tion arisingfrom the delectronsis denoted by M=Msn. The exchange interaction between itinerant and local- ized electrons can be written as Hsd=M/integraltext dr(n·σ), where the Pauli matrices σare spin operators of the itinerant electrons and Mis the coupling constant. In order to treat the magnetization dynamics with an ar- bitrary amplitude,9we define the temporal spinor oper- ators of the itinerant electrons a(t) = (a↑(t),a↓(t))Tin the rotation coordinate system with ↑(↓) labeling the2 spin orientation parallel (antiparallel) to n. With a uni- tary transformation matrix U(t), one can connect the operators a↑(↓)with those defined in the lattice coor- dinate system c↑(↓)bya(t) =U(t)c. Then, an SU(2) gauge field Aµ(t) =−iU(t)†(∂µU(t)) =Aµ(t)·σshould be introduced into the rotation framework to guarantee the invariance of the total Lagrangian.9In the slow and smooth precession limit, the gauge field can be treated perturbatively.9Besides, one needs a time-dependent 3×3 orthogonal rotation matrix R(t), which obeys U†σU=Rσ, to transform any vector between the two coordinate systems. More details can be found in Ref. 2. In the following, we restrict our derivation to a spa- tially homogeneous system, to obtain the Gilbert damp- ing torque. Up to the first order, the interaction Hamiltonian due to the gauge field is HA=/summationtext kA0·a† kσakand the spin- orbit couping reads Hso=1 2/summationdisplay khk·c†σc=1 2/summationdisplay k˜hk·a† kσak,(2) with˜h=Rh. Here, we take the Planck constant /planckover2pi1= 1. We start from the fully microscopic kinetic spin Bloch equations of the itinerant electrons derived from the non- equilibrium Green’s function approach,15,18 ∂tρk=∂tρk/vextendsingle/vextendsingle coh+∂tρk/vextendsingle/vextendsinglec scat+∂tρk/vextendsingle/vextendsinglef scat,(3)whereρkrepresenttheitinerantelectrondensitymatrices defined in the rotation coordinate system. The coherent term can be written as ∂tρk/vextendsingle/vextendsingle coh=−i[A·σ,ρk]−i[1 2˜hk·σ+ˆΣHF,ρk].(4) Here [,] is the commutator and A(t) =A0(t)+Mˆzwith A0andMˆzrepresenting the gauge field and effective magnetic filed due to s-dexchange interaction, respec- tively.ˆΣHFis the Coulomb Hartree-Fock term of the electron-electron interaction. ∂tρk/vextendsingle/vextendsinglec scatand∂tρk/vextendsingle/vextendsinglef scatin Eq.(3) include all the relevant spin-conserving and spin- flip scattering processes, respectively. The spin-flip term ∂tρk/vextendsingle/vextendsinglef scatresults in the damping ef- fect was studied in Ref. 9. Let us confirm this by considering the case of the magnetic disorder Vm imp= us/summationtext j˜Sj·a†σaδ(r−Rj). The spin-flip part then reads ∂tρk/vextendsingle/vextendsinglef scat=∂tρk/vextendsingle/vextendsinglef(0) scat+∂tρk/vextendsingle/vextendsinglef(1) scat, (5) with∂tρk/vextendsingle/vextendsinglef(i) scatstanding for the i-th order term with re- spect to the gauge field, i.e., ∂tρk/vextendsingle/vextendsinglef(0) scat=−πnsu2 sS2 imp 3/summationdisplay k1η1η2σαρ> k1(t)Tη1σαTη2ρ< k(t)δ(ǫk1η1−ǫkη2)−(>↔<)+H.c., (6) ∂tρk/vextendsingle/vextendsinglef(1) scat=i2πnsu2 sS2 imp 3εαβγAγ 0(t)/summationdisplay k1η1η2σαρ> k1(t)Tη1σβTη2ρ< k(t)d dǫk1η1δ(ǫk1η1−ǫkη2)−(>↔<)+H.c.,(7) whereTη(i,j) =δηiδηjfor the spin band η. Here ρ> k= 1−ρk,ρ< k=ρk. (>↔<) is obtained by inter- changing >and<from the first term in each equation. εijkis the Levi-Civita permutation symbol. The gauge field term, ∂tρk/vextendsingle/vextendsinglef(1) scat, results from the spin correlation of a single magnetic impurity at different times.9It induces a spin polarization proportional to ˆz×A⊥ 0(t) which gives a Gilbert torque. The damping coefficient is inversely proportional to the spin relaxation time τsdetemined by the spin-flip scattering ∂tρk/vextendsingle/vextendsinglef(0) scat. The spin-flip scattering term in Eq.(3) thus reproduces the result of Ref. 9. We now demonstrate that the Gilbert damping torque arises also from the spin-conserving scattering. For the discussion of the spin-conserving term, it is suffi- cient to approximate the spin-flip term as ∂tρk/vextendsingle/vextendsinglef scat= −(ρk−ρe k)/τs, withρe krepresenting the instantaneous equilibrium distribution (i.e., ρe kisρkwithout the gaugefield and Ps k). Equation(3) then reads ∂tρk=−i[A·σ,ρk]−i[1 2˜hk·σ,ρk] +∂tρk/vextendsingle/vextendsinglec scat−(ρk−ρe k)/τs+Ps k.(8) Here, we add an additional term, Ps k, to describe the source of a pure spin current due to the magnetization dynamic pumping4or electrically injection19,20in order to discuss the system with a pure spin current. We ne- glect the Coulomb Hartree-Fock effective magnetic field sinceitisapproximatelyparalleltothe s-dexchangefield, but with a smaller magnitude. By averaging density matrices over the momentum direction, one obtains the isotropic component ρi,k=/integraltextdΩk 4πρk. The anisotropic component is then expressed asρa,k=ρk−ρi,k. It is obvious that this anisotropic component does not give any spin torque in the absence of the SOC, since/summationtext kTr(σρa,k) = 0. Below, it is shown3 that this component leads to the damping when coupled to the spin-orbit interaction. By denoting the isotropic component of the equi- librium part ( ρe k) asρe i,kand representing the non- equilibrium isotropic part by δρi,k=ρi,k−ρe i,k, we write the kinetic spin Bloch equations of the non- equilibrium isotropic density matrices δρi,kand those of the anisotropic components ρa,kas ∂tρi,k=−δρi,k τs−i[A·σ,δρi,k]−i[1 2˜hk·σ,ρa,k] −i[A0·σ,ρe i,k], (9) ∂tρa,k=∂tρa,k/vextendsingle/vextendsinglec scat−i[A·σ,ρa,k]−i[1 2˜hk·σ,δρi,k] −i[1 2˜hk·σ,ρa,k]+i[1 2˜hk·σ,ρa,k]+Ps k,(10) respectively. The overline in these equations presents a angular average over the momentum space. We further define ρ(0) a,kas the anisotropic density in the absence of the gauge field, A0. As easily seen, it vanishes whenPs k= 0. The anisotropic component involving the gauge field is denoted by ρ(1) a,k=ρa,k−ρ(0) a,k. Equation (10) is expressed in terms of these components as ∂tρ(0) a,k=−i[M·σ,ρ(0) a,k]+∂tρ(0) a,k/vextendsingle/vextendsinglec scat+Ps k −i[1 2˜hk·σ,ρ(0) a,k]+i[1 2˜hk·σ,ρ(0) a,k],(11) ∂tρ(1) a,k=∂tρ(1) a,k/vextendsingle/vextendsinglec scat−i[A·σ,ρ(1) a,k]−i[1 2˜hk·σ,δρi,k] −i[1 2˜hk·σ,ρ(1) a,k]−i[A0·σ,ρ(0) a,k].(12) Within the elastic scattering approximation, the electron-phonon scattering as well as the electron-non- magnetic impurity scattering can be simply written as/summationtext l,mρ(1) a,k,lmYlm/τl, where the density matrices are ex- panded by the spherical harmonics functions Ylm, i.e., ρ(1) a,k,lm=/integraltextdΩk 4πρ(1) a,kYlm.τlis the effective momentum relaxation time. The exact calculation of the Coulomb scattering is more complicated. Nevertheless, one can still express this term in the form of ρ(1) a,k/Fk(ρ), where Fkis a function of the density matrices21and reflects many-body effects. For simplification, we just introduce a uniform momentum relaxation time τ∗ lin the following calculation. Expanding Eq.(12) by the spherical har- monics functions, one obtains ∂tρ(1) a,k,lm=−i[A·σ,ρ(1) a,k,lm]−i[1 2˜hk,lm·σ,δρi,k] −i[A0·σ,ρ(0) a,k,lm]−i[1 2˜hk·σ,ρ(1) a,k]lm−ρ(1) a,k,lm τ∗ l,(13) where the expanssion coefficient of any term fkis ex- pressed as fk,lm=/integraltextdΩk 4πfkYlm. In the strong scattering regime, i.e.,1 τ∗ l≫Mand1 τ∗ l≫ |hk|, the first and fourth terms are much smaller than the last term, hence can be discarded from the right side. By taking the fact that the time derivative is a higher order term into account, one also neglects ∂tρ(1) a,k,lm. The solution of Eq.(13) canbe written as ρ(1) a,k,lm=−iτ∗ l{[1 2˜hk,lm·σ,δρi,k]+[A0·σ,ρ(0) a,k,lm]}.(14) Substituting it into Eq.(14) and rewriting the equation in the leading order, one obtains ∂tρi,k=−i[A·σ,δρi,k]−i 2[˜hk·σ,ρ(0) a,k]−i[A0·σ,ρe i,k] −/summationdisplay lmτ∗ l 4/bracketleftBig ˜hk,lm·σ,[˜hk,lm·σ,δρi,k]/bracketrightBig −δρi,k τs.(15) The third term on the right hand side of the equation is proportional to the second order term of the SOC, which gives the spin dephasing channel due to the D’yakonov- Perel’ (DP) mechanism.22This term can be expressed byτ−1 DPδρi,kwithτ−1 DPstanding for the spin dephas- ing rate tensor, which can be written as ( τ−1 DP)i,j=/summationtext l,m/an}b∇acketle{tτ∗ l((hk,lm)2δij−hi k,lmhj k,lm)/an}b∇acket∇i}htby performingthe en- semble averaging over the electron distribution. In the following, we treat τDPas a scalar for simplification and the total spin lifetime is hence given by τr= 1/(τ−1 DP+τ−1 s). (16) Similar to the previous procedure, we discard ∂tρi,kin Eq. (15) and obtain i[A·σ,δρi,k]+δρi,k/τr=−i[1 2˜hk·σ,ρ(0) a,k]−i[A0·σ,ρe i,k]. (17) By taking δ˜si=1 2/summationtext kTr(σδρi,k),˜se i=1 2/summationtext kTr(σρe i,k) and˜s(0) a,k=1 2Tr(σρ(0) a,k), one can write the solution as δ˜si=˜v+2τrAטv+4τ2 r(˜v·A)A 1+4|A|2τ2r−˜se i,(18) where˜v=˜se i+τr/summationtext k˜hkטs(0) a,k.˜se iisjust the equilibrium spin density , which is parallel to the magnetization, i.e., ˜se i= ˜se iˆz. Now, we pick up the transverse component in theformof ˆz×A⊥ 0,δ˜s⊥, sinceonlythiscomponentresults in a Gilbert torque of the magnetization as mentioned above. We come to δ˜s⊥= 2˜vz(A⊥ 0׈z)τ2 exτr/(τ2 r+τ2 ex),(19) withτex= 1/(2M). By transforming it back to the lat- tice coordinate system with R(ˆz×A⊥ 0) =1 2∂tn,9one obtains δs⊥=−˜vz(∂tn)τ2 exτr/(τ2 r+τ2 ex),(20) This nonequilibrium spin polarization results in a spin torqueperformed on the magnetizationaccordingto T= −2Mn×δs, i.e., T= ˜vz(n×∂tn)τexτr/(τ2 r+τ2 ex).(21) Compared with Eq.(1), the modification of the Gilbert damping coefficient from this torque is α= ˜vzτexτr/(Msτ2 r+Msτ2 ex), (22)4 We fisrt discuss the case without the source term of the spin current. In this case, the anisotropic component ρ(0) a,kvanishesand ˜ vz= ˜se i. We seethat the Gilbert damp- ing then arises from 1 /τr[Eq. (16)], i.e., from both the spin-flip scattering and the DP mechanism.22Our main message is that this DP contribution is affected by the spin-conservingscattering processes such as the electron- electron interaction and phonons. The temperature de- pendenceoftheGilbertdampingandthecurrent-induced magnetization switching can thus be discussed quantita- tively by evaluating τr. We note that our result reduces to the results of previous works7,9when only the spin-flip scattering is considered. We should point out that our formalism applies also to metals, by considering the case1 τ∗ l≪M. In this case, the last term of Eq.(13) can be neglected and the effect of the spin-conserving scattering through τ∗ lbecomes ir- relevant. When the pure spin current is included, we found ad- ditional contribution due to the interplay of the spin cur- rent and the SOC, since we have ˜vz= ˜se i+/bracketleftBig R/parenleftBig τr/summationdisplay khk×s(0) a,k/parenrightBig/bracketrightBig z= ˜se i+ ˜ssc z,(23) with the spin current associated term ˜ ssc zdefined ac- cordingly. The origin of ˜ ssc zcan be understood as fol- lows. The anisotropic spin polarization s(0) a,karising from the pure spin current rotates around the SOC effective magnetic field hk, which is also anisotropic. This pre-cession finally results in an isotropic spin polarization ssc=τr/summationtext khk×s(0) a,kin the presence of spin relaxation. This term contributes to the spin polarization of the itin- erant electrons along the direction of the magnetization, i.e., ˜ssc z, thereby modifies the Gilbert damping term by ˜ssc z/˜se i. The additional Gilbert damping due to the spin cur- rent found here is different from the enhancement of the damping in the spin pumping systems, where the exis- tence of the interface is essential.4In other words, what contributes there is the divergence of the spin current, as is understood from the continuity equation for the spin, indicating that the spin damping is equal to ∇·js+ ˙s(s is the total spin density). In contrast, the damping found in the present paper arises even when the spin current is uniform if the spin-orbit interaction is there. In summary, we have shown that the spin-conserving scatterings in ferromagnetic semiconductors, such as the electron-electron, electron-phonon and electron-non- magnetic impurity scatterings, contribute to the Gilbert damping in the presence of the SOC because of the in- homogeneous broadening effect. We also predict that a Gilbert torque arises from a pure spin current when cou- pled to the spin-orbit interaction. This work wassupported by the Natural Science Foun- dation of China under Grant No. 10725417, the Na- tional Basic Research Program of China under Grant No. 2006CB922005,the KnowledgeInnovation Projectof Chinese Academy of Sciences, Kakenhi (1948027)MEXT Japan and the Hitachi Sci. Tech. Foundation. ∗Author to whom correspondence should be addressed; Electronic address: mwwu@ustc.edu.cn. 1Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 2G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213 (2008). 3T. L. Gilbert, Phys. Rev. 100, 1243 (1955); L. D. Landau, E.M.Lifshitz, andL.P.Pitaevski, Statistical Physics, Part 2, 3rd ed. (Pergamon, Oxford, 1980). 4Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 5M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45, 3889 (2006). 6J.-P. Adam, N. Vernier, J. Ferr´ e, A. Thiaville, V. Jeudy, A. Lemaˆ ıtre, L. Thevenard, and G. Faini, Phys. Rev. B 80, 193204 (2009). 7P. Pi´ echon and A. Thiaville, Phys. Rev. B 75, 174414 (2007). 8H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006). 9H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007). 10Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 74, 144405 (2006). 11J. Kuneˇ s and V. Kambersk´ y, Phys. Rev. B 65, 212411(2002). 12D. Steiauf and M. F¨ ahnle, Phys. Rev. B 72, 064450 (2005). 13G. Dresselhaus, Phys. Rev. 100, 580 (1955). 14Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984); JETP Lett. 39, 78 (1984). 15M. W. Wu and C. Z. Ning, Eur. Phys. J. B 18, 373 (2000); M. W. Wu, J. Phys. Soc. Jpn. 70, 2195 (2001). 16J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. K. Furdyna, W. A. Atkinson, and A. H. MacDonald, Phys. Rev. B 69, 085209 (2004). 17Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phys. Lett. 84, 5234 (2004) 18M. W. Wu and H. Metiu, Phys. Rev. B 61, 2945 (2000); For review: M. W. Wu, J. H. Jiang, and M. Q. Weng, arXiv:1001.0606; and references therein. 19F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel- mans, and B. J. van Wees, Nature 416, 713 (2002). 20X. H. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J. J. Zhang, K. S. M. Reddy, S. D. Flexner, C. J. Palmstrom, and P. A. Crowell, Nature Phys. 3, 197 (2007). 21M. M. Glazov and E. L. Ivchenko, JETP 99, 1279 (2004). 22M. I. D’yakonov and V. I. Perel’, Zh. ´Eksp. Teor. Fiz. 60, 1954(1971) [Sov.Phys.JETP 33, 1053(1971)]; Fiz.Tverd. Tela (Leningrad) 13, 3581 (1971) [Sov. Phys. Solid State 13, 3023 (1972)].

0807.5009v1.Scattering_Theory_of_Gilbert_Damping.pdf

arXiv:0807.5009v1 [cond-mat.mes-hall] 31 Jul 2008Scattering Theory of Gilbert Damping Arne Brataas,1,∗Yaroslav Tserkovnyak,2and Gerrit E. W. Bauer3 1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA 3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands The magnetization dynamics of a single domain ferromagnet i n contact with a thermal bath is studied by scattering theory. We recover the Landau-Lift shitz-Gilbert equation and express the effective fields and Gilbert damping tensor in terms of the sca ttering matrix. Dissipation of magnetic energy equals energy current pumped out of the system by the t ime-dependent magnetization, with separable spin-relaxation induced bulk and spin-pumping g enerated interface contributions. In linear response, our scattering theory for the Gilbert damp ing tensor is equivalent with the Kubo formalism. Magnetization relaxation is a collective many-body phenomenon that remains intriguing despite decades of theoretical and experimental investigations. It is im- portant in topics of current interest since it determines the magnetization dynamics and noise in magnetic mem- ory devices and state-of-the-art magnetoelectronic ex- periments on current-induced magnetization dynamics [1]. Magnetization relaxation is often described in terms of a damping torque in the phenomenological Landau- Lifshitz-Gilbert (LLG) equation 1 γdM dτ=−M×Heff+M×/bracketleftBigg˜G(M) γ2M2sdM dτ/bracketrightBigg , (1) whereMis the magnetization vector, γ=gµB//planckover2pi1is the gyromagnetic ratio in terms of the gfactor and the Bohr magnetonµB, andMs=|M|is the saturation magneti- zation. Usually, the Gilbert damping ˜G(M) is assumed to be a scalar and isotropic parameter, but in general it is a symmetric 3 ×3 tensor. The LLG equation has been derived microscopically [2] and successfully describes the measured response of ferromagnetic bulk materials and thin films in terms of a few material-specific parameters thatareaccessibletoferromagnetic-resonance(FMR) ex- periments [3]. We focus in the following on small fer- romagnets in which the spatial degrees of freedom are frozen out (macrospin model). Gilbert damping pre- dicts a striclylinear dependence ofFMR linewidts on fre- quency. This distinguishes it from inhomogenous broad- ening associated with dephasing of the global precession, which typically induces a weaker frequency dependence as well as a zero-frequency contribution. The effective magnetic field Heff=−∂F/∂Mis the derivative of the free energy Fof the magnetic system in an external magnetic field Hext, including the classi- cal magnetic dipolar field Hd. When the ferromagnet is part of an open system as in Fig. 1, −∂F/∂Mcan be expressed in terms of a scattering S-matrix, quite anal- ogous to the interlayer exchange coupling between ferro- magnetic layers [4]. The scattering matrix is defined in the space of the transport channels that connect a scat- tering region (the sample) to thermodynamic (left andleft reservoirF N Nright reservoir FIG. 1: Schematic picture of a ferromagnet (F) in contact with a thermal bath via metallic normal metal leads (N). right) reservoirs by electric contacts that are modeled by ideal leads. Scattering matrices also contain information to describe giant magnetoresistance, spin pumping and spin battery, and current-induced magnetization dynam- ics in layered normal-metal (N) |ferromagnet (F) systems [4, 5, 6]. In the following we demonstrate that scattering the- ory can be also used to compute the Gilbert damping tensor˜G(M).The energy loss rate of the scattering re- gion can be described in terms of the time-dependent S-matrix. Here, we generalize the theory of adiabatic quantum pumping to describe dissipation in a metallic ferromagnet. Our idea is to evaluate the energy pump- ingoutoftheferromagnetandtorelatethistotheenergy loss of the LLG equation. We find that the Gilbert phe- nomenology is valid beyond the linear response regime of small magnetization amplitudes. The only approxima- tion that is necessary to derive Eq. (1) including ˜G(M) is the (adiabatic) assumption that the frequency ωof the magnetization dynamics is slow compared to the relevant internal energy scales set by the exchange splitting ∆. The LLG phenomenology works so well because /planckover2pi1ω≪∆ safely holds for most ferromagnets. Gilbert damping in transition-metal ferromagnets is generally believed to stem from spin-orbit interaction in combinationwith impurityscatteringthattransfersmag- netic energy to itinerant quasiparticles [3]. The subse- quent drainage of the energy out of the electronic sys- tem,e.g.by inelastic scattering via phonons, is believed to be a fast process that does not limit the overall damp- ing. Our key assumption is adiabaticiy, meaning that the precession frequency goes to zero before letting the sample size become large. The magnetization dynam- ics then heats up the entire magnetic system by a tiny2 amount that escapes via the contacts. The leakage heat current then equals the total dissipation rate. For suf- ficiently large samples, bulk heat production is insensi- tive to the contact details and can be identified as an additive contribution to the total heat current that es- capes via the contacts. The chemical potential is set by the reservoirs, which means that (in the absence of an intentional bias) the sample is then always very close to equilibrium. The S-matrix expanded to linear order in the magnetization dynamics and the Kubo linear re- sponse formalisms should give identical results, which we will explicitly demonstrate. The role of the infinitesi- mal inelastic scattering that guarantees causality in the Kubo approach is in the scattering approach taken over by the coupling to the reservoirs. Since the electron- phonon relaxation is not expected to directly impede the overall rate of magnetic energy dissipation, we do not need to explicitly include it in our treatment. The en- ergy flow supported by the leads, thus, appears in our model to be carried entirely by electrons irrespective of whethertheenergyisactuallycarriedbyphonons, incase the electrons relax by inelastic scattering before reaching the leads. So we are able to compute the magnetization damping, but not, e.g., how the sample heats up by it . According to Eq. (1), the time derivative of the energy reads ˙E=Heff·dM/dτ= (1/γ2)˙ m/bracketleftBig ˜G(m)˙ m/bracketrightBig ,(2) in terms of the magnetization direction unit vector m= M/Msand˙ m=dm/dτ. We now develop the scatter- ing theory for a ferromagnet connected to two reservoirs by normal metal leads as shown in Fig. 1. The total energy pumping into both leads I(pump) Eat low tempera- tures reads [11, 12] I(pump) E= (/planckover2pi1/4π)Tr˙S˙S†, (3) where˙S=dS/dτandSis the S-matrix at the Fermi energy: S(m) =/parenleftbiggr t′ t r′/parenrightbigg . (4) randt(r′andt′) are the reflection and transmissionma- trices spanned by the transport channels and spin states for an incoming wave from the left (right). The gener- alization to finite temperatures is possible but requires knowledge of the energy dependence of the S-matrix around the Fermi energy [12]. The S-matrix changes parametrically with the time-dependent variation of the magnetization S(τ) =S(m(τ)). We obtain the Gilbert damping tensor in terms of the S-matrix by equating the energy pumping by the magnetic system (3) with the en- ergy loss expression (2), ˙E=I(pump) E. Consequently Gij(m) =γ2/planckover2pi1 4πRe/braceleftbigg Tr/bracketleftbigg∂S ∂mi∂S† ∂mj/bracketrightbigg/bracerightbigg ,(5)which is our main result. The remainder of our paper serves three purposes. We show that (i) the S-matrix formalism expanded to linear responseis equivalentto Kubolinearresponseformalism, demonstrate that (ii) energy pumping reduces to inter- face spin pumping in the absence ofspin relaxationin the scattering region, and (iii) use a simple 2-band toy model with spin-flip scattering to explicitly show that we can identify both the disorder and interface (spin-pumping) magnetization damping as additive contributions to the Gilbert damping. Analogous to the Fisher-Lee relation between Kubo conductivity and the Landauer formula [15] we will now prove that the Gilbert damping in terms of S-matrix (5) is consistent with the conventionalderivation of the mag- netization damping by the linear response formalism. To this end we chose a generic mean-field Hamiltonian that depends on the magnetization direction m:ˆH=ˆH(m) describes the system in Fig. 1. ˆHcan describe realistic band structures as computed by density-functional the- ory including exchange-correlation effects and spin-orbit couplingaswell normaland spin-orbitinduced scattering off impurities. The energy dissipation is ˙E=/angb∇acketleftdˆH/dτ/angb∇acket∇ight, where/angb∇acketleft.../angb∇acket∇ightdenotes the expectation value for the non- equilibriumstate. Inlinearresponse,weexpandthemag- netization direction m(t) around the equilibrium magne- tization direction m0, m(τ)=m0+u(τ). (6) The Hamiltonian can be linearized as ˆH=ˆHst+ ui(τ)∂iˆH, where ˆHst≡ˆH(m0) is the static Hamilto- nian and∂iˆH≡∂uiˆH(m0), where summation over re- peated indices i=x,y,zis implied. To lowest order ˙E= ˙ui(τ)/angb∇acketleft∂iˆH/angb∇acket∇ight, where /angb∇acketleft∂iˆH/angb∇acket∇ight=/angb∇acketleft∂iˆH/angb∇acket∇ight0+/integraldisplay∞ −∞dτ′χij(τ−τ′)uj(τ′).(7) /angb∇acketleft.../angb∇acket∇ight0denotes equilibrium expectation value and the re- tarded correlation function is χij(τ−τ′) =−i /planckover2pi1θ(τ−τ′)/angbracketleftBig [∂iˆH(τ),∂jˆH(τ′)]/angbracketrightBig 0(8) in the interaction picture for the time evolution. In order to arrive at the adiabatic (Gilbert) damping the magne- tization dynamics has to be sufficiently slow such that uj(τ)≈uj(t) + (τ−t) ˙uj(t). Since m2= 1 and hence ˙ m·m= 0 [7] ˙E=i∂ωχij(ω→0)˙ui˙uj, (9) whereχij(ω) =/integraltext∞ −∞dτχij(τ)exp(iωτ). Next, we use the scattering states as the basis for expressing the correlation function (8). The Hamiltonian consists of a free-electron part and a scattering potential: ˆH= ˆH0+ˆV(m). We denote the unperturbed eigenstates of3 the free-electron Hamiltonian ˆH0=−/planckover2pi12∇2/2mat en- ergyǫby|ϕs,q(ǫ)/angb∇acket∇ight, wheres=l,rdenotes propagation direction and qtransverse quantum number. The po- tentialˆV(m) scatters the particles between these free- electron states. The outgoing (+) and incoming wave (-) eigenstates |ψ(±) s,q(ǫ)/angb∇acket∇ightof the static Hamiltonian ˆHst fulfill the completeness conditions /angb∇acketleftψ(±) s,q(ǫ)|ψ(±) s′,q′(ǫ′)/angb∇acket∇ight= δs,s′δq,q′δ(ǫ−ǫ′) [10]. These wave functions can be ex- pressed as |ψ(±) s(ǫ)/angb∇acket∇ight= [1 +ˆG(±) stˆVst]|ϕs(ǫ)/angb∇acket∇ight, where the static retarded (+) and advanced (-) Green functions are ˆG(±) st(ǫ) = (ǫ±iη−ˆHst)−1andηis a positive infinites- imal. By expanding χij(ω) in the basis of the outgo- ing wave functions |ψ(+) s/angb∇acket∇ight, the low-temperature linear re- sponse leads to the followingenergydissipation (9) in the adiabatic limit ˙E=−π/planckover2pi1˙ui˙uj/angbracketleftBig ψ(+) s,q|∂iˆH|ψ(+) s′,q′/angbracketrightBig/angbracketleftBig ψ(+) s′,q′|∂jˆH|ψ(+) s,q/angbracketrightBig , (10) with wave functions evaluated at the Fermi energy ǫF. In order to compare the linear response result, Eq. (10), withthat ofthe scatteringtheory, Eq. (5), weintro- duce the T-matrix ˆTasˆS(ǫ;m) = 1−2πiˆT(ǫ;m), where ˆT=ˆV[1 +ˆG(+)ˆT] in terms of the full Green function ˆG(+)(ǫ,m) = [ǫ+iη−ˆH(m)]−1. Although the adiabatic energy pumping (5) is valid for any magnitude of slow magnetization dynamics, in order to make connection to the linear-response formalism we should consider small magnetization changes to the equilibrium values as de- scribed by Eq. (6). We then find ∂τˆT=/bracketleftBig 1+ˆVstˆG(+) st/bracketrightBig ˙ui∂iˆH/bracketleftBig 1+ˆG(+) stˆVst/bracketrightBig .(11) into Eq. (5) and using the completeness of the scattering states, we recover Eq. (10). Our S-matrix approach generalizes the theory of (non- local) spin pumping and enhanced Gilbert damping in thin ferromagnets [5]: by conservation of the total an- gular momentum the spin current pumped into the surrounding conductors implies an additional damping torque that enhances the bulk Gilbert damping. Spin pumping is an N |F interfacial effect that becomes impor- tant in thin ferromagnetic films [14]. In the absence of spin relaxation in the scattering region, the S-matrix can be decomposed as S(m) =S↑(1+ˆσ·m)/2+S↓(1−ˆσ· m)/2, where ˆσis a vector of Pauli matrices. In this case, Tr(∂τS)(∂τS)†=Ar˙ m2, whereAr= Tr[1−ReS↑S† ↓] and the trace is over the orbital degrees of freedom only. We recover the diagonal and isotropic Gilbert damping tensor:Gij=δijGderived earlier [5], where G=γMsα=(gµB)2 4π/planckover2pi1Ar. (12) Finally, we illustrate by a model calculation that we can obtain magnetization damping by both spin- relaxationandinterfacespin-pumpingfromtheS-matrix.We consider a thin film ferromagnet in the two-band Stoner model embedded in a free-electron metal ˆH=−/planckover2pi12 2m∇2+δ(x)ˆV(ρ), (13) where the in-plane coordinate of the ferromagnet is ρ and the normal coordinate is x.The spin-dependent po- tentialˆV(ρ) consists of the mean-field exchange interac- tion oriented along the magnetization direction mand magnetic disorder in the form of magnetic impurities Si ˆV(ρ) =νˆσ·m+/summationdisplay iζiˆσ·Siδ(ρ−ρi),(14) which are randomly oriented and distributed in the film atx= 0. Impurities in combination with spin-orbit cou- pling will give similar contributions as magnetic impuri- ties to Gilbert damping. Our derivation of the S-matrix closely follows Ref. [8]. The 2-component spinor wave function can be written as Ψ( x,ρ) =/summationtext k/bardblck/bardbl(x)Φk/bardbl(ρ), where the transverse wave function is Φ k/bardbl(ρ) = exp(ik/bardbl· ρ)/√ Afor the cross-sectional area A. The effective one- dimensional equation for the longitudinal part of the wave function is then /bracketleftbiggd2 dx2+k2 ⊥/bracketrightbigg ck/bardbl(x) =/summationdisplay k′ /bardbl˜Γk/bardbl,k′ /bardblck/bardbl(0)δ(x),(15) where the matrix elements are defined by ˜Γk/bardbl,k′ /bardbl= (2m//planckover2pi12)/integraltext dρΦ∗ k/bardbl(ρ)ˆV(ρ)Φk′ /bardbl(ρ)and the longitudinal wave vector k⊥is defined by k2 ⊥= 2mǫF//planckover2pi12−k2 /bardbl. For an incoming electron from the left, the longitudinal wave function is ck/bardbls=χs√k⊥/braceleftBigg eik⊥xδk/bardbls,k′ /bardbls′+e−ik⊥xrk/bardbls,k′ /bardbls′,x<0 eik⊥xtk/bardbls,k′ /bardbls′,x>0, (16) wheres=↑,↓andχ↑= (1,0)†andχ↓= (0,1)†. Inver- sion symmetry dictates that t′=tandr=r′. Continu- ity of the wave function requires 1+ r=t. The energy pumping (3) then simplifies to I(pump) E=/planckover2pi1Tr/parenleftbig˙t˙t†/parenrightbig /π. Flux continuity gives t= (1 +iˆΓ)−1, whereˆΓk/bardbls,k′ /bardbls′= χ† sˆΓk/bardbls,k′ /bardbls′χs′(4k⊥k⊥)−1/2. In the absence of spin-flip scattering, the transmis- sion coefficient is diagonal in the transverse momentum: t(0) k/bardbl= [1−iη⊥σ·m]/(1+η2 ⊥), whereη⊥=mν/(/planckover2pi12k⊥). The nonlocal (spin-pumping) Gilbert damping is then isotropic,Gij(m) =δijG′, G′=2ν2/planckover2pi1 π/summationdisplay k/bardblη2 ⊥ (1+η2 ⊥)2. (17) It can be shown that G′is a function of the ratio be- tween the exchange splitting versus the Fermi wave vec- tor,ηF=mν/(/planckover2pi12kF).G′vanishes in the limits ηF≪14 (nonmagnetic systems) and ηF≫1 (strong ferromag- net). We include weak spin-flip scattering by expanding the transmission coefficient tto second order in the spin- orbit interaction, t≈/bracketleftbigg 1+t0iˆΓsf−/parenleftBig t0iˆΓsf/parenrightBig2/bracketrightbigg t0, which inserted into Eq. (5) leads to an in general anisotropic Gilbert damping. Ensemble averaging over all ran- dom spin configurations and positions after considerable but straightforward algebra leads to the isotropic result Gij(m) =δijG G=G(int)+G′(18) whereG′is defined in Eq. (17). The “bulk” contribution to the damping is caused by the spin-relaxation due to the magnetic disorder G(int)=NsS2ζ2ξ, (19) whereNsis the number of magnetic impurities, Sis the impurity spin, ζis the average strength of the magnetic impurity scattering, and ξ=ξ(ηF) is a complicated ex- pression that vanishes when ηFis either very small or very large. Eq. (18) proves that Eq. (5) incorporates the “bulk” contribution to the Gilbert damping, which grows with the number of spin-flip scatterers, in addition to in- terface damping. We could have derived G(int)[Eq. (19)] as well by the Kubo formula for the Gilbert damping. The Gilbert damping has been computed before based on the Kubo formalism based on first-principles elec- tronic band structures [9]. However, the ab initio appeal is somewhat reduced by additional approximations such as the relaxation time approximation and the neglect of disorder vertex corrections. An advantage of the scatter- ingtheoryofGilbertdampingisitssuitabilityformodern ab initio techniques of spin transport that do not suffer from these drawbacks [16]. When extended to include spin-orbit coupling and magnetic disorder the Gilbert damping can be obtained without additional costs ac- cording to Eq. (5). Bulk and interface contributions can be readily separated by inspection of the sample thick- ness dependence of the Gilbert damping. Phononsareimportantforthe understandingofdamp- ing at elevated temperatures, which we do not explic- itly discuss. They can be included by a temperature- dependent relaxation time [9] or, in our case, structural disorder. A microscopic treatment of phonon excitations requires extension of the formalism to inelastic scatter- ing, which is beyond the scope of the present paper. In conclusion, we hope that our alternative formal- ism of Gilbert damping will stimulate ab initio electronic structure calculations as a function of material and dis- order. By comparison with FMR studies on thin ferro- magnetic films this should lead to a better understanding of dissipation in magnetic systems.This work was supported in part by the Re- search Council of Norway, Grants Nos. 158518/143 and 158547/431, and EC Contract IST-033749 “DynaMax.” ∗Electronic address: Arne.Brataas@ntnu.no [1] For a review, see M. D. Stiles and J. Miltat, Top. Appl. Phys.101, 225 (2006) , and references therein. [2] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta- tus Solidi 23, 501 (1967); V. Kambersky, Can. J. Phys. 48, 2906 (1970); V. Korenman, and R. E. Prange, Phys. Rev. B6, 2769 (1972); V. S. Lutovinov and M. Y. Reizer, Zh. Eksp. Teor. Fiz. 77, 707 (1979) [Sov. Phys. JETP 50, 355 1979]; V. L. Safonov and H. N. Bertram, Phys. Rev. B61, R14893 (2000). J. Kunes and V. Kambersky, Phys. Rev. B65, 212411 (2002); V. Kambersky Phys. Rev. B 76, 134416 (2007). [3] J. A. C. Bland and B. Heinrich, Ultrathin Mag- netic Structures III Fundamentals of Nanomagnetism , Springer Verlag, (Heidelberg, 2004). [4] P. Bruno, Phys. Rev. B 52, 411 (1995). [5] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002); Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys.77, 1375 (2005). [6] X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph, Phys. Rev. B 62, 12317 (2000); A. Brataas, Yu. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2006). [7] E. Simanek and B. Heinrich, Phys. Rev. B 67, 144418 (2003). [8] A. Brataas and G. E. W. Bauer, Phys. Rev. B 49, 14684 (1994). [9] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett.99, 027204 (2007). [10] P. A. Mello and N. Kumar, Quantum Transport in Meso- scopic Systems , Oxford University Press (New York, 2005). [11] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Phys. Rev. Lett., 87, 236601 (2001). [12] M. Moskalets and M. B¨ uttiker, Phys. Rev. B 66, 035306 (2002);Phys. Rev. B 66, 205320 (2002). [13] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B 66, 060404(R) (2002); X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas, and Y. Tserkovnyak, Phys. Rev. Lett. 97, 216602 (2006). [14] B. Heinrich, Y.Tserkovnyak, G. Woltersdorf, A.Brataa s, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett 90, 187601 (2003);M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett.97, 216603 (2006); G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H. Back, Phys. Rev. Lett 99, 246603 (2007). [15] D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981). [16] M. Zwierzycki et al., Phys. Stat. Sol. B 245, 623 (2008)

1307.7427v1.Theoretical_Study_of_Spin_Torque_Oscillator_with_Perpendicularly_Magnetized_Free_Layer.pdf

arXiv:1307.7427v1 [cond-mat.mes-hall] 29 Jul 20131 Theoretical Study of Spin-Torque Oscillator with Perpendicularly Magnetized Free Layer Tomohiro Taniguchi, Hiroko Arai, Hitoshi Kubota, and Hiros hi Imamura∗ Spintronics Research Center, AIST, Tsukuba, Ibaraki 305-8 568, Japan Abstract—The magnetization dynamics of spin torque oscilla- tor (STO) consisting of a perpendicularly magnetized free l ayer and an in-plane magnetized pinned layer was studied by solvi ng the Landau-Lifshitz-Gilbert equation. We derived the anal ytical formula of the relation between the current and the oscillat ion frequency of the STO by analyzing the energy balance between the work done by the spin torque and the energy dissipation du e to the damping. We also found that the field-like torque break s the energy balance, and change the oscillation frequency. Index Terms —spintronics, spin torque oscillator, perpendicu- larly magnetized free layer, the LLG equation I. INTRODUCTION SPIN torque oscillator (STO) has attracted much attention due to its potential uses for a microwave generator and a recording head of a high density hard disk drive. The self- oscillation of the STO was first discovered in an in-plane magnetized giant-magnetoresistive (GMR) system [1]. Afte r that, the self-oscillation of the STO has been observed not only in GMR systems [2]-[6] but also in magnetic tunnel junctions (MTJs) [7]-[11]. The different types of STO have been proposedrecently; for example,a point-contactgeome try with a confinedmagneticdomainwall [12]-[14]whichenables usto controlthe frequencyfroma few GHz to a hundredGHz. Recently, Kubota et al.experimentally developed the MgO- based MTJ consisting of a perpendicularly magnetized free layer and an in-plane magnetized pinned layer [15],[16]. Th ey also studied the self-oscillation of this type of MTJ, and observed a large power ( ∼0.5µW) with a narrow linewidth (∼50MHz) [17].These results are great advancesin realizing the STO device.However,the relationbetweenthe currentan d the oscillation frequency still remains unclear. Since a pr ecise control of the oscillation frequency of the STO by the curren t is necessary for the application, it is important to clarify the relation between the current and the oscillation frequency . In this paper, we derived the theoretical formula of the relation between the current and the oscillation frequency of the STO consisting of the perpendicularly magnetized free layer and the in-plane magnetized pinned layer. The derivation is based on the analysis of the energy balance between the work done by the spin torque and the energy dissipation due to the damping. We found that the oscillatio n frequencymonotonicallydecreases with increasing the cur rent by keeping the magnetization in one hemisphere of the free layer. The validity of the analytical solution was confirmed by numerical simulations. We also found that the field-like ∗Corresponding author. Email address: h-imamura@aist.go. jppmelectron (I>0)z xy spin torquedamping damping spin torque Fig. 1. Schematic view of the system. The directions of the sp in torque and the damping during the precession around the z-axis are indicated. torque breaks the energy balance, and change the oscillatio n frequency. The shift direction of the frequency, high or low , is determined by the sign of the field-like torque. This paper is organized as follows. In Sec. II, the current dependence of the oscillation frequency is derived by solvi ng the Landau-Lifshitz-Gilbert (LLG) equation. In Sec. III, t he effect of the field-like torque on the oscillation behaviour is investigated. Section IV is devoted to the conclusions. II. LLG STUDY OF SPIN TORQUE OSCILLATION The system we consider is schematically shown in Fig. 1. We denote the unit vectors pointing in the directions of the magnetization of the free and the pinned layers as m= (sinθcosϕ,sinθsinϕ,cosθ)andp, respectively. The x-axis is parallel to pwhile the z-axis is normal to the film plane. The variable θofmis the tilted angle from the z-axis whileϕis the rotation angle from the x-axis. The current I flows along the z-axis, where the positive current corresponds to the electron flow from the free layer to the pinned layer. We assume that the magnetization dynamics is well de- scribed by the following LLG equation: dm dt=−γm×H−γHsm×(p×m)+αm×dm dt.(1) The gyromagneticration and the Gilbert damping constant ar e denotedas γandα, respectively.The magnetic field is defined byH=−∂E/∂(Mm), where the energy density Eis E=−MHapplcosθ−M(HK−4πM) 2cos2θ.(2) Here,M,Happl, andHKare the saturation magnetization,the applied field along the z-axis, and the crystalline anisotropy field along the z-axis, respectively. Because we are interested in the perpendicularly magnetized system, the crystalline anisotropy field, HK, should be larger than the demagneti- zation field, 4πM. Since the LLG equation conserves the2 I = 1.2 ~ 2.0 (mA) mz mxmy1 -1 0 -1 -1 1 100(a) current (mA)34567(b) frequency (GHz) 1.2 1.4 1.6 1.8 2.0 Fig. 2. (a) The trajectories of the steady state precession o f the magne- tization in the free layer with various currents. (b) The dot s represent the dependence of the oscillation frequency obtained by numeri cally solving the LLG equation. The solid line is obtained by Eqs. (8) and (9). normofthe magnetization,the magnetizationdynamicscan b e described by a trajectory on an unit sphere. The equilibrium states of the free layer correspond to m=±ez. In following, the initial state is taken to be the north pole, i.e., m=ez. It should be noted that a plane normal to the z-axis, in which θ is constant, corresponds to the constant energy surface. The spin torque strength, Hsin Eq. (1), is [18]-[20] Hs=/planckover2pi1ηI 2e(1+λmx)MSd, (3) whereSanddare the cross section area and the thickness of the free layer. Two dimensionless parameters, ηandλ (−1< λ <1), determine the magnitude of the spin polariza- tion and the angle dependence of the spin torque, respective ly. Although the relation among η,λ, and the material parameters depends on the theoretical models [20]-[22], the form of Eq. (3) is applicable to both GMR system and MTJs. In particular, the angle dependence of the spin torque characterized by λis a key to induce the self-oscillation in this system. Figure 2 (a) shows the steady state precession of the mag- netization in the free layer obtained by numerically solvin g Eq. (1). The values of the parameters are M= 1313emu/c.c., HK= 17.9kOe,Happl= 1.0kOe,S=π×50×50nm2, d= 2.0nm,γ= 17.32MHz/Oe, α= 0.005,η= 0.33, andλ= 0.38, respectively [17]. The self-oscillation was observedforthe current I≥1.2mA.Althoughthe spintorque breaks the axial symmetry of the free layer along the z-axis, the magnetization precesses around the z-axis with an almost constanttilted angle. Thetilted angle fromthe z-axisincreases with increasing the current; however, the magnetization st ays in the northsemisphere( θ < π/2). The dotsin Fig. 2 (b) show the dependence of the oscillation frequency on the current. As shown, the oscillation frequencymonotonicallydecreases with increasing the current magnitude. Let us analytically derive the relation between the current and the oscillation frequency. Since the self-oscillation occurs due to the energysupply into the free layer by the spin torque , the energy balance between the spin torque and the damping should be investigated. By using the LLG equation, the time derivative of the energy density Eis given by dE/dt=Ws+ Wα, where the work done by spin torque, Ws, and the energydissipation due to the damping, Wα, are respectively given by Ws=γMHs 1+α2[p·H−(m·p)(m·H)−αp·(m×H)], (4) Wα=−αγM 1+α2/bracketleftBig H2−(m·H)2/bracketrightBig . (5) By assuming a steady precession around the z-axis with a constant tilted angle θ, the time averages of WsandWαover one precession period are, respectively, given by Ws=γM 1+α2/planckover2pi1ηI 2eλMSd/parenleftBigg 1/radicalbig 1−λ2sin2θ−1/parenrightBigg ×[Happl+(HK−4πM)cosθ]cosθ,(6) Wα=−αγM 1+α2[Happl+(HK−4πM)cosθ]2sin2θ.(7) The magnetization can move from the initial state to a point at which dE/dt= 0. Then, the current at which a steady precession with the angle θcan be achieved is given by I(θ) =2αeλMSd /planckover2pi1ηcosθ/parenleftBigg 1/radicalbig 1−λ2sin2θ−1/parenrightBigg−1 ×[Happl+(HK−4πM)cosθ]sin2θ.(8) The corresponding oscillation frequency is given by f(θ) =γ 2π[Happl+(HK−4πM)cosθ].(9) Equations (8) and (9) are the main results in this section. The solid line in Fig. 2 (b) shows the current dependence of the oscillation frequency obtained by Eqs. (8) and (9), wher e the good agreement with the numerical results confirms the validity of the analytical solution. The critical current f or the self-oscillation, Ic= limθ→0I(θ), is given by Ic=4αeMSd /planckover2pi1ηλ(Happl+HK−4πM).(10) The value of Icestimated by using the aboveparametersis 1.2 mA, showing a good agreement with the numerical simulation shown in Fig. 2 (a). The sign of Icdepends on that of λ, and the self-oscillation occurs only for the positive (nega tive) current for the positive (negative) λ. This is because a finite energy is supplied to the free layer for λ/negationslash= 0, i.e.,Ws>0. In the case of λ= 0, the average of the work done by the spin torque is zero, and thus, the self-oscillation does not occu r. It should be noted that I(θ)→ ∞in the limit of θ→π/2. This means the magnetization cannot cross over the xy-plane, and stays in the north hemisphere ( θ < π/2). The reason is as follows. The average of the work done by spin torque becomes zero in the xy-plane (θ=π/2) because the direction of the spin torque is parallel to the constant energy surface . On the other hand, the energy dissipation due to the damping is finite in the presence of the applied field [21]. Then, dE/dt(θ=π/2) =−αγMH2 appl/(1 +α2)<0, which meansthe dampingmovesthe magnetizationto the northpole. Thus, the magnetization cannot cross over the xy-plane. The controllable range of the oscillation frequency by the curr ent isf(θ= 0)−f(θ=π/2) =γ(HK−4πM)/(2π), which is independent of the magnitude of the applied field.3 Since the spin torque breaks the axial symmetry of the free layer along the z-axis, the assumption that the tilted angle is constant used above is, in a precise sense, not valid , and thez-component of the magnetization oscillates around a certain value. Then, the magnetization can reach the xy- plane and stops its dynamics when a large current is applied. However,thevalue ofsuch currentis morethan15mA forour parameter values, which is much larger than the maximum of the experimentallyavailable current. Thus, the above form ulas work well in the experimentally conventional current regio n. Contrary to the system considered here, the oscillation be- haviour of an MTJ with an in-plane magnetized free layer and a perpendicularly magnetized pinned layer has been widely investigated [23]-[26]. The differences of the two systems are as follows. First, the oscillation frequency decreases with increasing the current in our system while it increases in the latter system. Second, the oscillation frequency in our system in the large current limit becomes independent of the z-component of the magnetization while it is dominated by mz= cosθin the latter system. The reasons are as follows. In our system, by increasing the current, the magnetization moves away from the z-axis due to which the effect of the anisotropy field on the oscillation frequency decreases, an d the frequency tends to γHappl/(2π), which is independent of the anisotropy. On the other hand, in the latter system, the magnetization moves to the out-of-plane direction, due to which the oscillation frequency is strongly affected by t he anisotropy (demagnetization field). The macrospin model developed above reproduces the ex- perimentalresultswith the freelayerof2nmthick[17],for ex- ample the current-frequency relation, quantitatively. Al though onlythezero-temperaturedynamicsisconsideredinthispa per, the macrospin LLG simulation at a finite temperature also reproduces other properties, such as the power spectrum and its linewidth, well. However, when the free layer thickness further decreases, an inhomogeneousmagnetization due to t he roughnessattheMgOinterfacesmayaffectsthemagnetizati on dynamics: for example, a broadening of the linewidth. III. EFFECT OF FIELD -LIKE TORQUE The field-like torque arises from the spin transfer from the conductionelectrons to the local magnetizations,as is the spin torque. When the momentum average of the transverse spin of theconductionelectronsrelaxesinthefreelayerveryfast ,only the spin torque acts on the free layer [19]. On the other hand, when the cancellation of the transverse spin is insufficient , the field-like torque appears. The field-like torque added to the right hand side of Eq. (1) is TFLT=−βγHsm×p, (11) where the dimensionless parameter βcharacterizes the ratio between the magnitudes of the spin torque and the field-like torque. The value and the sign of βdepend on the system parameters such as the band structure, the thickness, the impurity density, and/or the surface roughness [22],[27]- [29]. The magnitude of the field-like torque in MTJ is much larger than that in GMR system [30],[31] because the band selectionmz mxmy1 -1 0 -1 -1 1 100(a) mz mxmy1 -1 0 -1 -1 1 100(b) mz mxmy1 -1 0 -1 -1 1 100(c) (d) 0 0.2 0.1 time (μs)0 -0.5 -1.00.51.0mzβ=0 β=0.5 β=-0.5 β=-0.5 β=0.5β=0 Fig. 3. The magnetization dynamics from t= 0with (a) β= 0, (b) β= 0.5, andβ=−0.5. The current magnitude is 2.0mA. (d) The time evolutions of mzfor various β. duringthe tunnelingleads to an insufficient cancellation o f the transverse spin by the momentum average. It should be noted that the effective energy density, Eeff=E−βM/planckover2pi1ηI 2eλMSdlog(1+λmx),(12) satisfying −γm×H+TFLT=−γm×[−∂Eeff/(Mm)], can be introduce to describe the field-like torque. The time derivative of the effective energy, Eeff, can be obtained by replacing the magnetic field, H, in Eqs. (4) and (5) with the effective field −∂Eeff/∂(Mm) =H+βHsp. Then, the average of dEeff/dtover one precession period around the z-axis consists of Eq. (6), (7), and the following two terms: W′ s=βγM 1+α2/parenleftbigg/planckover2pi1ηI 2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ (1−λ2sin2θ)3/2−1/bracketrightbigg , (13) W′ α=−αγM 1+α2/parenleftbiggβ/planckover2pi1ηI 2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ (1−λ2sin2θ)3/2−1/bracketrightbigg −2αβγM 1+α2/planckover2pi1ηI 2eλMSd/parenleftBigg 1/radicalbig 1−λ2sin2θ−1/parenrightBigg ×[Happl+(HK−4πM)cosθ]cosθ. (14) The constant energy surface of Eeffshifts from the xy-plane due to a finite |β|(≃1), leading to an inaccuracy of the calculation of the time average with the constant tilted ang le assumption. Thus, Eqs. (13) and (14) are quantitatively val id for only |β| ≪1. However, predictions from Eqs. (13) and (14) qualitatively show good agreements with the numerical simulations, as shown below. For positive β,W′ sis also positive, and is finite at θ=π/2. Thus,dEeff/dt(θ=π/2)can be positive for a sufficiently large current. This means, the magnetization can cross over thexy-plane, and move to the south semisphere ( θ > π/2). On the other hand, for negative β,W′ sis also negative. Thus, theenergysupplybythespintorqueissuppressedcomparedt o4 current (mA)34567frequency (GHz) 1.2 1.4 1.6 1.8 2.02 1 0β=0 β=0.5β=-0.5 Fig. 4. The dependences of the oscillation frequency on the c urrent for β= 0(red),β= 0.5(orange), and β=−0.5(blue), respectively. thecaseof β= 0.Then,arelativelylargecurrentisrequiredto induce the self-oscillation with a certain oscillation fre quency. Also, the magnetization cannot cross over the xy-plane. We confirmed these expectations by the numerical simu- lations. Figures 3 (a), (b) and (c) show the trajectories of the magnetization dynamics with β= 0,0.5, and−0.5 respectively,while thetime evolutionsof mzareshownin Fig. 3 (d). The current value is 2.0 mA. The current dependences of the oscillation frequency are summarized in Fig. 4. In the case of β= 0.5>0, the oscillation frequency is low compared to that for β= 0because the energy supply by the spin torque is enhanced by the field-like torque, and thus, th e magnetization can largely move from the north pole. Above I= 1.9mA, the magnetizationmovesto the southhemisphere (θ > π/2), and stops near θ≃cos−1[−Happl/(HK−4πM)] in the south hemisphere, which corresponds to the zero fre- quency in Fig. 4. On the other hand, in the case of β=−0.5<0, the magnetization stays near the north pole compared to the case ofβ= 0, because the energy supply by the spin torque is suppressed by the field-like torque. The zero frequency in Fig. 4 indicates the increase of the critical current of the s elf- oscillation. Compared to the case of β= 0, the oscillation frequency shifts to the high frequency region because the magnetization stays near the north pole. IV. CONCLUSIONS In conclusion, we derived the theoretical formula of the relation between the current and the oscillation frequency of STO consisting of the perpendicularly magnetized free laye r and the in-plane magnetized pinned layer. The derivation is based on the analysis of the energy balance between the work done by the spin torque and the energy dissipation due to the damping.The validityof the analyticalsolutionwas confirm ed by numerical simulation. We also found that the field-like torque breaks the energy balance, and changes the oscillati on frequency. The shift direction of the frequency, high or low , depends on the sign of the field-like torque ( β).ACKNOWLEDGMENT The authors would like to acknowledge T. Yorozu, H. Maehara, A. Emura, M. Konoto, A. Fukushima, S. Yuasa, K. Ando, S. Okamoto, N. Kikuchi, O. Kitakami, T. Shimatsu, K. Kudo, H. Suto, T. Nagasawa, R. Sato, and K. Mizushima. REFERENCES [1] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C, Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature, vol.425, pp.380-383, 2003. [2] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. , vol.92, pp.027201, 2004. [3] J. C. Sankey, I. N. Krivorotov, S. I. Kiselev, P. M. Baraga nca, N. C, Em- ley, R. A. Buhrman, and D. C. Ralph, Phys. Rev. B , vol.72, pp.224427, 2005. [4] I. N. Krivorotov, N. C, Emley, R. A. Buhrman, and D. C. Ralp h,Phys. Rev. B, vol.77, pp.054440, 2008. [5] W. H. Rippard, M. R. Pufall, M. L. Schneider, K. Garello, a nd S. E. Russek J. Appl. Phys. , vol.103, pp.053914, 2008. [6] J. Sinha, M. Hayashi, Y. K. Takahashi, T. Taniguchi, M. Dr apeko, S. Mitani, and K. Hono, Appl. Phys. Lett. , vol.99, pp.162508, 2011. [7] A. V. Nazarov, H. M. Olson, H. Cho, K. Nikolaev, Z. Gao, S. S tokes, and B. B. Pant, Appl. Phys. Lett. , vol.88, pp.162504, 2006. [8] A.M.Deac, A.Fukushima, H.Kubota, H.Maehara, Y. Suzuki , S.Yuasa, Y. Nagamine K. Tsunekawa, D. D. Djayaprawira and N. Watanabe ,Nat. 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Kubota, presented at 12thJoint Magnetism and Magnetic Materi- als/International Magnetics Conference, 2013. [18] J. C. Slonczewski, Phys. Rev. B , vol.39, pp.6995, 1989. [19] J. C. Slonczewski, J. Magn. Magn. Mater. , vol.159, pp.L1-L7, 1996. [20] J. C. Slonczewski, J. Magn. Magn. Mater. , vol.247, pp.324-338, 2002. [21] In the absence of the applied field, dE/dt(θ=π/2) = 0. In this case,I(θ)given by Eq. (8) satisfies Ic=I(θ= 0)> I(θ=π/2). These results mean that, above the critical current, the mag netization immediately moves from the initial state to the xy-plane, and stop its motion. Thus, the self-oscillation does not occur. [22] S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. , vol.88, pp.236601, 2002. [23] A. Kent, B. ¨Ozyilmaz, and E. del Barco, Appl. Phys. Lett. , vol.84, pp.3897, 2004. [24] K. J. Lee, O. Redon, and B. Dieny Appl. Phys. Lett. , vol.85, pp.022505, 2005. [25] W. Jin, Y. Liu, and H. Chen, IEEE.Trans. Magn. , vol.42, pp.2682, 2006. [26] U. Ebles, D. 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2204.10596v2.A_short_circuited_coplanar_waveguide_for_low_temperature_single_port_ferromagnetic_resonance_spectroscopy_set_up_to_probe_the_magnetic_properties_of_ferromagnetic_thin_films.pdf

arXiv:2204.10596v2 [cond-mat.mtrl-sci] 19 Jul 2022A short-circuited coplanar waveguide for low-temperature single-port ferromagnetic resonance spectroscopy set-up to probe the magnetic proper ties of ferromagnetic thin films Sayani Pal, Soumik Aon, Subhadip Manna and Chiranjib Mitra∗ Indian Institute of Science Education and Research Kolkata , West Bengal, India A coplanar waveguide shorted in one end is proposed, designe d, and implemented successfully to measure the properties of magnetic thin films as a part of the v ector network analyser ferromag- netic resonance (VNA-FMR) spectroscopy set-up. Its simple structure, potential applications and easy installation inside the cryostat chamber made it advan tageous especially for low-temperature measurements. It provides a wide band of frequencies in the g igahertz range essential for FMR measurements. Our spectroscopy set-up with short-circuit ed coplanar waveguide has been used to extract Gilbert damping coefficient and effective magnetizat ion values for standard ferromagnetic thin films like Py and Co. The thickness and temperature depen dent studies of those magnetic parameters have also been done here for the afore mentioned m agnetic samples. INTRODUCTION In recent years, extensive research on microwave mag- netization dynamics in magnetic thin films[1–3], planar nanostructures[4–6] and multi-layers[7–9] havebeen per- formedduetotheirpotentialapplicationsinvariousfields of science and technology. Spintronics is one such emerg- ing discipline that encompasses the interplay between magnetization dynamics and spin transport. It also in- cludes fields like spin-transfer torque [10–13], direct and inversespin hall effect [14–18], spin pumping [19, 20] etc., which are crucial in industrial applications for develop- ing devices like magnetic recording head[21], magnetic tunnel junction(MTJ) sensors [22, 23], magnetic memory devices[24, 25] andspin-torquedevices[26, 27]. Thus ex- ploring more about the static and dynamic properties of magnetic materials in itself is an interesting subject. Fer- romagnetic resonance spectroscopy(FMR) is a very ba- sic and well-understood technique that is used to study the magnetization dynamics of ferromagnets[28, 29, 31]. Nowadays, most advanced FMR spectroscopy methods use a vector network analyzer (VNA)[30, 31] as the mi- crowave source and detector. We have used VNA in our set-up too. To determine the magnetic parameters of the ferromag- netic materials using the VNA-FMR spectroscopy, one needs to carry out the measurements at a wide range of frequencies. Since the microwave magnetic field in the coplanar waveguide (CPW) is parallel to the plane, it servesthepurposeofexploringthemagneticpropertiesof the concernedsystem overabroadfrequencyrangein the GHz region. The advantage of using CPW in the spec- troscopy system lies in the fact that we no longer need to remount samples at different waveguides or cavities foreveryotherfrequency measurements, which consumes ∗Corresponding author:chiranjib@iiserkol.ac.ina lot of time and effort in an experiment[32, 33]. Re- searchers design and use different types of CPW for vari- ous other purposes like micron-sized CPW in microwave- assisted magnetic recording; two-port CPW in antenna; shorted CPW in ultra-wideband bandpass-filter and per- meability measurements [34–36]. However, in broadband FMR spectroscopy two-port CPW jigs have most com- monly been used till date. Using two-port CPW in FMR spectroscopy, one gets absorption spectra in terms of the transmissioncoefficient of scatteringparameters, and from there magnetic parameters of the samples can be determined. The use of two-port CPW in VNA-FMR can be replaced by one-port CPW where the reflection coefficient of scattering parameters of the FMR spectra can be used to determine the magnetic parameters of the sample. One port reflection geometry is a lot more convenient in terms of easy design, calibration, installa- tion, and sample loading. This is especially true when the whole CPW arrangement is kept inside the cryostat chamber for low-temperature measurements and the sys- tem becomes very sensitive to vibration and any kind of magnetic contacts, one port CPW seems very con- venient to operate rather than the two-port one. Previ- ously, manyhavedesignedandusedshort-circuitedCPW jigs for other purposes but to the best our knowledge it has not been used for low-temperature VNA-FMR spec- troscopy measurements before. In this work, we report the development of short- circuited CPW based low-temperature broadband VNA- FMR spectroscopy set-up to study the magnetic param- eters of standard ferromagnetic samples. For measure- ments, we chose the permalloy(Py) thin films as ferro- magnetic (FM) material which has greatly been used in researchfields like spintronics and industrial applications due to its interesting magnetic properties like high per- meability, large anisotropy magnetoresistance, low coer- civity, and low magnetic anisotropy. We have also con- sidered another standard magnetic thin film, Co of thick- ness 30nm as a standard for ascertaining the measure-2 ment accuracy. In our system, we swept the magnetic field keeping the frequencies constant, and got the FMR spectra for several frequencies. From there we found the variation of resonance fields and field linewidths with the resonance frequencies. We have used the linear fit for resonance frequencies vs field line-widths data to calculate the Gilbert damping coefficient( α). We fit- ted the set of resonance frequencies vs resonance fields data to the Kittel formula [59] to obtain the effec- tive magnetization(4 πMeff). Subsequently, we investi- gated the thickness and temperature-dependent studies of 4πMeffandαfor FM thin films of different thickness inthetemperaturerangeof7.5Kto300K.Tocharacterise the measurement set-up using short-circuited CPW, we compared the previous measurements in the literature with ourresults and there wasa good agreementbetween the two[36, 41]. EXPERIMENTAL DETAILS A short-circuited CPW has been designed and fab- ricated as a part of our low-temperature VNA-FMR spectroscopy set-up. To make the CPW we have used Rogers AD1000, a laminated PCB substrate with copper cladding on both sides of the dielectric. The thickness of the dielectric and the copper layer are 1.5 mm and 17.5 microns respectively and the dielectric constant of the substrate is 10.7. The main concern about the design of the CPW is to match its characteristic impedance with the impedance of the microwave transmission line con- nected to it. We haveused the line calculatorto calculate the dimensions of CPW. For a CPW with a characteris- tic impedance of 50 ohms, the line calculator calculated the width of the signal line and the gap to be 900 mi- crons and 500 microns respectively. The fabrication is done using optical lithography which is described in de- tail in the literature[49]. Other components of our mea- CryostatVNA ElectromagnetSample CPWCoaxial Transmission Line FIG. 1. The schematic diagram of measurement system and the arrangement inside the cryostat with the sample on top of the CPW surement system are a)Vector Network Analyser(VNA), which is a microwave source as well as a detector, b)theelectromagnet that generates the external magnetic field, i.e., Zeemanfieldand, c)optistatdrycryogen-freecooling system from Oxford instruments which is used for low- temperature measurements. One end of the CPW signal line is shorted to the ground, and the other end is con- nected to the VNA through a SMA connector and coax- ial cable (fig 3b). On top of the CPW, thin-film samples have been placed face down after wrapping them with an insulating tape to electrically isolate them. For low- temperature measurements, the sample has been glued to the CPW using a low-temperature adhesive to ensure contact of sample and resonator at all times, in spite of the vibration caused by the cryostat unit. This whole ar- rangementis then placed inside the twopole pieces of the electromagnet as we can see from the diagram in fig 1. Therearetwostandardmethods ofgettingFMR spectra: sweeping the frequency keeping the field constant and sweeping the magnetic field while keeping the frequency constant. We have adopted the second method. We have worked in the frequency range from 2.5GHz to 5.5GHz and in the magnetic field range from 0 Oe to roughly around 500 Oe. We have used 1mW of microwave power throughout the experiment. From the FMR spectra, we havedeterminedeffectivemagnetizationanddampingco- efficient of FM thin films and studied their variation with temperature and sample thickness. SAMPLE PREPARATION AND CHARACTERIZATION Py (Ni80Fe20) and Co thin films were fabricated by thermal evaporation technique on Si/SiO 2substrates, from commercially available pellets (99 .995%pure) at room temperature. The substrates were cleaned with acetone, IPA and DI water respectively in ultrasonica- tor and dried with a nitrogen gun. The chamber was pumped down to 1 ×10−7torr using a combination of a scroll pump and turbo pump. During the deposition, pressure reached upto 1 ×10−6torr. Thin films were fab- ricated at a rate of 1 .2˚A/swhere thickness can be con- trolled by Inficon SQM 160 crystal monitor. For our experiments a series of Py thin films of different thick- nesses were fabricated by keeping the other parameters like base pressure, deposition pressure and growth rate constant. Film thickness and morphology was measured by using atomic force microscopy technique as shown in fig 2(a). We have used Py films with thicknesses 10nm, 15nm, 34nm, 50nm, and 90nm with a surface roughness of around 1nm and one Co film of thickness 30nm. X-ray diffraction experiment confirms the polycrystalline struc- ture of the samples as shown in fig 2b and fig 2c for Py and Co respectively.3 2µm 2µm (a) /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53 /s54/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48 /s52/s52/s46/s51/s54/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41 /s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s80/s121/s32/s40/s49/s53/s110/s109/s41 (b) /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41 /s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s67/s111/s32/s40/s51/s48/s110/s109/s41 (c) FIG. 2. (a)Atomic force microscope (AFM) image of 30 nm thick Py thin film with a surface roughness of 1 nm . X-ray diffraction peak of (b)15nm thick Py film and (c)30nm Co prepared by thermal evaporation. RESULTS AND DISCUSSION We have calculated the dimensions of the short- circuited CPW using the line calculator of the CST Stu- dio Suite software as mentioned in the experimental de- tails section. Using those dimensions we have also done the full-waveelectromagneticsimulation in CST software to get the electric and magnetic field distribution of the CPW. One can see from the simulation result displayed in figure 3a that the farther it is from the gap, the weaker the intensity of the magnetic field, and the magnitude of the field in the gap area is one order of greater than that on the signal line. When placing the thin film sample on top of the CPW, the dimension of the sample shouldDielectricSampleSignal Line Gap Magnetic field lines Electric field lines a) b) c) FIG. 3. (a) Schematic diagram of the cross-sectional view of CPW. (b) Top view of the short-circuited CPW after fabri- cation. (c)Intensity distribution of microwave magnetic fi eld in the one end shorted CPW at 5GHz (top view) /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s83 /s49/s49/s40/s100/s66/s41 /s72/s32/s40/s79/s101/s41/s32/s32/s32 /s102/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s50/s46/s53/s71/s72/s122 /s32/s51/s46/s53/s71/s72/s122 /s32/s52/s46/s53/s71/s72/s122 /s32/s53/s46/s53/s71/s72/s122/s49/s53/s110/s109/s32/s80/s121 /s84/s61/s51/s48/s48/s75 FIG. 4. Ferromagnetic Resonance spectra of absorption at frequencies 2.5 GHz, 3.5 GHz, 4.5 GHz, 5.5 GHz for 15nm Py thin films at room temperature after background subtraction be such that it can cover the gap area on both sides of the signal line of the CPW because the magnetic field is most intense in that area. This microwave magnetic field circulatingthe signal line ofthe CPW is perpendicular to4 /s50 /s51 /s52 /s53 /s54 /s55/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s51/s48/s48/s75 /s32/s67/s111/s32/s40/s51/s48/s110/s109/s41 /s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s68 /s72/s32/s40/s79/s101/s41 /s102/s32/s40/s71/s72/s122/s41 (a)/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s97 /s116/s32 /s80/s121 /s32/s40/s110/s109/s41/s32/s84/s61/s51/s48/s48/s75 (b) /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s51/s52/s53/s54/s55 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s32/s51/s48/s48/s75 /s32/s67/s111/s32/s40/s51/s48/s110/s109/s41 /s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s102/s32/s40/s71/s72/s122/s41 /s72/s32/s40/s79/s101/s41 (c)/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s56/s57/s49/s48/s49/s49/s52 /s112 /s77 /s101/s102/s102/s32/s40/s107/s71/s41 /s116/s32/s45/s49 /s32/s80/s121/s32/s40/s110/s109/s45/s49 /s41/s32/s84/s61/s51/s48/s48/s75 (d) FIG. 5. a)Field linewidth variation with resonance frequen cies at 300K for 34nm Py and 30nm Co thin films. Equation 1 has been used for fitting the curve and to determine the Gilbert da mping coefficient; b)thickness dependence of Gilbert dampin g coefficient at room temperature for Py thin films; c)resonance field variation with resonance frequencies at 300K for 34 nm P y and 30 nm Co thin films. Kittel formula (eqn-3)has been used fo r fitting the curve and to determine the effective magnetizati on; d)thickness dependence of effective magnetization for Py th in films at room temperature. the external magnetic field and both the magnetic fields are parallel to the film surface as can be seen from fig 3a and 3b. On account of the static magnetic field, the magnetic moment will undergo a precession around the static magnetic field at a frequency called the Larmor precession frequency. Absorption of electromagnetic en- ergy happens when the frequency of the transverse mag- netic field (microwave) is equal to the Larmor frequency. Fig4exhibitsthe absorptionspectrafor15nmbarePy film after subtraction of a constant background for four different frequencies, 2.5 GHz, 3.5 GHz, 4.5 GHz and 5.5 GHz at room temperature in terms of S-parameter re- flection coefficient ( S11) vs. external magnetic field. We fitted these experimental results to the Lorentz equation [56]. We extracted the field linewidth at half maxima from the FMR spectra at different frequencies and fitted them using equation 1 to obtain αas one can see from fig 5a and fig 6a. The experimental values of the absorp-tion linewidth (∆ H) contains both the effect of intrinsic Gilbert damping and the extrinsic contribution to the damping. Linewidth due to Gilbert damping is directly proportional to the resonance frequency and follows the equation: ∆H= (2π γ)αf+∆H0 (1) whereγis the gyromagneticratio, αis the Gilbert damp- ing coefficient and ∆ H0is the inhomogeneous linewidth. A number of extrinsic contributions to the damping coef- ficient like magnetic inhomogeneities, surface roughness, defects of the thin films bring about the inhomogeneous linewidth broadening [55]. αhas been determined using the above equation only. Damping coefficient values ob- tainedhereareintherangeofabout0 .005to0.009forPy samplesofthicknessescoveringthe whole thin film region i.e., 10nm to 90nm at room temperature. These values5 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48 /s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75 /s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121/s68 /s72/s32/s40/s79/s101/s41 /s102/s32/s40/s71/s72/s122/s41/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75 /s32/s49/s53/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121 (a)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50 /s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s116 /s32 /s80/s121 /s32/s49/s53/s110/s109 /s32/s49/s48/s110/s109 (b) /s54/s48 /s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48 /s51/s54/s48 /s52/s50/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48 /s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75 /s32/s49/s53/s110/s109/s32/s84/s61/s32/s52/s53/s75/s80/s121/s102/s32/s40/s71/s72/s122/s41 /s72/s32/s40/s79/s101/s41/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75 /s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121 (c)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s56/s46/s48/s56/s46/s50/s56/s46/s52/s56/s46/s54/s56/s46/s56/s57/s46/s48/s57/s46/s50/s57/s46/s52/s57/s46/s54/s52 /s77 /s101/s102/s102/s32/s40/s107/s71/s41 /s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s116/s32 /s80/s121 /s32/s49/s53/s110/s109 /s32/s49/s48/s110/s109 (d) FIG. 6. a)Field linewidth variation with resonance frequen cies at 300K and 45K for 10 nm and 15nm Py films. Equation 1 has been used for fitting the curve and to determine the Gilber t damping coefficient; b)temperature dependence of damping coefficient for 10nm and 15nm Py thin films; c)resonance field va riation with resonance frequencies at 300K and 45K for 10nm and 15nm Py thin films. Kittel formula (eqn-3) has been used fo r fitting the curve and to determine the 4 πMeff; d)temperature dependence of 4 πMefffor 10nm and 15nm Py thin films. are pretty close to the values previously reported in the literature [39–41, 43, 44]. For the Co film of thickness 30 nm we have obtained the value of αto be 0.008 ±0.0004. Baratiet al.measured the damping value of 30nm Co film to be 0.004 [37, 38]. There are other literature also where Co multilayers have been studied where damping coefficient value increasesbecause ofspin pumping effect. αis a veryinterestingparameterto investigatebecause it is used in the phenomenological LLG equation [57], [58] to describe magnetization relaxation: d/vectorM dt=−γ/vectorM×/vectorHeff+α MS/vectorM×d/vectorM dt(2) where,µBisBohrmagneton, /vectorMisthemagnetizationvec- tor,MSis the saturation magnetization and Heffis the effectve magnetic field which includes the external field, demagnetization and crystalline anisotropy field. The in-troduction of the Damping coefficient in LLG equation is phenomenological in nature and the question of whether it has a physical origin or not has not been fully under- stood till date. We have measured 4 πMeffalso from the absorption spectra. We have fitted the Kittel for- mula (equation 3) into resonance field vs. the resonance frequency ( fres) data as shown in fig 5c and fig 6c. fres= (γ 2π)[(H+4πMeff)H]1 2 (3) where,His the applied magnetic field, and Meffis the effective magnetization which contains saturation mag- netization and other anisotropic contributions. We ob- tained the 4 πMeffvalue for 30nm thick Co and 34nm Py to be 17.4 ±0.2kG and 9.6 ±0.09kG respectively at room temperature. These values also agree quite well with the literature. For a 10nm Co film, Beaujour et al.measured the value to be around 16 kG[45] and for a 30nm Py the6 value is 10 .4kGas measured by Zhao et al[41]. We tried to address here the thickness and tempera- ture dependence of αand 4πMeffusing our measure- ment set-up. The variation of the αwith thickness is shown here in figure 5b. It increases smoothly as film thickness decreases and then shows a sudden jump below 15nm. Increased surface scattering could be the reason behind this enhanced damping for thinner films. It has been previously observed [60] that damping coefficient and electrical resistivity follows a linear relation at room temperature for Py thin film. It suggests a strong corre- lation between magnetization relaxation( α) and electron scattering. Magnetization relaxation could be explained by electron scattering by phonons and magnons. In the former case, αis proportional to the electron scatter- ing rate, τ−1and in the later case, α∼τ. Theoretical predictions by Kambersky [61] suggests that at higher temperature α∼τ−1as electron scattering by phonons are predominant there. So, here in our case we can elim- inate the possibility of electron scattering by magnons as thickness dependent study has only been done at room temperature where phonon scattering is prevalent. Ing- vasson et.al in their paper[60] also suggests that the re- laxation of magnetization is similar to bulk relaxation where phonon scattering in bulk is replaced by surface and defect scattering in thin films. Thicknessdependent studyof4 πMeffalsohasbeen done for Py thin films at room temperature. As we can see from fig 5d, Meffis linear for thinner films and becomes almost independent of thickness for thicker films. The change in Meffwith thickness mainly comes from the surface anisotropy, µ0Meff=µ0Ms−2Ks Msd(4) whereMsis the saturation magnetization and2Ks Msdis the surface anisotropy field. Surface anisotropy is higher for thinner films and the anisotropy reduces as one in- creases the film thickness. We have obtained saturation magnetization(4 πMs) value of Py to be 10 .86kGusing the linear fit (equation 4). Previously Chen et al.has re- ported the 4 πMeffvalue for a 30nm Py film to be 12 kG [54] which includes both 4 πMsand anisotropy field. Temperature dependence of αfor 15nm and 10nm Py film is represented in figure 6b. The αvalue decreases monotonically from room temperature value and reaches a minimum value at around 100K and then starts to in- crease with further decrease of temperature and reaches a maximum value at 45K. Zhao et al.have seen this kind of damping enhancement at around50Kin their low temperature experiment with Py thin films with differ- ent types of capping layers and Rio et al.observed the damping anomaly at temperature 25K when they have usedPtas a capping layer on Py thin film.[39, 41]. We did not use any capping layer on Py film in our mea- surement. So there is no question of interface effect for the enhanced damping at 45K. A possible reason for the strong enhancement of damping at 45K could be the/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48 /s32/s57/s71/s72/s122 /s32/s52/s71/s72/s122/s51/s48/s110/s109/s32/s67/s111/s68 /s72 /s32/s40/s79/s101/s41 /s84/s32/s40/s75/s41 FIG. 7. Temperature induced linewidth variation of 30nm Co thin film at two different frequencies 4GHz and 9GHz spin reorientation transition(SRT) on the Py surface at that particular temperature [41, 42]. Previously it has been established that the competition between different anisotropy energies: magnetocrystalline anisotropy, sur- face anisotropy, shape anisotropy decides the magnetiza- tion direction in magnetic films. For thin films, the vari- ation of temperature, film thickness, strain can alter the competition between shape and surface anisotropy. In our case, temperature variation could be the reason for the spin reorientation transition on Py surface at around 45K.Foradeeperunderstandingofthespinreorientation we investigated the temperature dependence of 4 πMeff for 15nm and 10nm Py film as shown in fig 6d. There Meffis showing an anomaly at around 45K, otherwise it is increasing smoothly with the decrease of temperature. Since there is no reason of sudden change in saturation magnetization at this temperature, the possible reason for the anomaly in Meffshould come from any change inmagneticanisotropy. Thatchangeofanisotropycanbe related to a spin reorientation at that particular temper- ature value. Sierra et.al., [42], have also argued that in the temperature dependent spin re-orientation (T-SRT), the central effect of temperature on the magnetic prop- erties of Py films was to increase the in-plane uniax- ial anisotropy and to induce a surface anisotropy which orients the magnetization out of plane in the Py sur- face. They have verified this using X-Ray diffraction experiments and high resolution transmission electron microscopy images. This establishes reasonably enough that it is a spin re-orientation transition around 45K. Lastly, for a 30nm Co thin film we have studied the temperature variation of FMR linewidth(∆ H) at mi- crowave frequencies 9GHz and 4GHz. One can see from fig7 that the linewidth does not change much in the tem- perature range 100 <T<300 but below 100K, ∆ Hstarts to increase significantly. This behaviour of ∆ Hhas been7 observed previously by Bhagat et al.[62]. They sug- gested that the increase of relaxation frequency at low temp might be related to the rapid variation in the elec- tronicmeanfreepath. Thiscomparisonsuggeststhatour set-up with a single port CPW has the same sensitivity of the previously reported set-up. CONCLUSIONS Short-circuited coplanar waveguide can successfully be used for low-temperature single-port broadband FMR spectroscopy measurements. The magnetic parameters obtained here for the standard ferromagnetic materials, Py and Co are in good agreement with other experimen- tal works and theoretical predictions. Temperature de- pendentstudiesof αand4πMeffforPyfilmshereexhibit spin reorientation phenomenon at low temperatures. We believe that the findings with Py sample will help in bet- ter understanding of magnetic phenomena for other fer- romagnetic materials at low temperature. Though this setup has limitations for angle-dependent FMR measure- ments, itcanreadilybe usedforstudiesonmagnetization dynamics for multi-layered films and planar nanostruc- tures and is currently under way. In future we aim to in- tegrate electrical measurement facilities with the existing set-up thereby extending the possibilities of inverse spinhall(ISHE) measurements and spin-torque ferromagnetic resonance(ST-FMR)spectroscopyexperimentswhich are very relevant for current scientific interests. We believe the short-circuited CPW will serve its purpose conve- niently there too. ACKNOWLEDGEMENTS TheauthorssincerelyacknowledgesMinistryofEduca- tion, 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 sup- port. S.P. acknowledges Department of Science and Technology(DST)-INSPIRE fellowship India, S. A. ac- knowledges Ministry of Education of Government of In- dia and S.M. acknowledges Council Of Scientific and Industrial Research(CSIR),India for research fellowship. The authors would like to thank Dr. Partha Mitra of the Department of Physical Sciences, Indian Institute of Sci- ence Education and Research Kolkata for providing the lab facilities for sample deposition. The authors would also like to thank Mr. Subhadip Roy, IISER Kolkata for his help in fabricating the CPW structure using home- built optical lithography set-up. 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0812.0832v1.Observation_of_ferromagnetic_resonance_in_strontium_ruthenate__SrRuO3_.pdf

arXiv:0812.0832v1 [cond-mat.mtrl-sci] 3 Dec 2008Observationofferromagnetic resonance instrontium ruthe nate (SrRuO 3) M.C. Langner,1,2C.L.S. Kantner,1,2Y.H. Chu,3L.M. Martin,2P. Yu,1R. Ramesh,1,4and J. 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, National Chiao Tung University, HsinChu, Taiwan, 30010 4Department of Materials Science and Engineering, University of California, Berkeley, CA 94720 (Dated: October 26, 2018) Abstract We report the observation of ferromagnetic resonance (FMR) in SrRuO 3using the time-resolved magneto-optical Kerr effect. The FMR oscillations in the ti me-domain appear in response to a sudden, optically induced change in the direction of easy-axis anis tropy. The high FMR frequency, 250 GHz, and large Gilbert damping parameter, α≈1, are consistent with strong spin-orbit coupling. We find th at the parameters associated with the magnetization dynamics, in cludingα, have a non-monotonic temperature dependence, suggestive of alink to theanomalous Hall effec t. PACS numbers: 76.50.+g,78.47.-p,75.30.-m 1Understanding and eventually manipulating the electron’s spin degree of freedom is a major goal of contemporary condensed matter physics. As a means to this end, considerable attention is focused on the spin-orbit (SO) interaction, which provid esa mechanism for control of spin po- larization by applied currents or electric fields [1]. Despi te this attention, many aspects of SO coupling are not fully understood, particularly in itinera nt ferromagnets where the same elec- trons are linked to both rapid current fluctuations and slow s pin dynamics. In these materials, SO coupling is responsible for spin-wave damping [2, 3], spi n-current torque [4, 5], the anoma- lous Hall effect (AHE) [6], and magnetocrystalline anisotr opy (MCA) [7]. Ongoing research is aimed toward a quantitative understanding of how bandstruc ture, disorder, and electron-electron interactionsinteracttodeterminethesizeandtemperatur edependenceoftheseSO-driveneffects. SrRuO 3(SRO) is a material well known for its dual role as a highly cor related metal and an itinerant ferromagnet with properties that reflect stron g SO interaction [8, 9, 10]. Despite its importance as a model SO-coupled system, there are no pre vious reports of ferromagnetic resonance (FMR) in SRO. FMR is a powerful probe of SO coupling in ferromagnets, providing a means to measure both MCA and the damping of spin waves in the small wavevector regime [11]. HerewedescribedetectionofFMRbytime-resolvedmag netoopticmeasurementsperformed on high-quality SRO thin films. We observe a well-defined reso nance at a frequency ΩFMR= 250 GHz. This resonant frequency is an order of magnitude hig her than in the transition metal ferromagnets,which accountsforthenonobservationbycon ventionalmicrowavetechniques. 10-200nmthickSROthinfilmsweregrownviapulsedlaserdepo sitionbetween680-700◦Cin 100 mTorr oxygen. High-pressure reflection high-energy ele ctron diffraction (RHEED) was used to monitor the growth of the SRO film in-situ. By monitoring RH EED oscillations, SRO growth was determined to proceed initially in a layer-by-layer mod e before transitioning to a step-flow mode. RHEED patterns and atomic force microscopy imaging co nfirmed the presence of pristine surfaces consisting of atomically flat terraces separated b y a single unit cell step ( 3.93 ˚A). X-ray diffractionindicatedfullyepitaxialfilmsandx-rayreflec tometrywasusedtoverifyfilmthickness. Bulk magnetization measurements using a SQUID magnetomete r indicated a Curie temperature, TC, of∼150K. Sensitive detection of FMR by the time-resolved magnetoopt ic Kerr effect (TRMOKE) has been demonstrated previously [12, 13, 14]. TRMOKE is an all o ptical pump-probe technique in whichtheabsorptionofan ultrashortlaserpulseperturbst hemagnetization, M, ofaferromagnet. The subsequent time-evolutionof Mis determined from the polarization state of a normally inci - 2dent, time-delayed probe beam that is reflected from the phot oexcited region. The rotation angle of the probe polarization caused by absorption of the pump, ∆ΘK(t), is proportional to ∆Mz(t), wherezisthedirectionperpendiculartotheplaneofthefilm[15]. Figs. 1a and 1b show ∆ΘK(t)obtained on an SRO film of thickness 200 nm. Very similar results are obtained in films with thickness down to 10 nm. Two distinct types of dynamics are observed,dependingonthetemperatureregime. Thecurvesi nFig. 1aweremeasuredattempera- turesnearT C. Therelativelyslowdynamicsagreewithpreviousreportsf orthisTregime[16]and are consistent with critical slowing down in the neighborho od of the transition [17]. The ampli- tudeofthephotoinducedchangeinmagnetizationhasalocal maximumnearT=115K.Distinctly differentmagnetizationdynamicsareobservedasTisreduc edbelowabout80K,asshowninFig. 1b. The TRMOKE signal increases again, and damped oscillati ons with a period of about 4 ps becomeclearly resolved. FIG. 1: Change in Kerr rotation as a function of time delay fol lowing pulsed photoexcitation, for several temperatures below the Curie temperature of 150 K. Top Panel : Temperature range 100 K <T<150 K. Bottom panel: Temperature range 5K <T<80 K.Signal amplitude and oscillations grow with decreasin g T. In order to test if these oscillations are in fact the signatu re of FMR, as opposed to another 3photoinduced periodic phenomenon such as strain waves, we m easured the effect of magnetic field on theTRMOKE signals. Fig. 2a shows ∆ΘK(t)for several fields up to 6 T applied normal to the film plane. The frequency clearly increases with incre asing magnetic field, confirming that theoscillationsare associatedwithFMR. ThemechanismfortheappearanceofFMRinTRMOKEexperiment siswellunderstood[14]. Beforephotoexcitation, Misorientedparallelto hA. Perturbationofthesystembythepumppulse (by local heating for example) generates a sudden change in t he direction of the easy axis. In the resulting nonequilibrium state, MandhAare no longer parallel, generating a torque that induces Mto precess at the FMR frequency. In the presence of Gilbert da mping,Mspirals towards the newhA, resultingin thedamped oscillationsof Mzthat appearin theTRMOKE signal. To analyze the FMR further we Fourier transform (FT) the time -domain data and attempt to extract thereal and imaginary parts of the transversesusce ptibility,χij(ω). Themagnetization in thetime-domainisgivenby therelation, ∆Mi(t) =/integraldisplay∞ 0χij(τ)∆hj A(t−τ)dτ, (1) whereχij(τ)is the impulse response function and ∆hA(t)is the change in anisotropy field. If∆hA(t)is proportional to the δ-function, ∆Mi(t)is proportional χij(τ)and the FT of the TRMOKE signal yields χij(ω)directly. However, for laser-induced precession one expec ts that ∆hA(t)will be more like the step function than the impulse function , as photoinduced local heating can be quite rapid compared with cooling via thermal conduction from the laser-excited region. When ∆hA(t)is proportional to the step function, χij(ω)is proportionalto the FT of the timederivativeof ∆Mi(t), ratherthan ∆Mi(t)itself. Inthiscase, theobservable ωIm{∆ΘK(ω)} shouldbecloselyrelated tothereal, ordissipativepart of χij(ω). InFig. 2bweplot ωIm{∆ΘK(ω)}foreachofthecurvesshowninFig. 2a. Thespectrashown in Fig. 2b do indeed exhibit features that are expected for Re χij(ω)near the FMR frequency. A well-definedresonancepeakisevident,whosefrequencyinc reaseswithmagneticfieldasexpected forFMR.TheinsettoFig. 2bshows ΩFMRasafunctionofappliedmagneticfield. Thesolidline throughthedatapointsisafitobtainedwithparameters |hA|=7.2Tandeasyaxisdirectionequal to 22 degrees from the film normal. These parameter values agr ee well with previous estimates based onequilibriummagnetizationmeasurements[8, 10]. Although the spectra in Fig. 2b are clearly associated with F MR, the sign change at low fre- quency is not consistent with Re χij(ω), which is positive definite. We have verified that the 4FIG. 2: Top panel: Change in Kerr rotation as a function of tim e delay following pulsed photoexcitation at T=5K,forseveral valuesofappliedmagneticfieldranging up to6Tesla. Bottompanel: Fourier transforms of signals shown intop panel. Inset: FMRfrequency vs. appli ed field. negativecomponentis always present in the spectra and is no t associated with errors in assigning thet=0pointinthetime-domaindata. Theoriginofnegativecomp onentoftheFTismadeclearer by referring back to the time domain. In Fig. 3a we show typica l time-series data measured in zero field at 5 K. Forcomparison we show theresponseto a step f unction changein theeasy axis direction predicted by the Landau-Lifschitz-Gilbert (LLG ) equations [18]. It is clear that, if the measured and simulated responses are constrained to be equa l at large delay times, the observed ∆ΘK(t)ismuchlarger thantheLLGpredictionat smalldelay. We have found that ∆ΘK(t)can be readily fit by LLG dynamics if we relax the assumption that∆hA(t)is a step-function, in particular by allowing the change in e asy axis direction to ”overshoot” at short times. The overshoot suggests that the easy-axis direction changes rapidly as the photoexcited electrons approach quasiequilibriumw ith the phonon and magnon degrees of freedom. The red line in Fig. 3a shows the best fit obtained by m odeling∆hA(t)byH(t)(φ0+ φ1e−t/τ), whereH(t)is the step function, φ0+φ1is the change in easy-axis direction at t= 0, andτis the time constant determining the rate of approach to the a symptotic value φ0. The fit is 5FIG. 3: Components of TRMOKE response in time (top panel) and frequency (bottom panel) domain. Blacklinesaretheobserved signals. Greenlineinthetoppa nel isthesimulated response toastepfunction change in easy-axis direction. Best fits to the ”overshoot” m odel described in the text are shown in red. Bluelines are thedifference between the measured and best- fit response. clearly much better when the possibility of overshoot dynam ics in∆hA(t)is included. The blue line shows the difference between measured and simulated re sponse. With the exception of this veryshortpulsecenterednear t=0,theobservedresponseisnowwelldescribedbyLLGdynami cs. Inprinciple,analternateexplanationforthediscrepancy withthestep-functionassumptionwould be to consider possible changes in the magnitude as well as di rection of M. However, we have found that fitting the data then requires |M(t)|to be larger at t >20ps than|M(t <0)|, a photoinducedincrease thatisunphysicalfora systemin ast ableFM phase. In Fig. 3b we compare data and simulated response in the frequ ency domain. With the al- lowance for an overshoot in ∆hA(t)the spectrum clearly resolves into two components. The peak at 250 GHz and the sign change at low frequency are the bot h part of the LLG response to ∆hA(t). The broad peak or shoulder centered near 600 GHz is the FT of t he short pulse compo- nentshowninFig. 3a. Wehavefoundthiscomponentisessenti allylinearinpumppulseintensity, 6and independent of magnetic field and temperature - observat ions that clearly distinguish it from the FMR response. Its properties are consistent with a photo induced change in reflectivity due to band-filling,whichiswell-knowntocross-coupleintotheT RMOKEsignalofferromagnets [19]. Byincludingovershootdynamicsin ∆hA(t),weareabletodistinguishstimulusfromresponse in the observed TRMOKE signals. Assuming LLG dynamics, we ca n extract the two parameters thatdescribetheresponse: ΩFMRandα;andthetwoparametersthatdescribethestimulus: φ1/φ0 andτ. In Fig. 4 we plot all four parameters as a function of tempera ture from 5 to 80 K. The T-dependence of the FMR frequency is very weak, with ΩFMRdeviating from 250 GHz by only about 5%overthe range ofthe measurement. TheGilbert damping param eterαis of order unity at all temperatures, avaluethatis approximatelyafactor 102largerthan found intransitionmetal ferromagnets. Over the same T range the decay of the easy axis overshoot varies from about 2 to 4 ps. We note that the dynamical processes that characteri ze the response all occur in strongly overlapping time scales, that is the period and damping time of the FMR, and the decay time of thehAovershoot,areeach inthe2-5ps range. WhileΩFMRisessentiallyindependentofT,theparameters α,φ1/φ0andτexhibitstructurein theirT-dependencenear40K.Thisstructureisreminiscent oftheT-dependenceoftheanomalous Hallcoefficient σxythathasbeenobservedinthinfilmsofSRO[20,21,22]. Forcom parison,Fig. 4dreproduces σxy(T)reportedinRef. [20]Thesimilaritybetween theT-dependen ceofAHEand parameters related to FMR suggests a correlation between th e two types of response functions. Recently Nagaosa and Ono [23] have discussed the possibilit y of a close connection between collective spin dynamics at zero wavevector (FMR) and the of f-diagonal conductivity (AHE). At a basic level,both effects are nonzero only in the presence o f both SO couplingand time-reversal breaking. However, the possibilityof a more quantitativec onnection is suggested by comparison of the Kubo formulas for the two corresponding functions. Th e off-diagonal conductivity can be writtenin theform [24], σxy(ω) =i/summationdisplay m,n,kJx mn(k)Jy nm(k)fmn(k) ǫmn(k)[ǫmn(k)−ω−iγ], (2) whereJi mn(k)is current matrix element between quasiparticle states wit h band indices n,mand wavevector k. The functions ǫmn(k)andfmn(k)are the energy and occupation difference, re- spectively,between such states, and γis a phenomenologicalquasiparticledamping rate. FMR is related to theimaginary part of theuniformtranverse susce ptibility,with thecorresponding Kubo 7FIG. 4: Temperature dependence of (a) FMR frequency (triang les) and damping parameter (circles), (b) overshoot decay time, (c) ratio of overshoot amplitude to st ep-response amplitude ( φ1/φ0), and (d) σxy (adapted from [20]). form, Imχxy(ω) =γ/summationdisplay m,n,kSx mn(k)Sy nm(k)fmn(k) [ǫmn(k)−ω]2+γ2, (3) whereSi mnisthematrixelementofthespinoperator. Ingeneral, σxy(ω)andχxy(ω)areunrelated, as they involvecurrent and spin matrix elements respective ly. However, it has been proposed that in several ferromagnets, including SRO, the k-space sums in Eqs. 2 and 3 are dominated by a small number of band-crossings near the Fermi surface [22, 2 5]. If the matrix elements Si mnand Ji mnvary sufficiently smoothly with k, thenσxy(ω)andχxy(ω)may be closely related, with both properties determined by thepositionofthechemical poten tialrelativeto theenergy at which the 8bandscross. Furthermore,asGilbertdampingisrelatedtot hezero-frequencylimitof χxy(ω),i.e., α=ΩFMR χxy(0)∂ ∂ωlim ω→∞Imχxy(ω), (4) and AHE is the zero-frequency limit of σxy(ω), the band-crossing picture implies a strong corre- lationbetween α(T)andσxy(T). In conclusion,we havereported the observationof FMR in the metallictransition-metaloxide SrRuO 3. Both the frequency and damping coefficient are significantl y larger than observed in transition metal ferromagnets. Correlations between FMR d ynamics and the AHE coefficient suggest that both may be linked to near Fermi surface band-cr ossings. Further study of these correlations, as Sr is replaced by Ca, or with systematic var iation in residual resistance, could be a fruitful approach to understanding the dynamics of magnet ization in the presence of strong SO interaction. Acknowledgments This research is supported by the US Department of Energy, Of fice of Science, under contract No. DE-AC02-05CH1123. Y.H.C. would also like to acknowledg e the support of the National Science Council,R.O.C., underContract No. NSC97-3114-M- 009-001. [1] I.ˆZuti´ c, J. Fabian, and S.DasSarma, Rev.Mod. Phys. 76, 323 (2004). [2] V. Korenman and R.E.Prange, Phys. Rev.B 6, 2769 (1972). [3] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970). [4] J. C.Slonczewski, J. Magn. Magn. Mater. 159, L1(1996). [5] L.Berger, Phys. Rev.B 54, 9353 (1996). [6] J. M.Luttinger and R. Karplus, Phys. Rev. 94, 782 (1954). [7] H.Brooks, Phys. Rev. 58, 909 (1940). [8] L. Klein, J. S. Dodge, C. H. Ahn, J. W. Reiner, L. Mieville, T. H. Geballe, M. R.Beasley, and A. Ka- pitulnik, J.Phys. Cond.-Matt. 8, 10111 (1996). [9] P.Kostic,Y.Okada,N.C.Collins,Z.Schlesinger, J.W.R einer,L.Klein,A.Kapitulnik, T.H.Geballe, and M. R.Beasley, Phys.Rev. Lett. 81, 2498 (1998). 9[10] A.F.Marshall, L.Klein, J.S.Dodge, C.H.Ahn,J.W.Rein er, L.Mieville, L.Antagonazza, A.Kapit- ulnik, T.H.Geballe, and M.R. Beasley, J.Appl. Phys. 85, 4131 (1999). [11] B.Heinrich and J. F.Cochran, Adv. Phys. 42, 523 (1993). [12] W.K.Hiebert, A.Stankiewicz, and M.R.Freeman, Phys. R ev.Lett.79, 1134 (1997). [13] Y. Acremann, C. H.Back, M. Buess, O.Portmann, A. Vaterl aus, D.Pescia, and H.Melchior, Science 290, 492 (2000). [14] M.vanKampen,C.Jozsa, J.T.Kohlhepp, P.LeClair, L.La gae,W.J.M.deJonge, andB.Koopmans, Phys. Rev. Lett. 88, 227201 (2002). [15] K. Shinagawa, in Magneto-optics , edited by S. Sugano and N. Kojima (Springer-Verlag, Berlin , Ger- many, 2000). [16] T. Ogasawara, K. Ohgushi, Y. Tomioka, K. S. Takahashi, H . Okamoto, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 94, 087202 (2005). [17] T. Kise, T. Ogasawara, M. Ashida, Y. Tomioka, Y. Tokura, and M. Kuwata-Gonokami, Phys. Rev. Lett.85, 1986 (2000). [18] W.F.Brown, Micromagnetics (Krieger, 1963). [19] B.Koopmans,M.vanKampen,J.T.Kohlhepp,andW.J.M.de Jonge,Phys.Rev.Lett. 85,844(2000). [20] R. Mathieu, A. Asamitsu, H. Yamada, K. S. Takahashi, M. K awasaki, Z. Fang, N. Nagaosa, and Y. Tokura, Phys. Rev. Lett. 93, 016602 (2004). [21] L. Klein, J. R. Reiner, T. H. Geballe, M. R. Beasley, and A . Kapitulnik, Phys. Rev. B 61, R7842 (2000). [22] Z. Fang, N. Nagaosa, K. Takahashi, A. Asamitsu, R. Mathi eu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura, and K.Terakura, Science 302, 92(2003). [23] M. Onoda, A.S.Mishchenko, and N. Nagaosa, J.Phys. Soc. Jap.77, 013702 (2008). [24] M. Onoda and N.Nagaosa, J. Phys.Soc. Jap. 71, 19 (2002). [25] X.Wang, J.R. Yates, I. Souza, and D.Vanderbilt, Phys.R ev. B.74, 195118 (2006). 10

2102.01137v2.Blow_up_and_lifespan_estimates_for_a_damped_wave_equation_in_the_Einstein_de_Sitter_spacetime_with_nonlinearity_of_derivative_type.pdf

arXiv:2102.01137v2 [math.AP] 20 Jan 2022BLOW-UP AND LIFESPAN ESTIMATES FOR A DAMPED WAVE EQUATION IN THE EINSTEIN-DE SITTER SPACETIME WITH NONLINEARITY OF DERIVATIVE TYPE MAKRAM HAMOUDA1, MOHAMED ALI HAMZA1AND ALESSANDRO PALMIERI2,3 Abstract. Inthis article, weinvestigatethe blow-upforlocalsolutionstoasem ilinear wave equation in the generalized Einstein - de Sitter spacetime with no nlinearity of derivative type. More precisely, we consider a semilinear damped wav e equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We pr ove in this work that the results obtained in a previous work, where the damping coe fficient takes two particular values 0 or 2, can be extended for any positive damping co efficient. We show the blow-up in finite time of local in time solutions and we establish u pper bound estimates for the lifespan, provided that the exponent in the nonlin ear term is below a suitable threshold and that the Cauchydata are nonnegativeand c ompactlysupported. 1.Introduction We are interested in the semilinear damped wave equation when the sp eed of propa- gation is depending on time, namely the damped wave equations in Einst ein - de Sitter spacetime, with time derivative nonlinearity which reads as follows: (1.1)/braceleftBigg utt−t−2k∆u+µ tut=|ut|p,inRN×[1,∞), u(x,1) =εf(x), ut(x,1) =εg(x), x∈RN, wherek∈[0,1),µ≥0,p >1,N≥1 is the space dimension, ε >0 is a parameter illustrating the size of the initial data, and f,gare supposed to be positive functions. Furthermore, we consider fandgwith compact support on B(0RN,R),R>0. The problem ( 1.1) with time derivative nonlinearity being replaced by power non- linearity is well understood in terms of blow-up phenomenon. Let us fi rst recall the equation in this case. Under the usual Cauchy conditions, the semilin ear wave equation with power nonlinearity is (1.2) utt−t−2k∆u+µ tut=|u|q,inRN×[1,∞). 2010Mathematics Subject Classification. 35L15, 35L71, 35B44. Key words and phrases. Blow-up, Einsten-de Sitter spacetime, Glassey exponent, Lifespa n, Critical curve, Nonlinear wave equations, Time-derivative nonlinearity. 1The blow-up phenomenon for ( 1.2) is related to two particular exponents. The first exponent,q0(N,k), is the positive root of ((1−k)N−1)q2−((1−k)N+1+2k)q−2(1−k) = 0, and the second exponent is given by q1(N,k) = 1+2 N(1−k). Hence, the positive number max/parenleftbigg q0(N+µ 1−k,k),q1(N,k)/parenrightbigg seems to be a serious candidate for the critical power stating thus the threshold betwe en the global existence and the blow-up regions, see e.g. [ 7,22,23,27,29]. Let us go back to ( 1.1) withk=µ= 0. This case is in fact connected to the Glassey conjecture in which the critical exponent pGis given by (1.3) pG=pG(N) := 1+2 N−1. The above value pGis creating a threshold (depending on p) between the region where we have the global existence of small data solutions (for p>pG) and another where the blow-up of the solutions under suitable sign assumptions for the Cau chy data occurs (for p≤pG); see e.g. [ 14,15,17,26,32,35]. Now, fork<0 andµ= 0, it is proven in [ 20] that the solution of ( 1.1), in the subcrit- ical case (1 <p≤pG(N(1−k))), blows up in finite time giving hence a lifespan estimate of the maximal existence time. This is equivalent to say that, for 1 <p≤pG(N(1−k)), we have the nonexistence of the solution of ( 1.1). However, the aforementioned result was recently improved in [ 18] thanks to the construction of adequate test functions. The new region obtained in [ 18] gives a plausible characterization of the critical exponent, namely (1.4) p≤pT(N,k) := 1+2 (1−k)(N−1)+k. Very recently, it is proved in [ 12] with different approaches, as an application of the case of mixed nonlinearities, that results similar to the above for the problem ( 1.1) with k<0 andµ= 0 hold. We consider now the case µ>0 andk= 0 in (1.2). Hence, for a small µ, the solution of (1.2) behaves like a wave. In fact, the damping produces a shifting by µ>0 on the dimensionNfor the value of the critical power, see e.g. [ 16,24,30,31], and [5,6] for the caseµ= 2 andN= 2,3. The global existence for µ= 2 is proven in [ 5,6,21]. 2However, for µlarge, the equation ( 1.2) is of a parabolic type and the behavior is like a heat-type equation; see e.g. [ 3,4,33]. On the other hand, for the solution of ( 1.1) withµ>0 andk= 0, in [19] a blow-up result is proved for 1 <p≤pG(N+2µ) and upper bound estimates for the lifespan are given as well. Later, this result was improved in [ 25], wherepG(N+µ) is found as upper bound forµ≥2. Recently, an improvement is obtained in [ 10] stating that the critical value forpis given by pG(N+µ) for allµ >0. This should be the optimal threshold value that needs to be rigorously proved by completing the present blow-up result with a global existence one when the exponent pis beyond the critical value. We focus in this article on the blow-up of the solution of ( 1.1) fork∈[0,1). Our target is to give the upper bound, denoted here by pE=pE(N,k,µ), delimiting a new blow-up region for the Einstein - de Sitter spacetime equation ( 1.1). First, as observed for the equation ( 1.2), where the damping produces a shift in q0in the dimensional parameter of magnitudeµ 1−k, we expect that the same phenomenon holds for ( 1.1). In other words, we predict that the upper bound pE=pE(N,k,µ) satisfies (1.5) pE(N,k,µ) =pE(N+µ 1−k,k,0). Using an explicit representation formula and Zhou’s approach to pro ving the blow-up on a certain characteristic line, in [ 13], we proved that (1.6) pE(N,k,0) =pT(N,k), wherepTis defined by ( 1.4). Now, in view of ( 1.5) and (1.6), we await, for the solution of ( 1.1) withk∈[0,1) and µ>0, that (1.7) pE=pE(N,k,µ) := 1+2 (1−k)(N−1)+k+µ. As we have mentioned, in [ 13] we proved that ( 1.7) holds true under some sign assumptions for the data for µ= 0, but also for µ= 2 (cf. Theorems 1.1 and 1.2). We aim in the present work to extend this result for all µ>0, and show that the upper bound value for pis in fact given by ( 1.7). We think that pE(N,k,µ), forksmall, characterizes the limiting value between the existence and nonexist ence regions of the solution of ( 1.1). However, it is clear that this limiting exponent does not reach the optimal one in view of the very recent results in [ 28]. Finally, we recall here that the wave in ( 1.1) has a speed of propagation dependent of time. Therefore, this time-dependent speed of propagation te rm can be seen, after 3rescaling (see ( 1.9) below), as a scale-invariant damping. Let v(x,τ) =u(x,t), where (1.8) τ=φk(t) :=t1−k 1−k. Hence, we can easily see that v(x,τ) satisfies the following equation: vττ−∆v+µ−k (1−k)τ∂τv=Ck,pτµk(p−2)|∂τv|p,inRN×[1/(1−k),∞), (1.9) whereµk:=−k 1−kandCk,p= (1−k)µk(p−2). Moreover, thanks to the above transforma- tion, we can use the methods carried out in some earlier works [ 2,9,10,11,12] to build the proof of our main result. The rest of the paper is arranged as follows. First, we state in Sect ion2the weak formulation of ( 1.1) in the energy space, and then we give the main theorem. Section 3is concerned with some technical lemmas that we will use to prove the main result. Finally, Section 4is assigned to the proof of Theorem 2.2which constitutes the main result of this article. 2.Nonexistence Result First, we define in the sequel the energy solution associated with ( 1.1). Definition 2.1. Letf∈H1(RN)andg∈L2(RN). The function uis said to be an energy solution of ( 1.1) on[1,T)if /braceleftBigg u∈ C([1,T),H1(RN))∩C1([1,T),L2(RN)), ut∈Lp loc((1,T)×RN), satisfies, for all Φ∈ C∞ 0(RN×[1,T))and allt∈[1,T), the following equation: (2.1)/integraldisplay RNut(x,t)Φ(x,t)dx−ε/integraldisplay RNg(x)Φ(x,1)dx −/integraldisplayt 1/integraldisplay RNut(x,s)Φt(x,s)dxds+/integraldisplayt 1s−2k/integraldisplay RN∇u(x,s)·∇Φ(x,s)dxds +/integraldisplayt 1/integraldisplay RNµ sut(x,s)Φ(x,s)dxds=/integraldisplayt 1/integraldisplay RN|ut(x,s)|pΦ(x,s)dxds, 4and the condition u(x,1) =εf(x)is fulfilled in H1(RN). A straightforward computation shows that (2.1)is equivalent to (2.2)/integraldisplay RN/bracketleftbig ut(x,t)Φ(x,t)−u(x,t)Φt(x,t)+µ tu(x,t)Φ(x,t)/bracketrightbig dx /integraldisplayt 1/integraldisplay RNu(x,s)/bracketleftbigg Φtt(x,s)−s−2k∆Φ(x,s)−∂ ∂s/parenleftBigµ sΦ(x,s)/parenrightBig/bracketrightbigg dxds =/integraldisplayt 1/integraldisplay RN|ut(x,s)|pψ(x,s)dxds+ε/integraldisplay RN/bracketleftbig −f(x)Φt(x,1)+(µf(x)+g(x))Φ(x,1)/bracketrightbig dx. Remark 2.1.Obviously, we can choose a test function Φ which is not compactly sup - ported in view of the fact that the initial data fandgare supported on BRN(0,R). In fact, we have supp( u)⊂ {(x,t)∈RN×[1,∞) :|x| ≤φk(t)+R}. The blow-up region and the lifespan estimate of the solutions of ( 1.1) constitute the objective of our main result which is the subject of the following theo rem. Theorem 2.2. Letµ >0,p∈(1,pE(N,k,µ)],N≥1andk∈[0,1). Suppose thatf∈H1(RN)andg∈L2(RN)are functions which are non-negative, with com- pact support on B(0RN,R), and non-vanishing everywhere. Then, there exists ε0= ε0(f,g,N,R,p,k,µ )>0such that for any 0< ε≤ε0the solution uto(1.1)which satisfies supp(u)⊂ {(x,t)∈RN×[1,∞) :|x| ≤φk(t)+R}, blows up in finite time Tε, and Tε≤/braceleftBigg Cε−2(p−1) 2−((1−k)(N−1)+k+µ)(p−1)for1<p<p E(N,k,µ), exp/parenleftbig Cε−(p−1)/parenrightbig forp=pE(N,k,µ), wherepE(N,k,µ)is given by (1.7)andCis a positive constant independent of ε. Remark 2.2.The results stated in Theorem 2.2hold true for k <0 andµ >0; see [1] where a more general model with mass term is studied. Remark2.3.After completing the first version of the present manuscript, we r eceived a draft version of [ 28], where problem ( 1.1) is studied, among other things. In particular, forn+1 n+2< k <1 andµ∈[0,(n+2)k−(n+1)) the upper bound for pin the blow-up result is improved in [ 28] by proving the nonexistence of global solutions to ( 1.1) for 1<p<1+1 (1−k)n+µ. 3.Auxiliary results It is worth mentioning that the choice of the test function, that we will use in the functionals that will be introduced later on, is crucial here. Natura lly, in terms of dynamics of the solution of ( 1.1), the more accurate the choice of the test function is, 5the better lifespan estimate we obtain. This is why we choose in the fo llowing to include all the linear terms inherited from ( 1.1). First, we introduce the function ρ(t) [22] given by (3.1) ρ(t) :=t1+µ 2Kµ−1 2(1−k)/parenleftbiggt1−k 1−k/parenrightbigg ,∀t≥1, whereKν(t) is the modified Bessel function of second kind defined as (3.2) Kν(t) =/integraldisplay∞ 0exp(−tcoshζ)cosh(νζ)dζ, ν∈R. It is easy to see that ρ(t) satisfies (3.3)d2ρ(t) dt2−t−2kρ(t)−d dt/parenleftBigµ tρ(t)/parenrightBig = 0,∀t≥1. Second, we define the function ϕ(x) by (3.4) ϕ(x) := /integraldisplay SN−1ex·ωdωforN≥2, ex+e−xforN= 1; note thatϕ(x) is introduced in [ 34] and satisfies ∆ ϕ=ϕ. Hence, the function ψ(x,t) :=ϕ(x)ρ(t) verifies the following equation: (3.5) ∂2 tψ(x,t)−t−2k∆ψ(x,t)−∂ ∂t/parenleftBigµ tψ(x,t)/parenrightBig = 0. In the following we enumerate some properties of the function ρ(t) that we will use later on in the proof of our main result. Lemma 3.1. The next properties hold true for the function ρ(t). (i)The function ρ(t)is positive on [1,∞). Moreover, for all t≥1, there exists a constantC1such thatρ(t)satisfies (3.6) C−1 1tk+µ 2exp(−φk(t))≤ρ(t)≤C1tk+µ 2exp(−φk(t)), whereφk(t)is given by (1.8). (ii)We have (3.7) lim t→+∞/parenleftbiggtkρ′(t) ρ(t)/parenrightbigg =−1. Proof.First, we recall here the definition of ρ(t), as in (3.1), and (1.8) (3.8) ρ(t) =t1+µ 2Kµ−1 2(1−k)(φk(t)),∀t≥1. 6Hence, the positivity of ρ(t) is straightforward thanks to ( 3.2). On the other hand, from [8], we have the following property for the function Kµ(t): (3.9) Kµ(t) =/radicalbiggπ 2te−t(1+O(t−1)),ast→ ∞. Combining ( 3.8) and(3.9), andagainremembering thedefinition of φk(t), given by ( 1.8), and the fact that k<1, we conclude ( 3.6). The assertion (i)is thus proven. Now, to prove (ii), using (3.8) we observe that (3.10)ρ′(t) ρ(t)=µ+1 2t+t−kK′ µ−1 2(1−k)(φk(t)) Kµ−1 2(1−k)(φk(t)). Exploiting the well-known identity for the modified Bessel function, (3.11)d dzKν(z) =−Kν+1(z)+ν zKν(z), and combining ( 3.10) and (3.11) yields (3.12)ρ′(t) ρ(t)=µ t−t−kK1+µ−1 2(1−k)(φk(t)) Kµ−1 2(1−k)(φk(t)). From (3.9) and (3.12), and using the fact that k∈[0,1), we deduce ( 3.7). This ends the proof of Lemma 3.1. /square Throughout this article, the use of a generic parameter Cis designed to denote a positive constant that might be dependent on p,q,k,N,R,f,g,µ but independent of ε. The value of the constant Cmay change from line to line. Nevertheless, when it is nec- essary, we will clearly mention the expression of Cin terms of the parameters involved in our problem. A classical estimate result for the function ψ(x,t) is stated in the next lemma. Lemma 3.2 ([34]).Letr>1. Then, there exists a constant C=C(N,µ,R,p,k,r )>0 such that (3.13)/integraldisplay |x|≤φk(t)+R/parenleftBig ψ(x,t)/parenrightBigr dx≤Cρr(t)erφk(t)(1+φk(t))(2−r)(N−1) 2,∀t≥1. Letube a solution to ( 1.1) for which we introduce the following functionals: (3.14) U(t) :=/integraldisplay RNu(x,t)ψ(x,t)dx, and (3.15) V(t) :=/integraldisplay RNut(x,t)ψ(x,t)dx. 7The first lower bounds for U(t) andV(t) are respectively given by the following two lemmas where, for tlarge enough, we will prove that ε−1t−kU(t) andε−1V(t) are two bounded from below functions by positive constants. Lemma 3.3. Letube a solution of (1.1). Assume in addition that the corresponding initial data satisfythe assumptionsas in Theorem 2.2. Then, there exists T0=T0(k,µ)> 2such that (3.16) U(t)≥CUεtk,for allt≥T0, whereCUis a positive constant that may depend on f,g,N,µ,Randk, but not on ε. Proof.Lett∈(1,T). Substituting in ( 2.2) Φ(x,t) byψ(x,t), we obtain (3.17)/integraldisplay RN/bracketleftbig ut(x,t)ψ(x,t)−u(x,t)ψt(x,t)+µ tu(x,t)ψ(x,t)/bracketrightbig dx =/integraldisplayt 1/integraldisplay RN|ut(x,s)|pψ(x,s)dxds+εC(f,g), where (3.18) C(f,g) :=ρ(1)/integraldisplay RN/bracketleftbig/parenleftbig µ−ρ′(1) ρ(1)/parenrightbig f(x)+g(x)/bracketrightbig φ(x)dx. Note thatC(f,g) is positive thanks to the fact that ρ(1) andµ−ρ′(1) ρ(1)are positive as well (in view of ( 3.12)) and the sign of the initial data. Hence, recall the definition of U, as in (3.14), and (3.4), (3.17) gives (3.19) U′(t)+Γ(t)U(t) =/integraldisplayt 1/integraldisplay RN|ut(x,s)|pψ(x,s)dxds+εC(f,g), where (3.20) Γ( t) :=µ t−2ρ′(t) ρ(t). Neglecting the nonlinear term in ( 3.19), then multiplying the resulting equation from (3.19) bytµ ρ2(t)and integrating on (1 ,t), we get U(t)≥ U(1)ρ2(t) tµρ2(1)+εC(f,g)ρ2(t) tµ/integraldisplayt 1sµ ρ2(s)ds. (3.21) From (3.1), the definition of φk(t), given by ( 1.8), and using the fact that U(1)>0, the estimate ( 3.21) implies that U(t)≥εC(f,g)tK2 µ−1 2(1−k)(φk(t))/integraldisplayt t/21 sK2 µ−1 2(1−k)(φk(s))ds,∀t≥2. (3.22) 8In view of ( 3.9), we deduce the existence of T0=T0(k,µ)>2 such that φk(t)K2 µ−1 2(1−k)(φk(t))>π 4e−2φk(t)andφk(t)−1K−2 µ−1 2(1−k)(φk(t))>1 πe2φk(t),∀t≥T0/2.(3.23) Inserting ( 3.23) in (3.22) and using ( 1.8), we obtain that U(t)≥εC(f,g) 4tke−2φk(t)/integraldisplayt t/2φ′ k(s)e2φk(s)ds (3.24) ≥εC(f,g) 8tk[1−e−2(φk(t)−φk(t/2))],∀t≥T0. Thanks to ( 1.8) and the fact that k <1, we observe that t/ma√sto→1−e−2(φk(t)−φk(t/2))is an increasing function on ( T0,∞), hence, its minimum is achieved at t=T0. Therefore we deduce that U(t)≥εκC(f,g)tk,∀t≥T0, (3.25) where κ:=1 8/parenleftbigg 1−exp/parenleftbigg −(2−2k)T1−k 0 1−k/parenrightbigg/parenrightbigg . Hence, Lemma 3.3is now proved. /square The next lemma gives the lower bound of the functional V(t). Lemma 3.4. Assume that the initial data are as in Theorem 2.2. Foruan energy solution of (1.1), there exists T1=T1(k,µ)>T0such that (3.26) V(t)≥CVε,for allt≥T1, whereCVis a positive constant depending on f,g,N,µ,Randk, but not on ε. Proof.Lett∈[1,T). Recall the definitions of UandV, given respectively by ( 3.14) and (3.15), (3.4) and the identity (3.27) U′(t)−ρ′(t) ρ(t)U(t) =V(t). Hence, the equation ( 3.19) yields (3.28) V(t)+/bracketleftbiggµ t−ρ′(t) ρ(t)/bracketrightbigg U(t) =/integraldisplayt 1/integraldisplay RN|ut(x,s)|pψ(x,s)dxds+εC(f,g). A differentiation in time of the equation ( 3.28) gives V′(t)+/bracketleftbiggµ t−ρ′(t) ρ(t)/bracketrightbigg U′(t)−/parenleftbiggµ t2+ρ′′(t)ρ(t)−(ρ′(t))2 ρ2(t)/parenrightbigg U(t) =/integraldisplay RN|ut(x,t)|pψ(x,t)dx.(3.29) 9Now, thanks to ( 3.3) and (3.27), we deduce from ( 3.29) that V′(t)+/bracketleftbiggµ t−ρ′(t) ρ(t)/bracketrightbigg V(t) =t−2kU(t)+/integraldisplay RN|ut(x,t)|pψ(x,t)dx, (3.30) that we rewrite as /parenleftbigg tµV(t) ρ(t)/parenrightbigg′ =tµ ρ(t)/parenleftbigg t−2kU(t)+/integraldisplay RN|ut(x,t)|pψ(x,t)dx/parenrightbigg ,∀t≥1. (3.31) An integration of ( 3.31) over (1,t) implies that tµV(t) ρ(t)=V(1) ρ(1)+/integraldisplayt 1sµ ρ(s)/parenleftbigg s−2kU(s)+/integraldisplay RN|ut(x,s)|pψ(x,s)dx/parenrightbigg ds,∀t≥1. (3.32) Thanks to the fact that V(1)≥0,ρ(1)>0 and using the lower bound of Uas in (3.16), we infer that V(t)≥ρ(t) tµ/integraldisplayt 1sµ ρ(s)/parenleftbigg CUεs−k+/integraldisplay RN|ut(x,s)|pψ(x,s)dx/parenrightbigg ds,∀t≥T0. (3.33) Therefore the estimate ( 3.33) gives V(t)≥CUερ(t) tµ/integraldisplayt t/2s−k+µ ρ(s)ds,∀t≥2T0. (3.34) For convenience, we rewrite ( 3.23) as follows: /radicalbig φk(t)Kµ−1 2(1−k)(φk(t))>√π 2e−φk(t)and1/radicalbig φk(t)K−1 µ−1 2(1−k)(φk(t))>1√πeφk(t),∀t≥T0/2.(3.35) Using the expressions of ρ(t) andφk(t), given respectively by ( 3.1) and (1.8), we deduce that V(t)≥εCU/parenleftbigg1 2/parenrightbiggµ 2+1 e−φk(t)/integraldisplayt t/2φ′ k(s)eφk(s)ds (3.36) ≥εCU/parenleftbigg1 2/parenrightbiggµ 2+1 [1−e−(φk(t)−φk(t/2))],∀t≥2T0. Analogously as in Lemma 3.3, we have V(t)≥CVε,∀t≥T1:= 2T0, (3.37) where CV:=CU/parenleftbigg1 2/parenrightbiggµ 2+1/parenleftbigg 1−exp/parenleftbigg −(1−2k−1)(2T0)1−k 1−k/parenrightbigg/parenrightbigg . This completes the proof of Lemma 3.4. /square 104.Proof of Theorem 2.2. This section is dedicated to proving the main result in Theorem 2.2which exposes the blow-up dynamics of the solution of ( 1.1). Hence, to prove the blow-up result for (1.1) we will use ( 3.28) and (3.30). For this purpose, we multiply ( 3.28) byαρ′(t) ρ(t), and subtract the resulting equation from ( 3.30). Therefore we obtain for a certain α≥0, whose range will be fixed afterward, (4.1) V′(t)+/bracketleftbiggµ t−(1+α)ρ′(t) ρ(t)/bracketrightbigg V(t) =−εαρ′(t) ρ(t)C(f,g)+/bracketleftbigg t−2k+αρ′(t) ρ(t)/parenleftbiggµ t−ρ′(t) ρ(t)/parenrightbigg/bracketrightbigg U(t) +/integraldisplay RN|ut(x,t)|pψ(x,t)dx−αρ′(t) ρ(t)/integraldisplayt 1/integraldisplay RN|ut(x,s)|pψ(x,s)dxds,∀t≥1. Using (3.7), we can choose ˜T2≥T1(T1is given in Lemma 3.4) such that (4.2)V′(t)+/bracketleftbiggµ t−(1+α)ρ′(t) ρ(t)/bracketrightbigg V(t)≥εαt−k 2C(f,g)+(1−4α)t−2kU(t) +/integraldisplay RN|ut(x,t)|pψ(x,t)dx+αt−k 2/integraldisplayt 1/integraldisplay RN|ut(x,s)|pψ(x,s)dxds,∀t≥˜T2. From now on the parameter αis chosen in (1 /7,1/4). Thanks to ( 3.16), the estimate (4.2) leads to the following lower bound: (4.3)V′(t)+/bracketleftbiggµ t−(1+α)ρ′(t) ρ(t)/bracketrightbigg V(t)≥εαt−k 2C(f,g)+/integraldisplay RN|ut(x,t)|pψ(x,t)dx +αt−k 2/integraldisplayt 1/integraldisplay RN|ut(x,s)|pψ(x,s)dxds,∀t≥˜T2. Now, we introduce the following functional: H(t) :=C2ε+1 16/integraldisplayt ˜T3/integraldisplay RN|ut(x,s)|pψ(x,s)dxds, whereC2:= min(αC(f,g)/4(1 +α),CV) (CVis given by Lemma 3.4) and we choose ˜T3>˜T2such that (4.4)α 2C(f,g)−C2tk/parenleftbiggµ t−(1+α)ρ′(t) ρ(t)/parenrightbigg ≥0, and (4.5)α 2−1 16tk/parenleftbiggµ t−(1+α)ρ′(t) ρ(t)/parenrightbigg ≥0, for allt≥˜T3(this is possible thanks to ( 3.7), the definition of C2and the fact that α∈(1/7,1/4)). Let F(t) :=V(t)−H(t), 11which satisfies (4.6)F′(t)+/bracketleftbiggµ t−(1+α)ρ′(t) ρ(t)/bracketrightbigg F(t)≥15 16/integraldisplay RN|ut(x,t)|pψ(x,t)dx +/bracketleftbiggα 2−1 16/parenleftbiggµ t1−k−(1+α)tkρ′(t) ρ(t)/parenrightbigg/bracketrightbigg t−k/integraldisplayt ˜T3/integraldisplay RN|ut(x,s)|pψ(x,s)dxds +/bracketleftbiggα 2C(f,g)−C2/parenleftbiggµ t1−k−(1+α)tkρ′(t) ρ(t)/parenrightbigg/bracketrightbigg εt−k,∀t≥˜T3. Thanks to ( 4.4) and (4.5), we easily conclude that (4.7) F′(t)+/bracketleftbiggµ t−(1+α)ρ′(t) ρ(t)/bracketrightbigg F(t)≥0,∀t≥˜T3. Multiplying ( 4.7) bytµ ρ1+α(t)and integrating over ( ˜T3,t), we get F(t)≥ F(˜T3)˜Tµ 3ρ1+α(t) tµρ1+α(˜T3),∀t≥˜T3. (4.8) Hence, we see that F(˜T3) =V(˜T3)−C2ε≥ V(˜T3)−CVε≥0 in view of Lemma 3.4and the definition of C2implying that C2≤CV. Therefore we deduce that (4.9) V(t)≥H(t),∀t≥˜T3. Now, employing the H¨ older inequality and the estimates ( 3.13) and (3.15), we obtain (4.10)H′(t)≥1 16Vp(t)/parenleftbigg/integraldisplay |x|≤φk(t)+Rψ(x,t)dx/parenrightbigg−(p−1) ≥CVp(t)ρ−(p−1)(t)e−(p−1)φk(t)(φk(t))−(N−1)(p−1) 2. In view of ( 3.6), we see that (4.11) H′(t)≥CVp(t)t−[(N−1)(1−k)+k+µ](p−1) 2,∀t≥˜T3. From the above estimate and ( 4.9), we have (4.12) H′(t)≥CHp(t)t−[(N−1)(1−k)+k+µ](p−1) 2,∀t≥˜T3. SinceH(˜T3) =C2ε >0, we easily obtain the blow-up in finite time for the functional H(t), and consequently the one for V(t) due to ( 4.9). The proof of Theorem 2.2is now achieved. aknowledgments The authors are deeply thankful to the anonymous reviewer for t he valuable re- marks that improved the paper. A. Palmieri is supported by the Jap an Society for the 12Promotion of Science (JSPS) – JSPS Postdoctoral Fellowship for Re search in Japan (Short-term) (PE20003). 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Sideris, Global behavior of solutions to nonlinear wave equations in three space dimensions , Comm. Partial Differential Equations, 8(12) (1983), 1291–1323. 2 [27] K. Tsutaya and Y. Wakasugi, Blow up of solutions of semilinear wave equations in Friedma nn- Lemaˆ ıtre-Robertson-Walker spacetime. J. Math. Phys., 61(2020), 091503. 2 [28] K. Tsutayaand Y. Wakasugi, On Glassey’s conjecture for semilinear wave equations in Fr iedmann- Lemaˆ ıtre-Robertson-Walker spacetime. Boundary Value Problems, 2021(2021), 94. 3,5 [29] K. Tsutaya and Y. Wakasugi, On heatlike lifespan of solutions of semilinear wave equati ons in Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime. J. Math. Anal. Appl., 500(2021), 125133. 2 [30] Z. Tu, and J. Lin, A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent, preprint, arXiv:1709.00866v2, 2017. 2 [31] Z. Tu, and J. Lin, Life-span of semilinear wave equations with scale-invaria nt damping: critical Strauss exponent case. Differential Integral Equations, 32(5-6) (2019), 249–264. 2 [32] N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac s ystem and wave equation with small data , Tsukuba J. Math., 22(1998), 193–211. 2 [33] K. Wakasugi, Critical exponent for the semilinear wave equation with sca le invariant damping. In: M. Ruzhansky , V. Turunen (Eds.) Fourier Analysis. Trends in Mathe matics. Birkh¨ auser, Cham (2014). https://doi.org/10.1007/978-3-319-02550-6 19. 3 [34] B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimen sions, J. Funct. Anal., 231(2006), 361–374. 6,7 [35] Y. Zhou, Blow-up of solutions to the Cauchy problem for nonlinear wav e equations , Chin. Ann. Math.,22B(3) (2001), 275–280. 2 141Department of Basic Sciences, Deanship of Preparatory Year and Supporting Stud- ies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 34212 Dammam, Saudi Ara- bia. 2Department of Mathematics, University of Pisa, Largo B. Pon tecorvo 5, 56127 Pisa, Italy. 3Current address: Mathematical Institute, Tohoku University, Aoba, Sendai 9 80-8578, Japan. Email address :mmhamouda@iau.edu.sa (M. Hamouda) Email address :mahamza@iau.edu.sa (M.A. Hamza) Email address :alessandro.palmieri.math@gmail.com (A. Palmieri) 15

2211.13486v1.Influence_of_non_local_damping_on_magnon_properties_of_ferromagnets.pdf

In uence of non-local damping on magnon properties of ferromagnets Zhiwei Lu,1,I. P. Miranda,2,Simon Streib,2Manuel Pereiro,2Erik Sj oqvist,2 Olle Eriksson,2, 3Anders Bergman,2Danny Thonig,3, 2and Anna Delin1, 4 1Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, AlbaNova University Center, SE-10691 Stockholm, Swedeny 2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden 3School of Science and Technology, Orebro University, SE-701 82, Orebro, Sweden 4SeRC (Swedish e-Science Research Center), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden (Dated: November 28, 2022) We study the in uence of non-local damping on magnon properties of Fe, Co, Ni and Fe 1xCox (x= 30%;50%) alloys. The Gilbert damping parameter is typically considered as a local scalar both in experiment and in theoretical modelling. However, recent works have revealed that Gilbert damping is a non-local quantity that allows for energy dissipation between atomic sites. With the Gilbert damping parameters calculated from a state-of-the-art real-space electronic structure method, magnon lifetimes are evaluated from spin dynamics and linear response, where a good agreement is found between these two methods. It is found that non-local damping aects the magnon lifetimes in dierent ways depending on the system. Specically, we nd that in Fe, Co, and Ni the non-local damping decreases the magnon lifetimes, while in Fe 70Co30and Fe 50Co50an opposite, non-local damping eect is observed, and our data show that it is much stronger in the former. INTRODUCTION In recent years, there has been a growing interest in magnonics, which uses quasi-particle excitations in mag- netically ordered materials to perform information trans- port and processing on the nanoscale. Comparing to the conventional information device, the magnonics device exhibits lower energy consumption, easier integrability with complementary metal-oxide semiconductor (CMOS) structure, anisotropic properties, and ecient tunability by various external stimuli to name a few [1{10]. Yttrium iron garnet (YIG) [11] as well as other iron garnets with rare-earth elements (Tm, Tb, Dy, Ho, Er) [12] are very promising candidates for magnonics device applications due to their low energy dissipation properties and, thus, long spin wave propagation distances up to tens of m. Contrary, the damping of other materials for magnonics, like CoFeB, is typically two orders of magnitude higher compared to YIG [12], leading to much shorter spin wave propagation distances. A clear distinction can be made between materials with an ultra-low damping parame- ter, like in YIG, and those with a signciantly larger, but still small, damping parameter. Materials like YIG are insulating, which hinders many of the microscopic mechanisms for damping, resulting in the low observed damping parameter. In contrast, materials like CoFeB are metallic. In research projects that utilize low damp- ing materials, YIG and similar non-metallic low damping systems are typically favored. However, metallic systems have an advantage, since magnetic textures can easily by in uenced by electrical currents. Hence, there is good These two authors contributed equally yCorresponding author: zhiweil@kth.sereason to consider metallic systems for low damping ap- plications, even though their damping typically is larger than in YIG. One can conclude that Gilbert damping is one of the major bottlenecks for the choice of mate- rial in magnonics applications and a detailed experimen- tal as well as theoretical characterisation is fundamen- tal for this eld of research, especially for metallic sys- tems. Thus, a more advanced and detailed understand- ing of Gilbert damping is called for, in order to overcome this obstacle for further development of magnonics-based technology. Whereas most studies consider chemical modications of the materials in order to tune damping [13, 14], only a few focus on the fundamental physical properties as well as dependencies of the Gilbert damping. Often Gilbert damping is considered as a phenomenological scalar pa- rameter in the equation of motion of localized atom- istic magnetic moments, i.e. the Landau-Lifshitz-Gilbert (LLG) equation [15]. However, from using the general Rayleigh dissipation function in the derivation proposed by Gilbert [16], it was theoretically found that the Gilbert damping should be anisotropic, a tensor, and non-local. Furthermore, it depends on the temperature and, thus, on underlying magnon as well as phonon congurations [17{20]. This is naturally built into the multiple theoret- ical methods developed to predict the damping parame- ter, including breathing Fermi surface model [21], torque correlation model [22], and linear response formulation [23]. For instance, the general Gilbert damping tensor as a function of the non-collinear spin conguration has been proposed in Ref. 24. Nonetheless, an experimental verication is still miss- ing due to lacking insights into the impact of the gen- eralised damping on experimental observables. In a re- cent experiment, however, the anisotropic behavior of the damping has been conrmed for Co 50Fe50thin lms andarXiv:2211.13486v1 [cond-mat.mtrl-sci] 24 Nov 20222 was measured to be of the order of 400% [25], with respect to changing the magnetization direction. Changes of Gilbert damping in a magnetic domain wall and, thus, its dependency on the magnetic conguration was measured in Ref. [26] and tted to the Landau-Lifshitz-Baryakhtar (LLBar) equation, which includes non-locality of the damping by an additional dissipation term proportional to the gradient of the magnetisation [27{29]. However, the pair-wise non-local damping ijhas not yet been measured. The most common experimental techniques of evaluat- ing damping are ferromagnetic resonance (FMR) [30] and time-resolved magneto-optical Kerr eect (TR-MOKE) [31]. In these experiments, Gilbert damping is related to the relaxation rate when (i)slightly perturbing the coherent magnetic moment out of equilibrium by an ex- ternal magnetic eld [32] or (ii)when disordered mag- netic moments remagnetise after pumping by an ultrafast laser pulse [33]. Normally, in case (i)the non-locality is suppressed due to the coherent precession of the atomic magnetic moments. However, this coherence can be per- turbed by temperature, making non-locality in principle measurable. One possible other path to link non-local damping with experiment is magnon lifetimes. Theoret- ically, the magnon properties as well as the impact of damping on these properties can be assessed from the dynamical structure factor, and atomistic spin-dynamics simulations have been demonstrated to yield magnon dis- persion relations that are in good agreement with exper- iment [34]. In experiment, neutron scattering [35] and electron scattering [36] are the most common methods for probing magnon excitations, where the linewidth broad- ening of magnon excitations is related to damping and provides a way to evaluate the magnon lifetimes [37]. It is found in ferromagnets that the magnon lifetimes is wave vector (magnon energy) dependent [38{40]. It has been reported that the magnon energy in Co lms is nearly twice as large as in Fe lms, but they have similar magnon lifetimes, which is related to the intrinsic damping mech- anism of materials [41]. However, this collective eect of damping and magnon energy on magnon lifetimes is still an open question. The study of this collective eect is of great interest for both theory and device applications. Here, we report an implementation for solving the stochastic Landau-Lifshitz-Gilbert (SLLG) equation in- corporating the non-local damping. With the dynamical structure factor extracted from the spin dynamics sim- ulations, we investigate the collective eect of non-local damping and magnon energy on the magnon lifetimes. We propose an ecient method to evaluate magnon life- times from linear response theory and verify its validity. The paper is organized as follows. In Sec. I, we give the simulation details of the spin dynamics, the adiabatic magnon spectra and dynamical structure factor, and the methodology of DFT calculations and linear response. Sec. II presents the non-local damping in real-space, non- local damping eects on the spin dynamics and magnon properties including magnon lifetimes of pure ferromag-nets (Fe, Co, Ni), and Fe 1xCox(x= 30%;50%) alloys. In Sec. III, we give a summary and an outlook. I. THEORY A. Non-local damping in atomistic spin dynamics The dynamical properties of magnetic materials at - nite temperature have been so far simulated from atom- istic spin dynamics by means of the stochastic Landau- Lifshitz-Gilbert equation with scalar local energy dissipa- tion. Here, the time evolution of the magnetic moments mi=mieiat atom site iis well described by: @mi @t=mi [Bi+bi(t)] + mi@mi @t ;(1) where is the gyromagnetic ratio. The eective eld Bi acting on each magnetic moment is obtained from: Bi=@H @mi: (2) The here considered spin-Hamiltonian Hconsists of a Heisenberg spin-spin exchange: H=X i6=jJijei
ej: (3) Here,Jij{ the Heisenberg exchange parameter { cou- ples the spin at site iwith the spin at site jand is cal- culated from rst principles (see Section I C). Further- more,is the scalar phenomenological Gilbert damp- ing parameter. Finite temperature Tis included in Eq. (1) via the uctuating eld bi(t), which is modeled by uncorrelated Gaussian white noise: hbi(t)i= 0 and b i(t)b j(t0) = 2Dij(tt0), whereis the Kro- necker delta, i;jare site and ;=fx;y;zgCartesian indices. Furthermore, the uctuation-dissipation theo- rem givesD=kBT mi[42], with the Boltzman constant kB. A more generalized form of the SLLG equation that includes non-local tensorial damping has been reported in previous studies [20, 43, 44] and is: @mi @t=mi0 @ [Bi+bi(t)] +X jij mj@mj @t1 A;(4) which can be derived from Rayleigh dissipation func- tional in the Lagrange formalism used by Gilbert [16]. In the presence of non-local damping, the Gaussian uc- tuating eld fullls [43, 45, 46] b i(t)b j(t0) = 2D ij(tt0); (5) withD ij= ijkBT mi. The damping tensor  ijmust be positive denite in order to be physically-dened. Along3 with spatial non-locality, the damping can also be non- local in time, as discussed in Ref. [47]. To prove the uctuation-dissipation theorem in Eq. (5), the Fokker- Planck equation has to be analysed in the presence of non-local damping, similar to Ref. [15]. This is, however, not the purpose of this paper. Instead, we will use the approximation  ij=1 3Trfiigijwithin the diusion constantD. Such an approximation is strictly valid only in the low temperature limit. To solve this SLLG equation incorporating the non- local damping, we have implemented an implicit mid- point solver in the UppASD code [48]. This iterative x-point scheme converges within an error of 1010B, which is typically equivalent to 6 iteration steps. More details of this solver are provided in Appendix A. The initial spin conguration in the typical N= 202020 supercell with periodic boundary conditions starts from totally random state. The spin-spin exchange interac- tions and non-local damping parameters are included up to at least 30 shells of neighbors, in order to guarantee the convergence with respect to the spatial expansion of these parameters (a discussion about the convergence is given in Section II A). Observables from our simulations are typically the average magnetisation M=1 NPN imi as well as the magnon dispersion. B. Magnon dispersion Two methods to simulate the magnon spectrum are applied in this paper: i)the dynamical structure factor andii)frozen magnon approach. For the dynamical structure factor S(q;!) at nite temperature and damping [34, 49], the spatial and time correlation function between two magnetic moments iat positionrandjat positionr0as well as dierent time 0 andtis expressed as: C(rr0;t) =hm r(t)m r0(0)ihm r(t)ihm r0(0)i:(6) Hereh
idenotes the ensemble average and are Carte- sian components. The dynamical structure factor can be obtained from the time and space Fourier transform of the correlation function, namely: S(q;!) =1p 2NX r;r0eiq
(rr0)Z1 1ei!tC(rr0;t)dt: (7) The magnon dispersion is obtained from the peak positions of S(q;!) along dierent magnon wave vectors qin the Brillouin zone and magnon energies !. It should be noted that S(q;!) is related to the scattering intensity in inelastic neutron scattering experiments [50]. The broadening of the magnon spectrum correlates to the lifetime of spin waves mediated by Gilbert damping as well as intrinsic magnon-magnon scattering processes. Good agreement between S(q;!) and experiment hasbeen found previously [34]. The second method { the frozen magnon approach { determines the magnon spectrum directly from the Fourier transform of the spin-spin exchange parameters Jij[51, 52] and non-local damping ij. At zero tempera- ture, a time-dependent external magnetic eld is consid- ered, B i(t) =1 NX qB qeiq
Rii!t; (8) whereNis the total number of lattice sites and B q= Bx qiBy q. The linear response to this eld is then given by M q=(q;!)B q: (9) We obtain for the transverse dynamic magnetic suscep- tibility [53, 54] (q;!) = Ms !!qi!q; (10) with saturation magnetization Ms, spin-wave frequency !q=E(q)=~and damping q=X j0jeiq
(R0Rj): (11) We can extract the spin-wave spectrum from the imagi- nary part of the susceptibility, Im(q;!) = Msq! [!!q]2+2q!2; (12) which is equivalent to the correlation function S(q;!) due to the uctuation-dissipation theorem [55]. We nd that the spin-wave lifetime qis determined by the Fourier transform of the non-local damping (for q1), q= q!q: (13) The requirement of positive deniteness of the damping matrixijdirectly implies q>0, sinceijis diago- nalized by Fourier transformation due to translational invariance. Hence, q>0 is a criterion to evaluate whether the damping quantity in real-space is physically consistent and whether rst-principles calculations are well converged. If q<0 for some wave vector q, energy is pumped into the spin system through the correspon- dent magnon mode, preventing the system to fully reach the saturation magnetization at suciently low temper- atures. The eective damping 0of the FMR mode at q= 0 is determined by the sum over all components of the damping matrix, following Eqn.11, tot0=X j0j: (14) Therefore, an eective local damping should be based ontotif the full non-local damping is not taken into account.4 C. Details of the DFT calculations The electronic structure calculations, in the framework of density functional theory (DFT), were performed us- ing the fully self-consistent real-space linear mun-tin orbital in the atomic sphere approximation (RS-LMTO- ASA) [56, 57]. The RS-LMTO-ASA uses the Haydock recursion method [58] to solve the eigenvalue problem based on a Green's functions methodology directly in real-space. In the recursion method, the continued frac- tions have been truncated using the Beer-Pettifor termi- nator [59], after a number LLof recursion levels. The LMTO-ASA [60] is a linear method which gives precise results around an energy E, usually taken as the center of thes,panddbands. Therefore, as we calculate ne quantities as the non-local damping parameters, we here consider an expression accurate to ( EE)2starting from the orthogonal representation of the LMTO-ASA formalism [61]. For bcc FeCo alloys and bcc Fe we considered LL= 31, while for fcc Co and fcc Ni much higher LLvalues (51 and 47, respectively), needed to better describe the density of states and Green's functions at the Fermi level. The spin-orbit coupling (SOC) is included as a l
s [60] term computed in each variational step [62]. All calculations were performed within the local spin den- sity approximation (LSDA) exchange-functional (XC) by von Barth and Hedin [63], as it gives general magnetic information with equal or better quality as, e.g., the generalized gradient approximation (GGA). Indeed, the choice of XC between LSDA and GGA [64] have a mi- nor impact on the onsite damping and the shape of the qcurves, when considering the same lattice parame- ters (data not shown). No orbital polarization [65] was considered here. Each bulk system was modelled by a big cluster containing 55000 (bcc) and696000 (fcc) atoms located in the perfect crystal positions with the re- spective lattice parameters of a= 2:87A (bcc Fe and bcc Fe1xCox, suciently close to experimental observations [66]),a= 3:54A (fcc Co [20, 67]), and a= 3:52A (fcc Ni [68]). To account for the chemical disorder in the Fe70Co30and Fe 50Co50bulks, the electronic structure calculated within the simple virtual crystal approxima- tion (VCA), which has shown to work well for the fer- romagnetic transition metals alloys (particularly for el- ements next to each other in the Periodic Table, such as FeCo and CoNi) [69{76], and also describe in a good agreement the damping trends in both FeCo and CoNi (see Appendix C). As reported in Ref. [77], the total damping of site i, in uenced by the interaction with neighbors j, can be decomposed in two main contributions: the onsite (fori=j), and the non-local (for i6=j). Both can be calculated, in the collinear framework, by the followingexpression,  ij=CZ1 1()Tr ^T i^Aij(^T j)y^Aji dT!0K! CTr ^T i^Aij(F+i)(^T j)y^Aji(F+i) ; (15) where we dene ^Aij(+i) =1 2i(^Gij(+i)^Gy ji(+i)) the anti-Hermitian part of the retarded physical Green's functions in the LMTO formalism, and C=g mtia pre-factor related to the i-th site magnetization. The imaginary part, , is obtained from the terminated con- tinued fractions. Also in Eq. 15, ^T i= [ i;Hso] is the so-called torque operator [20] evaluated in each Cartesian direction;=fx;y;zgand at site i,() =@f() @is the derivative of the Fermi-Dirac distribution f() with respect to the energy ,g= 2 1 +morb mspin theg-factor (not considering here the spin-mixing parameter [78]), are the Pauli matrices, and mtiis the total magnetic moment of site i(mti=morbi+mspini). This results in a 33 tensor with terms  ij. In the real-space bulk calculations performed in the present work, the ij(with i6=j) matrices contain o-diagonal terms which are can- celled by the summation of the contributions of all neigh- bors within a given shell, resulting in a purely diagonal damping tensor, as expected for symmetry reasons [15]. Therefore, as in the DFT calculations the spin quanti- zation axis is considered to be in the z([001]) direction (collinear model), we can ascribe a scalar damping value ijas the average ij=1 2(xx ij+yy ij) =xx ijfor the systems investigated here. This scalar ijis, then, used in the SLLG equation (Eq. 1). The exchange parameters Jijin the Heisenberg model were calculated by the Liechtenstein-Katsnelson- Antropov-Gubanov (LKAG) formalism [79], according to the implementation in the RS-LMTO-ASA method [61]. Hence all parameters needed for the atomistic LLG equa- tion have been evaluated from ab-initio electronic struc- ture theory. II. RESULTS A. Onsite and non-local dampings Table I shows the relevant ab-initio magnetic prop- erties of each material; the TCvalues refer to the Curie temperature calculated within the random-phase approx- imation (RPA) [80], based on the computed Jijset. De- spite the systematic totvalues found in the lower limit of available experimental results (in similar case with, e.g., Ref. [81]), in part explained by the fact that we analyze only the intrinsic damping, a good agreement between theory and experiment can be seen. When the whole VCA Fe 1xCoxseries is considered (from x= 0% tox= 60%), the expected Slater-Pauling behavior of5 the total magnetic moment [73, 82] is obtained (data not shown). For all systems studied here, the dissipation is domi- nated by the onsite ( ii) term, while the non-local pa- rameters (ij,i6=j) exhibit values at least one order of magnitude lower; however, as it will be demonstrated in the next sections, these smaller terms still cause a non- negligible impact on the relaxation of the average magne- tization as well as magnon lifetimes. Figure 1 shows the non-local damping parameters for the investigated ferro- magnets as a function of the ( i;j) pairwise distance rij=a, together with the correspondent Fourier transforms q over the rst Brillouin Zone (BZ). The rst point to no- tice is the overall strong dependence of on the wave vectorq. The second point is the fact that, as also re- ported in Ref. [20], ijcan be an anisotropic quantity with respect to the same shell of neighbors, due to the broken symmetry imposed by a preferred spin quantiza- tion axis. This means that, in the collinear model and for a given neighboring shell, ijis isotropic only for equiva- lent sites around the magnetization as a symmetry axis. Another important feature that can be seen in Fig. 1 is the presence of negative ijvalues. Real-space neg- ative non-local damping parameters have been reported previously [20, 77, 97]. They are related to the decrease of damping at the -point, but may also increase qfrom the onsite value in specic qpoints inside the BZ; there- fore, they cannot be seen as ad hoc anti-dissipative con- tributions. In the ground-state, these negative non-local dampings originate from the overlap between the anti- Hermitian parts of the two Green's functions at the Fermi level, each associated with a spin-dependent phase factor (=";#) [20, 80]. Finally, as shown in the insets of Fig. 1, a long-range convergence can be seen for all cases investigated. An illustrative example is the bcc Fe 50Co50bulk, for which the eective damping can be 60% higher than the con- vergedtotif only the rst 7 shells of neighbors are con- sidered in Eq. 14. The non-local damping of each neigh- boring shell is found to follow a1 r2 ijtrend, as previously argued by Thonig et al. [20] and Umetsu et al. [97]. Explicitly, ij/sin(k"
rij+ ") sin(k#
rij+ #) jrijj2; (16) which also qualitatively justies the existence of negative ij's. Thus, the convergence in real-space is typically slower than other magnetic quantities, such as exchange interactions ( Jij/1 jrijj3) [80], and also depends on the imaginary part (see Eq. 15) [20]. The dierence in the asymptotic behaviour of the damping and the Heisenberg exchange is distinctive; the rst scales with the inverse of the square of the distance while the latter as the inverse of the cube of the distance. Although this asymptotic behaviour can be derived from similar arguments, both using the Greens function of the free electron gas, the results are dierent. The reason for this dierence issimply that the damping parameter is governed by states close to the Fermi surface, while the exchange parameter involves an integral over all occupied states [20, 79]. From bcc Fe to bcc Fe 50Co50(Fig. 1(a-f)), with in- creasing Co content, the average rst neighbors ijde- creases to a negative value, while the next-nearest neigh- bors contributions reach a minimum, and then increase again. Similar oscillations can be found in further shells. Among the interesting features in the Fe 1xCoxsystems (x= 0%;30%;50%), we highlight the low qaround the high-symmetry point H, along the HPandHN directions, consistently lower than the FMR damping. Bothvalues are strongly in uenced by non-local con- tributions &5 NN. Also consistent is the high qob- tained forq=H. For long wavelengths in bcc Fe, some qanisotropy is observed around , which resembles the same trait obtained for the corresponding magnon dis- persion curves [80]. This anisotropy changes to a more isotropic behavior by FeCo alloying. Far from the more noticeable high-symmetry points, qpresents an oscillatory behavior along BZ, around the onsite value. It is noteworthy, however, that these oscil- latoryqparameters exhibit variations up to 2 times ii, thus showing a pronounced non-local in uence in specicqpoints. In turn, for fcc Co (Fig. 1(g,h)) the rst values are characterized by an oscillatory behavior around zero, which also re ects on the damping of the FMR mode, q=0. In full agreement with Ref. [20], we compute a peak ofijcontribution at rij3:46a, which shows the long-range character that non-local damping can ex- hibit for specic materials. Despite the relatively small magnitude of ij, the multiplicity of the nearest neigh- bors shells drives a converged qdispersion with non- negligible variations from the onsite value along the BZ, specially driven by the negative third neighbors. The maximum damping is found to be in the region around the high-symmetry point X, where thus the lifetime of magnon excitations are expected to be reduced. Simi- lar situation is found for fcc Ni (Fig. 1(i,j)), where the rst neighbors ijare found to be highly negative, con- sequently resulting in a spectrum in which q> q=0 for everyq6= 0. In contrast with fcc Co, however, no notable peak contributions are found. B. Remagnetization Gilbert damping in magnetic materials determines the rate of energy that dissipates from the magnetic to other reservoirs, like phonons or electron correlations. To ex- plore what impact non-local damping has on the energy dissipation process, we performed atomistic spin dynam- ics (ASD) simulations for the aforementioned ferromag- nets: bcc Fe 1xCox(x= 0%;30%;50%), fcc Co, and fcc Ni, for the (i)fully non-local ijand (ii)eective tot(dened in 14) dissipative case. We note that, al- though widely considered in ASD calculations, the adop-6 TABLE I. Spin ( mspin) and orbital ( morb) magnetic moments, onsite ( ii) damping, total ( tot) damping, and Curie temper- ature (TC) of the investigated systems. The theoretical TCvalue is calculated within the RPA. In turn, mtdenotes the total moments for experimental results of Ref. [82]. mspin(B)morb(B)ii(103) tot(103) TC(K) bcc Fe (theory) 2.23 0.05 2.4 2.1 919 bcc Fe (expt.) 2.13 [68] 0 :08 [68] 1:97:2 [33, 83{89] 1044 bcc Fe 70Co30(theory) 2.33 0.07 0.5 0.9 1667 bcc Fe 70Co30(expt.) mt= 2:457 [82] 0:51:7a[33, 83, 90] 1258 [92] bcc Fe 50Co50(theory) 2.23 0.08 1.5 1.6 1782 bcc Fe 50Co50(expt.) mt= 2:355 [82] 2:03:2b[25, 33, 83] 1242 [93] fcc Co (theory) 1.62 0 :08 7.4 1.4 1273 fcc Co (expt.) 1 :68(6) [94] 2:8(5) [33, 89] 1392 fcc Ni (theory) 0 :61 0 :05 160.1 21.6 368 fcc Ni (expt.) 0 :57 [68] 0 :05 [68] 23:664 [22, 83, 87{89, 95, 96] 631 aThe lower limit refers to polycrystalline Fe 75Co2510 nm-thick lms from Ref. [33]. Lee et al. [90] also found a low Gilbert damping in an analogous system, where tot<1:4103. For the exact 30% of Co concentration, however, previous results [33, 84, 91] indicate that we should expect a slightly higher damping than in Fe 75Co25. bThe upper limit refers to the approximate minimum intrinsic value for a 10 nm-thick lm of Fe 50Co50jPt (easy magnetization axis). tion of a constant totvalue (case (ii)) is only a good ap- proximation for long wavelength magnons close to q= 0. First, we are interested on the role of non-local damp- ing in the remagnetization processes as it was already discussed by Thonig et al. [20] and as it is important for,e.g., ultrafast pump-probe experiments as well as all- optical switching. In the simulations presented here, the relaxation starts from a totally random magnetic con- guration. The results of re-magnetization simulations are shown in Figure 2. The fully non-local damping (i) in the equation of motion enhances the energy dissipa- tion process compared to the case when only the eective damping (ii)is used. This eect is found to be more pro- nounced in fcc Co and fcc Ni compared to bcc Fe and bcc Fe50Co50. Thus, the remagnetization time to 90% of the saturation magnetisation becomes 58 times faster for case (i)compared to the case (ii). This is due to the increase of qaway from the point in the whole spectrum for Co and Ni (see Fig. 1), where in Fe and Fe50Co50it typically oscillates around tot. For bcc Fe 70Co30, the eect of non-local damping on the dynamics is opposite to the data in Fig. 2; the re- laxation process is decelerated. In this case, almost the entireqspectrum is below q=0, which is an interest- ing result given the fact that FMR measurements of the damping parameter in this system is already considered an ultra-low value, when compared to other metallic fer- romagnets [33]. Thus, in the remagnetization process of Fe70Co30, the majority of magnon modes lifetimes is un- derestimated when a constant totis considered in the spin dynamics simulations, which leads to a faster overall relaxation rate. Although bcc Fe presents the highest Gilbert damp-ing obtained in the series of the Fe-Co alloys (see Table I) the remagnetization rate is found to be faster in bcc Fe50Co50. This can be explained by the fact that the ex- change interactions for this particular alloy are stronger (80% higher for nearest-neighbors) than in pure bcc Fe, leading to an enhanced Curie temperature (see Table I). In view of Eq. 13 and Fig. 1, the dierence in the remagnetization time between bcc Fe 50Co50and elemen- tal bcc Fe arises from qvalues that are rather close, but where the magnon spectrum of Fe 50Co50has much higher frequencies, with corresponding faster dynamics and hence shorter remagnetization times. From our calculations we nd that the sum of non-local dampingP i6=jij contributes with 13%,81%, 87%, +80%, and +7% to the local damping in bcc Fe, fcc Co, fcc Ni, bcc Fe 70Co30, and bcc Fe 50Co50, respec- tively. The high positive ratio found in Fe 70Co30indi- cates that, in contrast to the other systems analyzed, the non-local contributions act like an anti-damping torque, diminishing the local damping torque. A similar anti- damping eect in antiferromagnetic (AFM) materials have been reported in theoretical and experimental in- vestigations ( e.g., [98, 99]), induced by electrical current. Here we nd that an anti-damping torque eect can have an intrinsic origin. To provide a deeper understanding of the anti-damping eect caused by a positive non-local contribution, we an- alytically solved the equation of motion for a two spin model system, e.g. a dimer. In the particular case when the onsite damping 11is equal to the non-local con- tribution12, we observed that the system becomes un- damped (see Appendix B). As demonstrated in Appendix B, ASD simulations of such a dimer corroborate the re-7 FIG. 1. Non-local damping ( ij) as a function of the nor- malized real-space pairwise ( i;j) distance computed for each neighboring shell, and corresponding Fourier transform q (see Eq. 11) from the onsite value ( ii) up to 136 shells of neighbors (136 NN) for: (a,b) bcc Fe; (c,d) bcc Fe 70Co30; (e,f) bcc Fe 50Co50in the virtual-crystal approximation; and up to 30 shells of neighbors (30 NN) for: (g,h) fcc Co; (i,j) fcc Ni. The insets in subgures (a,c,e,g,i) show the convergence oftotin real-space. The obtained onside damping values are shown in Table I. In the insets of the left panel, green full lines are guides for the eyes. sult of undamped dynamics. It should be further noticed that this proposed model system was used to analyse the stability of the ASD solver, verifying whether it can preserve both the spin length and total energy. Full de- tail of the analytical solution and ASD simulation of a spin-dimer and the anti-damping eect are provided inAppendix B. FIG. 2. Remagnetization process simulated with ASD, con- sidering fully non-local Gilbert damping ( ij, blue sold lines), and the eective damping ( tot, red dashed lines), for: (a) fcc Ni; (b) fcc Co; and (c) bcc Fe 1xCox(x= 0%;30%;50%). The dashed gray lines indicate the stage of 90% of the satu- ration magnetization. C. Magnon spectra In order to demonstrate the in uence of damping on magnon properties at nite temperatures, we have per- formed ASD simulations to obtain the excitation spectra from the dynamical structure factor introduced in Sec- tion I. Here, we consider 16 NN shells for S(q;!) calcula- tions both from simulations that include non-local damp- ing as well as the eective total damping (see Appendix D for a focused discussion). In Fig. 3, the simulated magnon spectra of the here investigated ferromagnets are shown. We note that a general good agreement can be observed between our computed magnon spectra (both from the the frozen magnon approach as well as from the dynamical structure factor) and previous theoretical as well as experimental results [34, 52, 80, 100{103], where deviations from experiments is largest for fcc Ni. This exception, however, is well known and has already been discussed elsewhere [104]. The main feature that the non-local damping causes to the magnon spectra in all systems investigated here, is in changes of the full width at half maximum (FWHM) 4q ofS(q;!). Usually,4qis determined from the super- position of thermal uctuations and damping processes. More specically, the non-local damping broadens the FWHM compared to simulations based solely on an eec- tive damping, for most of the high-symmetry paths in all of the here analyzed ferromagnets, with the exception of Fe70Co30. The most extreme case is for fcc Ni, as qex- ceeds the 0:25 threshold for q=X, which is comparable to the damping of ultrathin magnetic lms on high-SOC metallic hosts [105]. As a comparison, the largest dier- ence of FWHM between the non-local damping process and eective damping process in bcc Fe is 2 meV, while in fcc Ni the largest dierence can reach 258 meV. In contrast, the dierence is 1 meV in Fe 70Co30and the8 largest non-local damping eect occurs around q=N and in the HPdirection, corroborating with the dis- cussion in Section II A. At the point, which corresponds to the mode measured in FMR experiments, all spins in the system have a coherent precession. This implies that @mj @tin Eq. 4 is the same for all moments and, thus, both damping scenarios discussed here (eecive local and the one that also takes into account non-local contributions) make no dierence to the spin dynamics. As a conse- quence, only a tiny (negligible) dierence of the FWHM is found between eective and non-local damping for the FMR mode at low temperatures. The broadening of the FWHM on the magnon spec- trum is temperature dependent. Thus, the eect of non- local damping to the width near can be of great in- terest for experiments. More specically, taking bcc Fe as an example, the dierence between width in eective damping and non-local damping process increases with temperature, where the dierence can be enhanced up to one order of magnitude from T= 0:1 K toT= 25 K. Note that this enhancement might be misleading due to the limits of nite temperature assumption made here. This temperature dependent damping eect on FWHM suggests a path for the measurement of non-local damp- ing in FMR experiments. We have also compared the dierence in the imaginary part of the transverse dynamical magnetic susceptibility computed from non-local and eective damping. Dened by Eq. 12, the imaginary part of susceptibility is re- lated to the FWHM [15]. Similar to the magnon spectra shown in Fig. 3, the susceptibility dierence is signi- cant at the BZ boundaries. Taking the example of fcc Co, Im(q;!) for eective damping processes can be 11:8 times larger than in simulations that include non- local damping processes, which is consistent to the life- time peak that occurs at high the symmetry point, X, depicted in Fig. 4. In the Fe 1xCoxalloy, and Fe 70Co30, the largest ratio is 1 :7 and 2:7 respectively. The intensity at point is zero since qis independent on the coupling vector and equivalent in both damping modes. The ef- fect of non-local damping on susceptibility coincides well with the magnon spectra from spin dynamics. Thus, this method allows us to evaluate the magnon properties in a more ecient way. D. Magnon lifetimes By tting the S(q;!) curve at each wave vector with a Lorentzian curve, the FWHF and hence the magnon lifetimes,q, can be obtained from the simple relation [15] q=2 4q: (17) Figure 4 shows the lifetimes computed in the high- symmetry lines in the BZ for all ferromagnets here in-vestigated. As expected, qis much lower at the qvec- tors far away from the zone center, being of the order of 1 ps for the Fe 1xCoxalloys (x= 0%;30%;50%), and from0:011 ps in fcc Co and Ni. In view of Eq. 13, the magnon lifetime is inversely proportional to both damping and magnon frequency. In the eective damping process, qis a constant and independent of q; thus, the lifetime in the entire BZ is dictated only by !q. The situation becomes more complex in the non- local damping process, where the qis in uenced by the combined eect of changing damping and magnon fre- quency. Taking Fe 70Co30as an example, even though theqis higher around the , the low magnon frequency compensates the damping eect, leading to an asymp- totically divergent magnon lifetime as !q!0. However, this divergence becomes nite when including e.g. mag- netocrystalline anisotropy or an external magnetic eld to the spin-Hamiltonian. In the HNpath, the magnon energy of Fe 70Co30is large, but qreaches4104 atq=1 4;1 4;1 2
, resulting in a magnon lifetime peak of 10 ps. This value is not found for the eective damping model. In the elemental ferromagnets, as well as for Fe 50Co50, it is found that non-local damping decreases the magnon lifetimes. This non-local damping eect is signicant in both Co and Ni, where the magnon lifetimes from the ij model dier by an order of magnitude from the eective model (see Fig. 4). In fact, considering qobtained from Eq. 13, the eective model predicts a lifetime already higher by more than 50% when the magnon frequencies are33 meV and14 meV in the K path ( i.e., near ) of Ni and Co, respectively. This dierence mainly arises, in real-space, from the strong negative contriu- tions ofijin the close neighborhood around the refer- ence site, namely the NN in Ni and third neighbors in Co. In contrast, due to the qspectrum composed of almost all dampings lower than tot, already discussed in Section II A, the opposite trend on qis observed for Fe 70Co30: the positive overall non-local contribution guide an anti- damping eect, and the lifetimes are enhanced in the non-local model. Another way to evaluate the magnon lifetimes is from the linear response theory. As introduced in Section I B, we have access to magnon lifetimes at low temperatures from the imaginary part of the susceptibility. The q calculated from Eq. 13 is also displayed in Fig. 4. Here the spin-wave frequency !qis from the frozen magnon method. The magnon lifetimes from linear response have a very good agreement with the results from the dynam- ical structure factor, showing the equivalence between both methods. Part of the small discrepancies are re- lated to magnon-magnon scattering induced by the tem- perature eect in the dynamical structure factor method. We also nd a good agreement on the magnon lifetimes of eective damping in pure Fe with previous studies [106]. They are in the similar order and decrease with the increasing magnon energy. However, their results are more diused since the simulations are performed at9 FIG. 3. Magnon spectra calculated with non-local Gilbert damping and eective Gilbert damping in: (a) bcc Fe; (b) bcc Fe70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni. The black lines denote the adiabatic magnon spectra calculated from Eq. 7. Full red and open blue points denote the peak positions of S(q;!) at each qvector fortotandijcalculations, respectively, at T= 0:1 K. The width of transparent red and blue areas corresponds to the full width half maximum (FWHM) on the energy axis tted from a Lorentzian curve, following the same color scheme. To highlight the dierence of FWHM between the two damping modes, the FWHMs shown in the magnon spectrum of Fe 1xCox, Co, and Ni are multiplied by 20, 5, and1 2times, in this order. The triangles represent experimental results: in (a), Fe at 10 K [102] (yellow up) and Fe with 12% Si at room-temperature [101] (green down); in (d), Co(9 ML)/Cu(100) at room-temperature [103] (green down); in (e) Ni at room-temperature (green down) [100]. The standard deviation of the peaks are represented as error bars. room-temperature. III. CONCLUSION We have presented the in uence of non-local damping on spin dynamics and magnon properties of elemental fer- romagnets (bcc Fe, fcc Co, fcc Ni) and the bcc Fe 70Co30 and bcc Fe 50Co50alloys in the virtual-crystal approxima- tion. It is found that the non-local damping has impor- tant eects on relaxation processes and magnon prop- erties. Regarding the relaxation process, the non-local damping in Fe, Co, and Ni has a negative contribution to the local (onsite) part, which accelerates the remagne- tization. Contrarily, in uenced by the positive contribu- tion ofij(i6=j), the magnon lifetimes of Fe 70Co30and Fe50Co50are increased in the non-local model, typically at the boundaries of the BZ, decelerating the remagneti- zation. Concerning the magnon properties, the non-local damping has a signicant eect in Co and Ni. More specically, the magnon lifetimes can be overestimated by an order of magnitude in the eective model for these two materials. In real-space, this dierence arises as a result of strong negative non-local contributions in theclose neighborhood around the reference atom, namely the NN in Ni and the third neighbors in Co. Although the eect of non-local damping to the stochastic thermal eld in spin dynamics is not included in this work, we still obtain coherent magnon lifetimes comparing to the analytical solution from linear response theory. Notably, it is predicted that the magnon lifetimes at certain wave vectors are higher for the non-local damp- ing model in some materials. An example is Fe 70Co30, in which the lifetime can be 3 times higher in the HN path for the non-local model. On the other hand, we have proposed a fast method based on linear response to evaluate these lifetimes, which can be used to high- throughput computations of magnonic materials. Finally, our study provides a link on how non-local damping can be measured in FMR and neutron scat- tering experiments. Even further, it gives insight into optimising excitation of magnon modes with possible long lifetimes. This optimisation is important for any spintronics applications. As a natural consequence of any real-space ab-initio formalism, our methodology and ndings also open routes for the investigation of other materials with preferably longer lifetimes caused by non- local energy dissipation at low excitation modes. Such materials research could also include tuning the local10 FIG. 4. Magnon lifetimes qof: (a) bcc Fe; (b) bcc Fe 70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni as function of q, shown in logarithmic scale. The color scheme is the same of Fig. 3, where blue and red represents qcomputed in the eective and non-local damping models. The transparent lines and opaque points depict the lifetimes calculated with Eq. 13 and by the FWHM of S(q;!) atT= 0:1 K (see Eq. 17). The lifetime asymptotically diverges around the -point due to the absence of anisotropy eects or external magnetic eld in the spin-Hamiltonian. chemical environments by doping or defects. IV. ACKNOWLEDGMENTS Financial support from Vetenskapsr adet (grant num- bers VR 2016-05980 and VR 2019-05304), and the Knut and Alice Wallenberg foundation (grant number 2018.0060) is acknowledged. Support from the Swedish Research Council (VR), the Foundation for Strategic Re-search (SSF), the Swedish Energy Agency (Energimyn- digheten), the European Research Council (854843- FASTCORR), eSSENCE and STandUP is acknowledged by O.E. . Support from the Swedish Research Coun- cil (VR) is acknowledged by D.T. and A.D. . The China Scholarship Council (CSC) is acknowledged by Z.L.. The computations/data handling were enabled by resources provided by the Swedish National Infrastruc- ture for Computing (SNIC) at the National Supercom- puting Centre (NSC, Tetralith cluster), partially funded by the Swedish Research Council through grant agree- ment No. 2016-07213. [1] A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye, M. Krawczyk, J. Gr afe, C. Adelmann, S. Cotofana, A. Naeemi, V. I. 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Phys. 117, 013912 (2015).13 Appendix A: Numerical solver In this Appendix, the numerical method to solve Eq. 1 is described. In previous studies, several numerical approaches have been proposed to solve the local LLG equations, including HeunP method, implicit midpoint method, Depondt-Merten's method [107], semi-implicit A (SIA) and semi-implicit B (SIB) methods [42]. To solve this non-local LLG equation, we use the xed-point iter- ation midpoint method. We have done convergence tests on this method and nd that it preserve the energy and spin length of the system, which is demonstrated in Fig. 5 for the case of a dimer. With stable outputs, the solver allows for a relatively large time step size, typically of the order of
t0:11 fs. Following the philosophy of an implicit midpoint method, the implemented algorithm can be described as follows. Let mt ibe the magnetic moment of site iat a given time step t. Then we can dene the quantity mmidand the time derivative of mi, respectively, as mmid=mt+1 i+mt i 2; @mi @t=mt+1 imt i
t:(A1) Using this denition in Eq. 4, the equation of motion of thei-th spin becomes: @mi @t=mmid0 @ [Bi(mmid) +bi(t)] +X jij mj@mj @t1 A: (A2) Thus, with a xed-point scheme, we can do the follow- ing iteration mt+1(k+1) i =mt i+
t0 @ mt+1(k) i +mt i 2! 0 @ " Bi mt+1(k) i +mt i 2! +bi(t)# +X jij mjmt+1(k) jmt j
t1 A1 A: (A3) Ifmt+1(k+1) imt+1(k) i , the self-consistency con- verges. Typically, about 6 iteration steps are needed. This solver was implemented in the software package Up- pASD [48] for this work. Appendix B: Analytical model of anti-damping in dimers In the dimer model, there are two spins on site 1 and site 2 denoted by m1andm2, which are here supposed to be related to the same element { so that, naturally, 11=22>0. Also, let's consider a suciently low temperature so that bi(t)!0, which is a reasonable assumption, given that damping has an intrinsic origin [108]. This simple system allows us to provide explicit expressions for the Hamiltonian, the eective magnetic elds and the damping term. From the analytical solu- tion, it is found that the dimer spin system becomes an undamped system when local damping is equal to non- local damping, i.e.the eective damping of the system is zero. Following the denition given by Eq. 4 in the main text, the equation of motion for spin 1 reads: @m1 @t=m1 B1+11 m1@m1 @t+12 m2@m2 @t ;(B1) and an analogous expression can be written for spin 2. For sake of simplicity, the Zeeman term is zero and theeective eld only includes the contribution from Heisen- berg exchange interactions. Thus, we have B1= 2J12m2 andB2= 2J21m1. Withjijj 1, we can take the LL form@mi @t= miBito approximate the time- derivative on the right-hand side of the LLG equation. Letm1=m2and12=11. SinceJ12=J21and m1m2=m2m1, then we have @m1 @t=2 J12m1 m2+ (1)11 m1(m1m2) : (B2) Therefore, when 12=21=11(i.e.,= 1), Eq. B1 is reduced to: @m1 @t=2 J12m1m2; (B3) and the system becomes undamped. It is however straightforward that, for the opposite case of a strong negative non-local damping ( =1), Eq. B2 describes a common damped dynamics. A side (and related) con- sequence of Eq. B2, but important for the discussion in Section II B, is the fact that the eective onsite damp- ing term 11= (1)11becomes less relevant to the dynamics as the positive non-local damping increases (!1), or, in other words, as tot= (11+12) strictly increases due to the non-local contribution. Exactly the same reasoning can be made for a trimer, for instance, composed by atoms with equal moments and exchange interactions ( m1=m2=m3,J12=J13=J23), and same non-local dampings ( 13=12=11).14 The undamped behavior can be directly observed from ASD simulations of a dimer with 12=11, as shown in Fig. 5. Here the magnetic moment and the exchange are taken the same of an Fe dimer, m1= 2:23BandJ12= 1:34 mRy. Nevertheless, obviously the overall behavior depicted in Fig. 5 is not dependent on the choice of m1andJ12. Thezcomponent is constant, while the x andycomponents of m1oscillate in time, indicating a precessing movement. In a broader picture, this simple dimer case exemplies the connection between the eigenvalues of the damping matrix= (ij) and the damping behavior. The occur- rence of such undamped dynamics has been recently dis- cussed in Ref. [109], where it is shown that a dissipation- free mode can occur in a system composed of two sub- systems coupled to the same bath. 0.00 0.02 0.04 0.06 0.08 0.10 t(ps)0.2 0.00.20.40.60.81.0Magnetization 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Energy(mRy)mxmymzmEnergy FIG. 5. Spin dynamics at T= 0 K of an undamped dimer in which12=21=11(see text). The vector m1is normalized and its Cartesian components are labeled in the gure asmx,myandmz. The black and grey lines indicate the length of spin and energy (in mRy), respectively. Appendix C: Eective and onsite damping in the FeCo and CoNi alloys As mentioned in Section I, the simple VCA model al- lows us to account for the disorder in 3 d-transition-metal alloys in a crude but ecient way which avoids the use of large supercells with random chemical distributions. With exactly the same purpose, the coherent potential approximation (CPA) [110] has also been employed to analyze damping in alloys ( e.g., in Refs. [84, 111, 112]), showing a very good output with respect to trends, when compared to experiments [33, 81]. In Fig. 6 we show the normalized calculated local (onsite, ii) and eec- tive damping ( tot) parameters for the zero-temperature VCA Fe 1xCoxalloy in the bcc structure, consistent with a concentration up to x60% of Co [33]. The computed values in this work (blue, representing ii, and red points, representing tot) are compared to previous theoretical CPA results and room-temperature experimental data. The trends with VCA are reproduced in a good agree-ment with respect to experiments and CPA calculations, showing a minimal totwhen the Co concentration is x30%. This behavior is well correlated with the local density of states (LDOS) at the Fermi level, as expected by the simplied Kambersk y equation [113], and the on- site contribution. Despite the good agreement found, the values we have determined are subjected to a known error of the VCA with respect to the experimental results. This discrepancy can be partially explained by three reasons: ( i) the signicant in uence of local environ- ments (local disorder and/or short-range order) to tot [25, 77]; ( ii) the fact that the actual electronic lifetime (i.e., the mean time between two consecutive scattering events) is subestimated by the VCA average for random- ness in the FeCo alloy, which can have a non-negligible impact in the damping parameter [22, 114]; and ( iii) the in uence on damping of noncollinear spin congurations in nite temperature measurements [54, 115]. On top of that, it is also notorious that damping is dependent on the imaginary part of the energy (broadening) [22, 114], , which can be seen as an empirical quantity, and ac- counts for part of the dierences between theory and ex- periments. 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0 10 20 30 40 50 60 0 5 10 15 20 25Damping value DOS at EF (states/Ry−atom) Co concentration (%)onsite (αii) total (αtot) Turek et al. (αtot) Mankovsky et al. [2013] (αtot) Mankovsky et al. [2018] (αtot) Schoen et al. n(EF) FIG. 6. (Color online) Left scale : Computed Gilbert eec- tive (tot, red circles) and onsite ( ii, blue squares) damping parameters as a function of Co the concentration ( x) for bcc Fe1xCoxbinary alloy in the virtual-crystal approximation. The values are compared with previous theoretical results us- ing CPA, from Ref. [84] (gray full triangles), Ref. [54] (black open rhombus), Ref. [112] (yellow open triangles), and room- temperature experimental data [33]. Right scale : The calcu- lated density of states (DOS) at the Fermi level as a function ofx, represented by the black dashed line. In the spirit of demonstrating the eectiveness of the simple VCA to qualitatively (and also, to some extent, quantitatively) describe the properties of Gilbert damp- ing in suitable magnetic alloys, we also show in Fig. 7 the results obtained for Co xNi1xsystems. The CoNi alloys15 are known to form in the fcc structure for a Ni concen- tration range of 10% 100%. Therefore, here we mod- eled CoxNi1xby a big fcc cluster containing 530000 atoms in real-space with the equilibrium lattice parame- ter ofa= 3:46A. The number of recursion levels consid- ered isLL= 41. A good agreement with experimental results and previous theoretical calculations can be no- ticed. In particular, the qualitative comparison with the- ory from Refs. [81, 84] indicates the equivalence between the torque correlation and the spin correlation models for calculating the damping parameter, which was also investigated by Sakuma [116]. The onsite contribution for each Co concentration, ii, is omitted from Fig. 7 due to an absolute value 2 4 times higher than tot, but follows the same decreasing trend. Again, the over- all eective damping values are well correlated with the LDOS, and re ect the variation of the quantity1 mtwith Co concentration (see Eq. 15). 0 0.005 0.01 0.015 0.02 0.025 0 10 20 30 40 50 60 70 10 15 20 25 30Damping value DOS at EF (states/Ry−atom) Co concentration (%)total (αtot) Mankovsky et al. [2013] (αtot) Starikov et al. (αtot) Schoen [2017] et al. n(EF) FIG. 7. (Color online) Left scale : Computed Gilbert eective (tot, red circles) damping parameters as a function of the Co concentration ( x) for fcc Co xNi1xbinary alloy in the virtual- crystal approximation. The values are compared with previ- ous theoretical results using CPA, from Ref. [84] (gray full triangles), Ref. [81] (gold full circles), and room-temperature experimental data [89]. Right scale : The calculated density of states (DOS) at the Fermi level as a function of x, represented by the black dashed line. Appendix D: Eect of further neighbors in the magnon lifetimes When larger cuto radii ( Rcut) ofijparameters are included in ASD, Eq. A3 takes longer times to achieve a self-consistent convergence. In practical terms, to reach a sizeable computational time for the calculation of a given system,Rcutneeds to be chosen in order to preserve the main features of the magnon properties as if Rcut!1 . A good quantity to rely on is the magnon lifetime q,as it consists of both magnon frequency and q-resolved damping (Eq. 13). In Section II C, we have shown the equivalence between Eq. 13 and the inverse of FWHM on the energy axis of S(q;!) for the ferromagnets inves- tigated here. Thus, the comparison of two qspectra for dierentRcutcan be done directly and in an easier way using Eq. 13. FIG. 8. (Color online) Magnon lifetimes calculated using Eq. 13 for: (a) bcc Fe; and (b) bcc Fe 50Co50, using a reduced set of 16 NN shells (opaque lines), and the full set of 136 NN shells (transparent lines). An example is shown in Figure 8 for bcc Fe and bcc Fe50Co50. Here we choose the rst 16 NN ( Rcut3:32a) and compare the results with the full calculated set of 136 NN (Rcut= 10a). It is noticeable that the reduced set of neighbors can capture most of the features of the qspectrum for a full NN set. However, long-range in- uences of small magnitudes, such as extra oscillations around the point q=Hin Fe, can occur. In particu- lar, these extra oscillations arise mainly due to the pres- ence of Kohn anomalies in the magnon spectrum of Fe, already reported in previous works [52, 80]. In turn, for the case of Fe 50Co50, the long-range ijreducestot, and causes the remagnetization times for non-local and eec-16 tive dampings to be very similar (see Fig. 2). For the other ferromagnets considered in the present research,comparisons of the reduced Rcutwith analogous quality were reached.

1012.1371v1.Turbulence_damping_as_a_measure_of_the_flow_dimensionality.pdf

Turbulence damping as a measure of the ow dimensionality M. Shats,D. Byrne, and H. Xia Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia (Dated: October 23, 2018) The dimensionality of turbulence in uid layers determines their properties. We study electro- magnetically driven ows in nite depth uid layers and show that eddy viscosity, which appears as a result of three-dimensional motions, leads to increased bottom damping. The anomaly coecient, which characterizes the deviation of damping from the one derived using a quasi-two-dimensional model, can be used as a measure of the ow dimensionality. Experiments in turbulent layers show that when the anomaly coecient becomes high, the turbulent inverse energy cascade is suppressed. In the opposite limit turbulence can self-organize into a coherent ow. PACS numbers: 47.27.Rc, 47.55.Hd, 42.68.Bz Fluid layers represent a broad class of ows whose depths are much smaller than their horizontal extents, for example, planetary atmospheres and oceans. A dis- covery of the upscale energy transfer in two-dimensional (2D) turbulence [1] gave new insight into the energy bal- ance in turbulent layers. The inverse cascade transfers energy from smaller to larger scales thus allowing for tur- bulence self-organization. This is in contrast with three- dimensional (3D) turbulence where energy is nonlinearly transferred towards small scales (direct cascade). Real physical layers dier from the ideal 2D model since they have nite depths and non-zero dissipation. The eect of the layer thickness on turbulence driven by 2D forcing has been studied in 3D numerical simulations [2, 3]. It has been shown that in \turbulence in more than two and less than three dimensions", the injected energy ux splits into the direct and inverse parts. At ratios of the layer depth hover the forcing scale lfabove h=lf0:5 the inverse energy cascade is greatly reduced. When the inverse energy ux is suppressed, the energy injected into the ow is transferred towards small scales by the direct cascade, developing the Kolmogorov k5=3 spectrum at k >k f. This result illustrates that 2D and 3D turbulence may coexist. 2D/3D eects have been studied in electromagnetically driven ows using two main schemes to force the uid mo- tion. In liquid metals placed in the vertical homogeneous magnetic eld the ow is forced by applying spatially varying electric eld which generates JBforces. In such magnetohydrodynamic (MHD) ows 2D properties are enforced by the magnetic eld and the 3D behav- ior is restricted to a very thin Hartmann layer [4]. The deviations from 2D in such ows may be due to the - nite resistivity in very thick layers [5, 6]. Another class of experiments employs spatially periodic magnetic eld crossed with the constant horizontal electric current to produce interacting vortices [7{9]. In this case the thick- Electronic address: Michael.Shats@anu.edu.auness of the Hartmann layer exceeds the layer depth and 2D/3D eects are determined by the factors which are dierent from those in MHD ows, for example, by a density stratication. The 3D eects are closely related to the energy dissi- pation in the layers. This connection however is not fully understood in experiments. The measured ow damp- ing rates are often compared with those derived from a quasi-2D model [10, 11] which assumes no vertical mo- tions within the layer. In thin layers, the agreement is usually within a factor of 2 [8, 12]. However in some experiments a much better agreement with the quasi-2D model was observed [13]. This contradicts recent claims about the intrinsic three-dimensionality of the ows in thin layers of electrolytes [14, 15]. There is a need to clarify this. Physical three-dimensionality of the ow is determined by the amount of 3D motion in the layer. This motion may naturally develop in the layer, as in [3], but it can also be injected into the ow by non-2D forcing or it can be generated by the shear-driven instabilities in the boundary layer. In this case, the critical layer thickness cannot be used as a practical criterion of the 2D/3D tran- sition since it will vary depending on the source of 3D motion. The transition from 2D to 3D, which marks a fundamental change in the energy transfer, needs to be characterized quantitatively, in other words, it is neces- sary to nd a measure of the ow dimensionality which would help to predict turbulence behavior. In this Letter we show that eddy viscosity increases damping in nite-depth uid layers compared with the quasi-2D model prediction. This increase can be used as the measure of the ow dimensionality which allows to evaluate the likelihood of the inverse energy cascade and of turbulence self-organization. We also show that the increased degree of three-dimensionality leads to the suppression of the turbulent cascades. In these experiments turbulence is generated via the in- teraction of a large number of electromagnetically driven vortices [9, 16, 17]. The electric current owing through a conducting uid layer interacts with the spatially variablearXiv:1012.1371v1 [physics.flu-dyn] 7 Dec 20102 vertical magnetic eld produced by arrays of magnets placed under the bottom. In this paper we use a 30 30 array of magnetic dipoles (8 mm apart) for the turbu- lence studies requiring large statistics. For the studies of vertical motions, a 6 6 array of larger magnets (25 mm separation) is used. The ow is visualized using seeding particles, which are suspended in the uid, illuminated using a horizontal laser slab and lmed from above. Par- ticle image velocimetry (PIV) is used to derive turbulent velocity elds. The ow is generated either in a single layer of electrolyte ( Na2SO4water solution), or in two immiscible layers of uids (electrically neutral heavier liq- uid at the bottom, electrolyte on top). Shortly after the current is switched on, JBdriven vortices interact with each other forming complex turbulent motion character- ized by a broad wave number spectrum. The steady state is reached within tens of seconds. To study vertical motions in single electrolyte layers, vertical laser slabs are used to illuminate the ow in the yzplane. Streaks of the seeding particles within the slab are lmed with the exposure time of 1 s. Quantita- tive measurements of the horizontal and vertical veloci- ties are performed using defocusing PIV technique. This technique, was rst described in [18], but had never been used in turbulence studies. It allows measurements of 3D velocity components of seeding particles using a single video camera with a multiple pinhole mask (three pin- holes constituting a triangle are used here). A schematic of the method is shown in Fig. 1. An image of a particle placed in the reference plane at z=0 (where the parti- cle is in focus) corresponds to a single dot in the image plane. As the particle moves vertically away from the reference plane, the light passes through each pinhole in the mask and reaches three dierent positions on the im- age plane. The distances between the triangle vertices in the image plane are used to decode z-positions of the particles. The xy-components of velocity are determined using a PIV/PTV hybrid algorithm to match particle pairs from frame to frame. This process is illustrated in Fig. 1. The technique allows to resolve vertical veloci- ties above<Vz>RMS0:5 mm/s. The imaged area in this experiment is 5 5 cm2. On average about 50 par- ticles (triangles) are tracked in two consecutive frames. Derived velocities are then averaged over about 100 of the frame pairs to generate converged statistics of the mean-square-root velocities <Vx;y;z>RMS. Figures 2(a-c) show particle streaks and corresponding vertical velocity proles Vz(z) for dierent layer depths. To keep forcing approximately constant, the electric cur- rent is increased proportionally to the layer thickness (constant current density). To obtain better vertical spa- tial resolution, a 6 6 array of larger magnets is used. For the layers thicknesses of up to 30 mm, a range of h=lf= 0:21:2 is achieved. Particle streaks show reason- ably 2D motion in a thin (5 mm) layer, Fig. 2(a). Vertical velocity is small over most of the layer thickness and is z=zImage Plane LensMask Lf Reference Plane z=0m m z=3m m z=5m mz=0 026810 0 2 4 6 81 04 x(mm)y(mm)(a) (b)FIG. 1: Schematic of the defocusing particle image velocime- try technique. close to the resolution of the technique, <Vz>RMS0:5 mm/s. As the layer thickness is increased, 3D motions develop. The corresponding vertical velocities increase up to4 mm/s, Figs. 2(b,c). Fig. 2(d) shows the ra- tio of vertical to horizontal velocities as a function of the normalized layer thickness. In single layers this ra- tio increases approximately linearly with h=lfreaching over< V z> = < V x;y>= 0:3 ath=lf= 0:8. In stratied double layers this ratio is substantially smaller, < Vz> = < V x;y>0:08 (solid squares in Fig. 2(d)), suggesting that the ow in a double layer conguration is much closer to 2D. In the absence of 3D motions, the ow in the layer is damped due to molecular viscosity. A decay of hori- zontal velocity Vx;y(z;t) in the quasi-2D ow due to the bottom friction is described by the diusive type equa- tion@Vx;y=@t=@2Vx;y=@z2, which together with the boundary conditions Vx;y(z= 0;t) = 0 and @Vx;y(z= h;t)=@z= 0 gives the characteristic inverse time of the energy decay, e.g. [10]: L=2=2h2: (1) Hereis the kinematic viscosity. The onset of 3D turbulent eddies in thicker layers should lead to a vertical ux of horizontal momentum and faster dissipation of the ow. Such a ux is related to the mean vertical velocity gradient @Vx;y=@z[19]: <~Vx;y~Vz>=K@Vx;y @z: (2) HereKis the eddy (turbulent) viscosity coecient. By assuming that uctuations of vertical and horizontal ve-3 012345 0123Vz(mm s )-1h(mm)hl/ = 0.2f(a) 05101520 0123hl/ = 0.8fh(mm) Vz(mm s )-1(c) 051015 0123hl/ = 0.6f(b) Vz(mm s )-1h(mm) 00.10.20.30.4 0.0 0.5 1.0 1.5h/lf(d) V/zVx,y 01234 0 0.5 1 1.5h/lf(e) /c97/c97t/L FIG. 2: Particle streaks lmed with an exposure time of 1 s (top panels) and the distribution of the vertical velocity uctuations (rms) over the layer thickness (bottom panels) in single layers: (a) h= 5 mm; (b) h= 15 mm; (c) h= 20 mm. (d) Ratio of rms vertical to the rms horizontal veloc- ity as a function of the normalized layer thickness h=lfin a single (open circles) and in a double (solid squares) layer congurations. (e) t=Lversush=lf. locities are well correlated, we can estimate the eddy viscosity coecient using the defocusing PIV data as K<~Vx;y>< ~Vz>(@Vx;y=@z)1. Then the damp- ing rate can be estimated using the contribution of both molecular and the eddy viscosities, t= (+K)2=2h2. The ratio of thus calculated damping rate to the linear dampingL(1) is shown in Fig. 2(e). The damping should become anomalous ( t=L>1) above some critical layer thickness of h=lf0:3. Accord- ing to Fig. 2(e) this anomaly should increase linearly with the increase in h=lf. Direct measurements of damping were performed to test that eddy viscosity increases the dissipation above its quasi-2D value (1) in layers thicker than h=lf>0:2. The ow is forced by a 30 30 magnet array . The bot- tom drag is derived from the energy decay of the steady ow. After forcing is switched o, the mean ow energy exponentially decays in time with a characteristic time constant, as shown in Fig. 3(a). We compare the en- 02468 0 5 10 15 20t(s)E0(10 m s )-4 2 -2(a) t0E=E t () e0-( t - 0)/c97t 01234567 0 0.5 1 1.5hl/fa/D=/c97/c97L(b) 012345 0 0.06 0.12hl/f(c) a/D=/c97/c97LFIG. 3: (a) Decay of the ow energy in a single layer, h= 10 mm; (b) Energy damping rate normalized by the viscous quasi-2D damping rate aD==L, as a function of h=lf. Open circles refer to single layers, solid squares were obtained in the double layer congurations. (c) The damping anomaly coecientaDversush=lffor the case of a strong large-scale vortex (100 mm diameter, Vmax x;y = 16 mm/s). ergy damping rate measured in a single layer of dierent depths with the linear damping rate. Fig. 3(b) shows the anomaly coecient aD==Las a function of the normalized layer thickness h=lf. In the thinnest layer (h1:7 mm,h=lf0:21) the damping rate coincides with the linear damping rate (1). However for thicker layers the damping anomaly is higher, such that aDin- creases linearly with hreachingaD= 6 ath=lf= 1:25. Measurements of the damping show that the anomaly coecient aDin Fig. 3(b) agrees very well with the anomaly estimated using the eddy viscosity derived from (2), Fig. 2(e). In the double layer experiments however, aDis substantially lower, as shown by the solid squares in Fig. 3(b). This is not surprising in the light of the result of Fig. 2(d) (solid squares) which shows substantially less 3D motion in double layers. The above results are related to low forcing levels, when 3D eddies are generated due to the nite layer thickness, as in [3]. However, electromagnetic forcing, which is maximum near the bottom in the single layer experiments (magnets underneath the uid cell), may inject 3D eddies into the ow from the bottom bound- ary layer at higher forcing levels. Figure 3(c) shows the damping anomaly coecient aDmeasured in the ow driven by a single strong large magnetic dipole. A single4 S3(10 m s )-7 3 -3 -1123 0 0.02 0.04 0.06l(m)h=3m m h=1 0m m FIG. 4: Third-order structure functions measured in a thin layer,h= 3 mm (solid squares), and in a thick layer h= 10 mm (open diamonds). The forcing scale lf8 mm. large-scale vortex is produced, whose diameter is about 100 mm and the maximum horizontal velocity is about 16 mm/s. As the layer thickness is increased from 2 to 10 mm (h=lf= 0:020:1) while keeping the current density constant, the anomaly coecient increases up to aD= 3:6 due to the increase in the vertical velocity uc- tuations. Thus, turbulent bottom drag may occur in rel- atively thin layers at stronger forcing. Now we test if the increased three-dimensionality, as characterized by aD, leads to the suppression of the in- verse energy cascade. The inverse energy cascade can be detected by measuring the third-order structure function S3and by using the Kolmogorov ux relation which pre- dicts linear dependence of S3on the separation distance l,S3=l. Hereis the energy ux in k-space. It has been shown that in thin stratied layers S3is positive and it is a linear function of l, as expected for 2D turbu- lence [9]. Figure 4 shows third-order structure functions measured in a single layer of electrolyte for two layer depths,h= 3 and 10 mm. In the 3 mm layer, S3is a positive linear function of l, while in the 10 mm layer S3is much smaller, indicating very low energy ux in the inverse energy cascade. The damping anomaly in the 3 mm layer is aD2, while for the 10 mm layer it is high,aD5. Since in this experiment, the forcing is 2D and it is relatively weak (no secondary instabilities in the boundary layer), this result is in agreement with nu- merical simulations [3] which show strong suppression of the inverse energy cascade above h=lf0:5. The 3 mm layer corresponds to h=lf0:38, while for the 10 mm layerh=lf1:25. We do not observe however any sig- natures of the direct energy cascade range, Ek/k5=3 atk > k fin the 10 mm layer. Instead, the spectrum is much steeper than the usual k3enstrophy range. This is probably due to the fact that the Reynolds numberin this experiment is not sucient to sustain 3D direct turbulent cascade. Summarizing, we demonstrate for the rst time that increased three-dimensionality of ows in layers can be characterized by the anomalous damping coecient aD. We show that the increase in aDcorrelates with the sup- pression of the inverse energy cascade. On the other hand, a strong reduction in aD, which can be achieved in the double layer conguration, correlates well with the observation of the inverse energy cascade and spectral condensation of turbulence into a ow coherent over the entire domain [7, 9, 16]. The authors are grateful to H. Punzmann and V. Stein- berg for useful discussions. This work was supported by the Australian Research Council's Discovery Projects funding scheme (DP0881544). [1] R. Kraichnan, Phys. Fluids 10, 1417 (1967). [2] L.M. Smith, J. R. Chasnov and F. Walee, Phys. Rev. Lett77, 2467 (1996). [3] A. Celani, S. Musacchio and D. Vincenzi, Phys. Rev. Lett. 104, 184506 (2010). [4] J. Sommeria, J. Fluid Mech. 170,139 (1986) [5] A. Thess and O. Zikanov, J. Fluid Mech. 579,383 (2007) [6] R. Klein and A. Potherat, Phys. Rev. Lett. 104, 34502 (2010). [7] J. Paret & P. Tabeling, Phys. Fluids 10, 3126 (1998). [8] G. Boetta, A. Cenedese, S. Espa and S. Musacchio, Eu- rophys. Lett. 71, 590 (2005). [9] H. Xia, M. Shats and G. Falkovich, Phys. Fluids 21, 125101 (2009). [10] F.V. Dolzhanskii, V.A. Krymov and D.Yu. Manin, J. Fluid Mech 241, 705(1992). [11] A.E. Hansen, D. Marteau and P. Tabeling, Phys. Rev. E 58, 7261 (1998). [12] S. Danilov, F.V. Dolzhanskii, V.A. Dovzhenko and V.A. Krymov, Phys. Rev. E 65, 036316 (2002). [13] H.J.H. Clercx, G.J.F. van Heijst and M.L. Zoeteweij, Phys. Rev. E 67, 066303 (2003). [14] R. A. D. Akkermans, A. R. Cieslik, L. P. J. Kamp, R. R. Trieling, H. J. H. Clercx, and G. J. F. van Heijst, Phys. Fluids ,20, 116601(2008) [15] R. A. D. Akkermans, L. P. J. Kamp, H. J. H. Clercx, and G. J. F. van Heijst, Phys. Rev. E ,82, 026314 (2010) [16] M.G. Shats, H. Xia and H. Punzmann, Phys. Rev. E 71, 046409 (2005). [17] H. Xia, H. Punzmann, G. Falkovich and M.G. Shats, Phys. Rev. Lett. 101, 194504 (2008). [18] C.E. Willert and M. Gharib, Experiments in Fluids 10, 181 (1991). [19] Glossary of meteorology, ed. E. R. Huschke, Boston, American Meteorological Society, 1959.

2206.04899v1.Spin_Pumping_into_Anisotropic_Dirac_Electrons.pdf

Spin Pumping into Anisotropic Dirac Electrons Takumi Funato1;2, Takeo Kato3, Mamoru Matsuo2;4;5;6 1Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan 2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China. 3Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan 4CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and 6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: June 13, 2022) We study spin pumping into an anisotropic Dirac electron system induced by microwave irra- diation to an adjacent ferromagnetic insulator theoretically. We formulate the Gilbert damping enhancement due to the spin current owing into the Dirac electron system using second-order perturbation with respect to the interfacial exchange coupling. As an illustration, we consider the anisotropic Dirac system realized in bismuth to show that the Gilbert damping varies according to the magnetization direction in the ferromagnetic insulator. Our results indicate that this setup can provide helpful information on the anisotropy of the Dirac electron system. I. INTRODUCTION In spintronics, spin currents are crucial in using elec- trons' charge and spin. Spin pumping, the spin current generation of conduction electrons from nonequilibrium magnetization dynamics at magnetic interfaces, is a pop- ular method for generating and manipulating spin cur- rents. In previous experimental reports on spin pumping, the enhancement of Gilbert damping in ferromagnetic resonance (FMR) was observed due to the loss of angu- lar momentum associated with the spin current injection into the nonmagnetic layer adjacent to the ferromagnetic layer1{9. Mizukami et al. measured the enhancement of the Gilbert damping associated with the adjacent non- magnetic metal. They reported that the strong spin-orbit coupling in the nonmagnetic layer strictly aected the enhancement of the Gilbert damping3{5. Consequently, electric detection by inverse spin Hall eect, in which the charge current is converted from the spin current, led to spin pumping being used as an essential technique for studying spin-related phenomena in nonmagnetic mate- rials10{24. Saitoh et al. measured electric voltage in a bilayer of Py and Pt under microwave application. They observed that charge current converted because of inverse spin Hall eect from spin current injected by spin pump- ing11. In the rst theoretical report on spin pumping, Berger predicted an increase in Gilbert damping due to the spin current owing interface between the ferromagnetic and nonmagnetic layers25,26. Tserkovnyak et al. calculated the spin current owing through the interface27{29based on the scattering-matrix theory and the picture of adi- abatic spin pumping30{32. They introduced a complex spin-mixing conductance that characterizes spin trans- port at the interfaces based on spin conservation and no spin loss. The spin mixing conductance can represent the spin pumping-associated phenomena and is quanti- tatively evaluated using the rst principle calculation33. Nevertheless, microscopic analysis is necessary to under-stand the detailed mechanism of spin transport at the in- terface34{44. It was claried that spin pumping depends on the anisotropy of the electron band structure and spin texture. Spin pumping is expected to be one of the probes of the electron states41{44. Bismuth has been extensively studied because of its at- tractive physical properties, such as large diamagnetism, largeg-factor, high ecient Seebeck eect, Subrikov-de Haas eect, and de Haas-van Alphen eect45,46. The electrons in the conduction and valence bands near the L-point in bismuth, which contribute mainly to the vari- ous physical phenomena, are expressed as eective Dirac electrons. Thus, electrons in bismuth are called Dirac electrons45{47. The doping antimony to bismuth is known to close the gap and makes it a topological insulator48,49. Because of its strong spin-orbit interaction, bismuth has attracted broad attention in spintronics as a high ecient charge-to-spin conversion material50{55. The spin current generation at the interface between the bismuth oxide and metal has been studied since a signicant Rashba MicrowaveDirac electron system Interfacial exchange Ferromagnetic insulator FIG. 1. Schematic illustration of a bilayer system composed of the Dirac electron system and ferromagnetic insulator. The applied microwave excited precession of the localized spin in the ferromagnetic insulator and spin current is injected into the Dirac electron system.arXiv:2206.04899v1 [cond-mat.mes-hall] 10 Jun 20222 spin-orbit interaction appears at the interface56. The spin injection into bismuth was observed due to spin pumping from yttrium iron garnet or permalloy57{59. Nevertheless, microscopic analysis of spin pumping into bismuth has not been performed. The dependence of the spin pumping on the crystal and band structure of bis- muth remains unclear. This study aims at a microscopic analysis of spin in- jection due to spin pumping into an anisotropic Dirac electron system, such as bismuth, and investigates the dependence of spin pumping on the band structure. We consider a bilayer system comprising an anisotropic Dirac electron system and a ferromagnetic insulator where a microwave is applied (see Fig. 1). The eect of the inter- face is treated by proximity exchange coupling between the Dirac electron spins and the localized spins of the ferromagnetic insulator34{44. We calculate the Gilbert damping enhancement due to spin pumping from the fer- romagnetic insulator into the Dirac electron system up to the second perturbation of the interfacial exchange cou- pling. For illustarion, we calculate the enhancement of the Gilbert damping for an anisotropic Dirac system in bismuth. This paper is organized as follows: Sec. II describes the model. Sec. III shows the formulation of the Gilbert damping enhancement and discuss the eect of the inter- facial randomness on spin pumping. Sec. IV summarizes the results and demonstration of the Gilbert damping enhancement in bismuth. Sec. V presents the conclu- sion. The Appendices show the details of the calcula- tion. Appendix A denes the magnetic moment of elec- trons in a Dirac electron system. Appendix B provides the detailed formulation of the Gilbert damping modu- lation, and Appendix C presents the detailed derivation of Gilbert damping modulation. II. MODEL We consider a bilayer system composed of an anisotropic Dirac electron system and a ferromagnetic insulator under a static magnetic eld. We evaluate a microscopic model whose Hamiltonian is given as ^HT=^HD+^HFI+^Hex; (1) where ^HD,^HFI, and ^Hexrepresent an anisotropic Dirac electron system, a ferromagnetic insulator, and an inter- facial exchange interaction, respectively. A. Anisotropic Dirac system The following Wol Hamiltonian models the anisotropic Dirac electron system46,47,50: ^HD=X kcy k(~k
v2+
3)ck; (2)where 2
(6= 0) is the band gap, cy k(ck) is the electrons' four-component creation (annihilation) operator, and v is the velocity operator given by vi=P wiwith wibeing the matrix element of the velocity operator. = (x;y;z) are the Pauli matrices in the spin space and= (1;2;3) are the Pauli matrices specifying the conduction and valence bands. For this anisotropic Dirac system, the Matsubara Green function of the electrons is given by gk(in) =in+~~k
2+
3 (in+)22 k; (3) wheren= (2n+ 1)=is the fermionic Matsubara fre- quencies with nbeing integers, (>
) is the chem- ical potential in the conduction band ~kis dened by ~k
=~k=k
v, andkis the eigenenergy given by k=p
2+ (~kiwi)2=q
2+~2~k2: (4) The density of state of the Dirac electrons per unit cell per band and spin is givcen by () =n1 DX k;(k); (5) =jj 22~3s 2
2
3detij(jj
); (6) wherenDis the number of unit cells in the system and ij is the inverse mass tensor near the bottom of the band, which characterize the band structure of the anisotropic Dirac electron system: ij=1 ~2@2k @ki@kj k=0=1
X wiwj: (7) The spin operator can be dened as ^sq=X kcy kq=2sck+q=2; (8) si=m
Mi3;(i=x;y;z ); (9) whereMiare the matrix elements of the spin magnetic moment given as50,51 Mi= ijkwiwj =2: (10) The detailed derivation of the spin magnetic moment can be found in Appendix A. B. Ferromagnetic insulator The bulk ferromagnetic insulator under a static mag- netic eld is described by the quantum Heisenberg model as ^HFI=2JX hi;jiSi
SjgBhdcX iSX i; (11)3 FIG. 2. Relation between the original coordinates ( x;y;z ) and the magnetization-xed coordinates ( X;Y;Z ). The direction of the ordered localized spin hSi0is xed to the X-axis.is the polar angle and is the azimuthal angle. whereJis an exchange interaction, gis g-factor of the electrons,Bis the Bohr magnetization, and hi;jirepre- sents the pair of nearest neighbor sites. Here, we have in- troduced a magnetization-xed coordinate ( X;Y;Z ), for which the direction of the ordered localized spin hSi0is xed to the X-axis. The localized spin operators for the magnetization-xed coordinates are related to the ones for the original coordinates ( x;y;z ) as 0 @Sx Sy Sz1 A=R(;)0 @SX SY SZ1 A; (12) whereR(;) =Rz()Ry() is the rotation matrix com- bining the polar angle rotation around the y-axisRy() and the azimuthal angle rotation around the z-axis Rz(), given by R(;) =0 @coscossinsincos cossincossinsin sin 0 cos 1 A:(13) By applying the spin-wave approximation, the spin op- erators are written as S k=SY kiSZ k=p 2Sbk(by k) and SX k=Sby kbkusing magnon creation/annihilation op- erators,by kandbk. Then, the Hamiltonian is rewritten as ^HFI=X k~!kby kbk; (14) where ~!k=Dk2+~!0withD=zJSa2being the spin stiness and zbeing the number of the nearest neighbor sites, and ~!0=gBhdcis the Zeeman energy.C. Interfacial exchange interaction The proximity exchange coupling between the electron spin in the anisotropic Dirac system and the localized spin in the ferromagnetic insulator is modeled by ^Hex=X q;k(Tq;k^s+ qS k+ h.c.); (15) whereTq;kis a matrix element for spin transfer through the interface and ^ s q= ^sY qi^sZ qare the spin ladder operators of the Dirac electrons. According to the re- lation between the original coordinate ( x;y;z ) and the magnetization-xed coordinate ( X;Y;Z ), the spin oper- ators of the Dirac electrons are expressed as 0 @sX sY sZ1 A=R1(;)0 @sx sy sz1 A; (16) whereR1(;) =Ry()Rz() is given by R1(;) =0 @coscoscossinsin sin cos 0 sincossinsincos1 A:(17) The spin ladder operators are given by s+=m
aiMi; s=m
a iMi; (18) whereai(i=x;y;z ) are dened by 0 @ax ay az1 A=0 @sin+isincos cos+isinsin icos1 A: (19) III. FORMULATION Applying a microwave to the ferromagnetic insulator includes the localized spin's precession. The Gilbert damping constant can be read from the retarded magnon Green function dened by GR k(!) =i ~Z1 0dtei(!+i)th[S+ k(t);S k]i; (20) withS+ k(t) =ei^HT=~S+ kei^HT=~being the Heisenberg representation of the localized spin, since one can prove that the absorption rate of the microwave is proportional to ImGR k=0(!) (see also Appendix B). By considering the second-order perturbation with respect to the matrix el- ement for the spin transfer Tq;k, the magnon Green func- tion is given by34{44 GR 0(!) =2S=~ (!!0) +i(+)!: (21) Here, we introduced a term, i!, in the denominator to express the spin relaxation within a bulk FI, where4 indicates the strength of the Gilbert damping. The enhancement of the damping, , is due to the adjacent Dirac electron system, calculated by =2S ~!X qjTq;0j2ImR q(!); (22) whereR q(!) is the retarded component of the spin sus- ceptibility (dened below). We assume that the FMR peak described by Im GR k=0(!) is suciently sharp, i.e., +1. Then, the enhancement of the Gilbert damp- ing can be regarded as almost constant around the peak (!'!0), allowing us to replace !inwith!0. The retarded component of the spin susceptibility for the Dirac electrons: R q(!) =i ~Z1 1dtei(!+i)t(t)h[s+ q(t);s q]i: (23) The retarded component of the spin susceptibility is derived from the following Matsubara Green function through analytic continuation i!l!~!+i: q(i!l) =Z 0dei!lh^s+ q()^s qi; (24) where!l= 2l= is the bosonic Matsubara frequency withlbeing integers. According to Wick's theorem, the Matsubara representation of the spin susceptibility is given by q(i!l) =1X k;intr[s+gk+q(in+i!l)sgk(in)];(25) whereP inindicates the sum with respect to the fermionic Matsubara frequency, n= (2+ 1)n=. The imaginary part of the spin susceptibility is given by ImR q(!) =F(;)X kX ;0=1 2+0 62
2+2 k kk+q h f(0k+q)f(k)i (~!0k+q+k);(26) wheref() = (e()+ 1)1is the Fermi distribution function,=is a band index (see Fig. 3), and F(;) is the dimensionless function which depends on the di- rection of the ordered localized spin, dened by F(;) =2m
2X aiMia jMj: (27) For detailed derivation, see Appendix C. In this paper, we model the interfacial spin transfer as a combination of the clean and dirty processes. The former corresponds to the momentum-conserved spin transfer and the latter to the momentum-nonconserved one41,44. By averaging over the position of the localized spin at FIG. 3. Schematic illustration of the band structure of the anisotropic Dirac electron system. The red band represents the conduction band with = +, and the blue band repre- sents the valence band with =. The chemical potential is in the conduction band. the interface, we can derive the matrix elements of the interfacial spin-transfer process as jTq;0j2=T2 1q;0+T2 2; (28) whereT1andT2are the averaged matrix elements con- tributing to the clean and dirty processes, respectively. Then, the enhancement of the Gilbert damping is given by =2S ~!F(;)n T1Im ~R uni(!0) +T2Im ~R loc(!0)o ; (29) whereR uni(!) andR uni(!) are the local and uniform spin susceptibilities dened by ~R loc(!0) =F1(;)X qR q(!0); (30) ~R uni(!0) =F1(;)R 0(!0); (31) respectively. From Eq. (26), their imaginary parts are calculated as Im ~R loc(!0) =n2 DZ d()(+~!0) 1 2+2
2+2 6(+~!0)h f(+~!0)f()i ; (32) Im ~R uni(!0) =nD~!0 2
~2!2 04
2 3~2!2 0 h f(~!0 2)f(~!0 2)i : (33) The enhancement of the Gilbert damping, , depends on the direction of the ordered localized spin through the5 FIG. 4. FMR frequency dependence of the (a) local and (b) uniform spin susceptibilities. The local spin sus- ceptibility is normalized by n2 D2 0and scaled by 106, and the uniform spin susceptibility is normalized by nD0with 01=22~3p detij. Note that kBis the Boltzmann con- stant. The line with kBT=
= 0:001 is absent in (a) because the local spin susceptibility approaches zero at low tempera- ture. dimensionless function F(;) regardless of the interfa- cial condition. By contrast, the FMR frequency dependence of re- ects the interfacial condition; for a clean interface, it is determined mainly by Im R uni(!0), whereas for a dirty interface, it is determined by Im R loc(!0). The FMR fre- quency dependence of the local and uniform spin sus- ceptibilities, Im R loc(!0) and ImR uni(!0), are plotted in Figs. 4 (a) and (b), respectively. The local and uniform spin susceptibilities are normalized by n2 D2 0andnD0, respectively, where 01=22~3p detijis dened. In the calculation, the ratio of the chemical potential to the energy gap was set to =
'4:61, which is the value in the bismuth46. According to Fig. 4 (a), the local spin sus- ceptibility increases linearly with the frequency !in the low-frequency region. This !-linear behavior can be re- produced analytically for low temperatures and ~!: Im ~loc(!0)'~!0 2n2 D[()]2 1 +2
2+2 32 :(34)Fig. 4 (b) indicates a strong suppression of the uniform spin susceptibility below a spin-excitation gap ( !0<2). This feature can be checked by its analytic form at zero temperature: Im ~R uni(!0) =nD~!0 2
~2!2 04
2 3~2!2 0(~!02): (35) Thus, the FMR frequency dependence of the enhance- ment of the Gilbert damping depends on the interfacial condition. This indicates that the measurement of the FMR frequency dependence may provide helpful infor- mation on the randomness of the junction. IV. RESULT We consider bismuth, which is one of the anisotropic Dirac electron systems45,46,52,60,61. The crystalline struc- ture of pure bismuth is a rhombohedral lattice with the space group of R3msymmetry, see Figs. 5 (a) and (b). It is reasonable to determine the Cartesian coordinate system in the rhombohedral structure using the trigonal axis withC3symmetry, the binary axis with C2symme- try, and the bisectrix axis, which is perpendicular to the trigonal and binary axes. Hereafter, we choose the x-axis as the binary axis, the y-axis as the bisectrix axis, and thez-axis as the trigonal axis. Note that the trigonal, bi- nary, and bisectrix axes are denoted as [0001], [1 210], and [1010], respectively, where the Miller-Bravais indices are used. The bismuth's band structure around the Fermi surface consists of three electron ellipsoids at L-points and one hole ellipsoid at the T-point. It is well known that the electron ellipsoids are the dominant contribu- tion to the transport phenomena since electron's mass is much smaller than that of the hole, see Fig. 5 (c). Therefore, the present study considers only the electron systems at the L-points. The electron ellipsoids are sig- nicantly elongated, with the ratio of the major to minor axes being approximately 15 : 1. Each of the three elec- tron ellipsoids can be converted to one another with 2 =3 rotation around the trigonal axis. The electron ellipsoid along the bisectrix axis is labeled as e1, and the other two-electron ellipsoids are labeled e2 ande3. The in- verse mass tensor for the e1 electron ellipsoids is given by $ e1=0 B@10 0 024 0431 CA: (36) The inverse mass tensor of the electron ellipsoids e2 and e3 are obtained by rotating that of e1 by 2=3 rotation6 as below: $ e2;e3=1 40 BB@1+ 32p 3(12)2p 34 p 3(12) 31+224 2p 3424 431 CCA: (37) Let us express the dimensionless function F(;) rep- resenting the localized spin direction dependence of the damping enhancement on the inverse mass tensors. F(;) =2m
2X h (sin2+ sin2cos2)M2 x +(cos2+ sin2sin2)M2 y + cos2(M2 zsin 2MxMy) + sin 2Mz(Mxcos+Mysin)i : (38) Here, we use the following calculations: X M2 x=
2 4(yyzz2 yz)total=
2 4m2?;(39) X M2 y=
2 4(zzxx2 zx)total=
2 4m2?;(40) X M2 z=
2 4(xxyy2 xy)total=
2 4m2k;(41) X MiMj=
2 4(ikjkijkk)total= 0;(42) wherei;j;k are cyclic. (


)totalrepresents the summa- tion of the contributions of the three electron ellipsoids, and k, ?(>0) are the total Gaussian curvature of the three electron ellipsoids normalized by the electron mass m, given by k= 3m212; (43) ?=3 2m2[(1+2)32 4]: (44) Hence, the dimensionless function Fis given by F() = (1 + sin2)?+ cos2k: (45) The results suggest that the variation of the damping enhancement depends only on the polar angle , which is the angle between the direction of the ordered localized spinhSi0and the trigonal axis. It is also found that the  dependence of the damping enhancement originates from the anisotropy of the band structure. The dimensionless functionF() is plotted in Fig. 6 by varying the ratio of the total Gaussian curvatures x= ?=k, which cor- responds to the anisotropy of the band structure. Fig- ure 6 shows that the -dependence of the damping en- hancement decreases with smaller xand the angular de- pendence vanishes in an isotropic Dirac electron system BinaryBisectrixTrigonal e�e�e�(c) Binary(x)(a) Bisectrix(y)Trigonal(z) BinaryBisectrixTrigonal(b)FIG. 5. (a) The rhombohedral lattice structure of bismuth. Thex-axis,y-axis, andz-axis are chosen as the binary axis withC2symmetry, the bisectrix axis, and the trigonal axis withC3symmetry, respectively. The yellow lines represents the unit cell of the rhombohedral lattice. (b) The rhombohe- dral structure viewed from the trigonal axis. (c) Schematic illustration of the band structure at the Fermi surface. The three electron ellispoids at L-points are dominant contribu- tion to the spin transport. x= 1. Bismuth is known to have a strongly anisotropic band structure. The magnitude of the matrix elements of the inverse mass 1-4was experimentally determined as m1= 806,m2= 7:95,m3= 349, and m4= 37:6. The total Gaussian curvatures are evaluated as46 k'1:92104; (46) ?'4:24105: (47) The ratio of the total Gaussian curvature is estimated asx'22:1. Therefore, the damping enhancement is expected to depend strongly on the polar angle in a bi- layer system composed of single-crystalline bismuth and ferromagnetic insulator. Conversely, the -dependence of the damping enhancement is considered to be suppressed for polycrystalline bismuth. The damping enhancement is independent of the az- imuthal angle . Therefore, it is invariant even on ro- tating the spin orientation around the trigonal axis. The reason is that the azimuthal angular dependence of the damping enhancement cancels out when the contribu- tions of the three electron ellipsoids are summed over, although each contribution depends on the azimuthal an- gle. The azimuthal angular dependence of the damping enhancement is expected to remain when strain breaks the in-plane symmetry. Additionally, suppose the spin can be injected into each electron ellipsoid separately, e.g., by interfacial manipulation of the bismuth atoms. In that case, the damping enhancement depends on the azimuthal angle of the spin orientation of the ferromag- netic insulator39. This may be one of the probes of the electron ellipsoidal selective transport phenomena.7 - /2 0 /2 theta1.01.52.0damping_modulation FIG. 6. The -dependence of the damping enhancement for dierent x. The ratio of the total Gaussian curvatures x= ?=krepresents the anisotropy of the band structure. The blue line with x= 22:1 corresponds to the damping en- hancement in single-crystalline bismuth, and the other lines correspond to that in the weakly anisotropic band structure. As can be seen from the graph, the -dependence of the damp- ing enhancement decreases as the more weakly anisotropic band structure, and the angular dependence turns out to van- ish in an isotropic Dirac electron system with x= 1. It is also noteworthy that the damping enhancement varies according to the ordered localized spin direction with both clean and dirty interfaces; that is independent of whether momentum is conserved in interfacial spin transport. Conversely, it was reported that the spin ori- entation dependence of the damping enhancement due to the Rashba and Dresselhaus spin-orbit interaction turned out to vanish by interfacial inhomogeneity42,43. V. CONCLUSION We theoretically studied spin pumping from a ferro- magnetic insulator to an anisotropic Dirac electron sys- tem. We calculated the enhancement of the Gilbert damping in the second perturbation concerning the prox- imity interfacial exchange interaction by considering the interfacial randomness. For illustration, we calcu- lated the enhancement of the Gilbert damping for an anisotropic Dirac system realized in bismuth. We showed that the Gilbert damping varies according to the polar angle between the ordered spin hSi0and the trigonal axis of the Dirac electron system whereas it is invariant in its rotation around the trigonal axis. Our results indicate that the spin pumping experiment can provide helpful in- formation on the anisotropic band structure of the Dirac electron system. The Gilbert damping is invariant in the rotation around the trigonal axis because the contributions of each electron ellipsoid depend on the in-plane direction of theordered spinhSi0. Nevertheless, the total contribution becomes independent of the rotation of the trigonal axis after summing up the contributions from the three elec- tron ellipsoids that are related to each other by the C3 symmetry of the bismuth crystalline structure. If the spin could be injected into each electron ellipsoid separately, it is expected that the in-plane direction of the ordered localized spin would in uence the damping enhancement. This may be one of the electron ellipsoid selective spin in- jection probes. The in-plane direction's dependence will also appear when a static strain is applied. A detailed discussion of these eects is left as a future problem. ACKNOWLEDGMENTS The authors would like to thank A. Yamakage and Y. Ominato for helpful and enlightening discussions. The continued support of Y. Nozaki is greatly appreciated. We also thank H, Nakayama for the daily discussions. This work was partially supported by JST CREST Grant No. JPMJCR19J4, Japan. This work was supported by JSPS KAKENHI for Grants (Nos. 20H01863, 20K03831, 21H04565, 21H01800, and 21K20356). MM was sup- ported by the Priority Program of the Chinese Academy of Sciences, Grant No. XDB28000000. Appendix A: Magnetic moment of electrons in Dirac electron system In this section, we dene the spin operators in the Dirac electron systems. The Wol Hamiltonian around the L point is given by HD=3
2
v, where vi=P wiwithwibeing the matrix component of the velocity vectors and =p+e cAis the momen- tum operator including the vector potential. It is rea- sonable to determine the magnetic moment of electrons in an eective Dirac system as the coecient of the Zee- man term. The Wol Hamiltonian is diagonalized by the Schrieer-Wol transformation up to v=
as below: eiHDei'
+1 2
(
v)2 3; (A1) where=1 2

vis chosen to erase the o-diagonal matrix for the particle-hole space. We can proceed cal- culation as follows: (
v)2=ijwiwj(+i  ); = (iwi)2+i 2  []iijkwjwk; =
 

+~e c
MiBi ; (A2) where we used ( ) =e~ cirAandMiis dened as Mi=1 2 ijkwjwk : (A3)8 Finally, we obtain eiHDei'
+
$
 2 Bis;i; (A4) wheres;iis a magnetic moment of the Dirac electrons dened as s;i=~e 2c
Mi3=~e 2c
Mi 0 0 : (A5) In the main text, we dened the spin operator sas the magnetic moment sdivided by the Bohr magnetization B=~e=2mc, i.e., si=s;i B=m
Mi 0 0 : (A6) For an isotropic Dirac system, the matrix component is given bywi=viand Eq. (A6) reproduces the well- known form of the spin operator s=g 2 0 0 ; (A7) whereg= 2m=mis the eective g-factor with m=
=v2being eective mass. Appendix B: Linear Response Theory In this section, we brie y explain how the microwave absorption rate is written in terms of the uniform spincorrelation function. The Hamiltonian of an external circular-polarized microwave is written as ^Hrf=gBhrf 2X i(S iei!t+S+ iei!t) =gBhrfpnF 2(S 0ei!t+S+ 0ei!t); (B1) wherehrfis an amplitude of the magnetic eld of the microwave, S kare the Fourier transformations dened as S k=1pnFX iS ieik
Ri; (B2) andRiis the position of the locazed spin i. Using the lin- ear response theory with respect to ^Hrf, the expectation value of the local spin is calculated as hS+ 0i!=GR 0(!)gBhrfpnF 2; (B3) whereGR k(!) is the spin correlation function dened in Eq. (20). Since the microwave absorption is determined by the dissipative part of the response function, it is proportional to Im GR 0(!), that reproduces a Lorentzian- type FMR lineshape. As explained in the main text, the change of the linewidth of the microwave absorption, , gives information on spin excitation in the Dirac system via the spin susceptibility as shown in Eq. (22). Appendix C: Spin susceptibility of Dirac electrons In this section, we give detailed derivation of Eq. (26). The trace part in Eq. (25) is calculated as tr[s+gk+q(in+i!l)sgk(in)] =[(in+i!l+)(in+) +
2]tr[s+s]tr[s+~(~k+~q)
s~~k
] [(in+i!l+)22 k+q][(in+)22 k]; (C1) where ( ~k+~q)
= (k+q)
v. Using the following relations tr[s+s] =2m
2X aiMia jMj; (C2) tr[s+~(~k+~q)
s~~k
] =2m
2X (2aiMi~~ka jMj~~k~2~k2aiMia jMj); (C3) the spin susceptibility is given by q(i!l) =2F(;)X k1X in(in+i!l+)(in+) +
2+~2~k2=3 [(in+i!l+)22 k+q][(in+)22 k]; (C4) where we dropped the terms proportional to ~k~k(6=) because they vanish after the summation with respect to the wavenumber k. Here, we introduced a dimensionless function, F(;) = (2m=
)2P aiMia jMj, which9 depends on the direction of the magnetization of the FI. Representing the Matsubara summation as the following contour integral, we derive q(i!l) =2F(;)X kIdz 4itanh(z) 2z(z+i!l) +
2+~2~k2=3 [(z+i!l)22 k+q][z22 k]; (C5) = 2F(;)X kIdz 2if(z)z(z+i!l) +
2+~2~k2=3 [(z+i!l)22 k+q][z22 k]; (C6) We note that tanh( (z)=2) has poles at z=in+and is related to the Fermi distribution function f(z) as tanh[(z)=2] = 12f(z). Using the following identities 1 z22 k=1 2kX = zk; (C7) z z22 k=1 2X =1 zk; (C8) the spin susceptibility is given by q(i!l) =F(;)X kIdz 2if(z)X ;0=" 1 2+(
2+~2~k2=3)0 2kk+q# 1 zk1 z+i!l0k+q; (C9) =F(;)X kX ;0=1 2+0 62
2+2 k kk+qf(0k+q)f(k) i!l0k+q+k: (C10) By the analytic continuation i!l=~!+i, we derive the retarded spin susceptibility as below: R q(!) =F(;)X kX ;0=1 2+0 62
2+2 k kk+qf(0k+q)f(k) ~!+i0k+q+k: (C11) The imaginary part of the spin susceptibility is given by ImR q(!) =F(;)X kX ;0=1 2+0 62
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2009.14143v1.Structural_Phase_Dependent_Giant_Interfacial_Spin_Transparency_in_W_CoFeB_Thin_Film_Heterostructure.pdf

1 Structural Phase Dependent Giant Interfacial Spin Transparency in W/CoFeB Thin Film Heterostructure Surya Narayan Panda, Sudip Majumder, Arpan Bhattacharyya, Soma Dutta, Samiran Choudhury and Anjan Barman* Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India *E-mail: abarman@bose.res.in Keywords: (Thin Film Heterostructures, Interface Properties, Spin Pumping, Spin Transparency, Spin-Mixing Conductance, Gilbert Damping, Time-resolved Magneto-optical Kerr Effect) Abstract Pure spin current has transfigured the energy-efficient spintronic devices and it has the salient characteristic of transport of the spin angular momentum. Spin pumping is a potent method to generate pure spin current and for its increased efficiency high effective spin-mixing conductance ( Geff) and interfacial spin transparency ( T) are essential. Here, a giant T is reported in Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) heterostructures in beta-tungsten (β-W) phase by employing all-optical time-resolved magneto-optical Kerr effect technique. From the variation of Gilbert damping with W and CoFeB thicknesses, the spin diffusion length of W and spin- mixing conductances are extracted. Subsequently, T is derived as 0.81 ± 0.03 for the β- W/CoFeB interface. A sharp variation of Geff and T with W thickness is observed in consonance with the thickness-dependent structural phase transition and resistivity of W. The spin memory loss and two-magnon scattering effects are found to have negligible contributions to damping modulation as opposed to spin pumping effect which is reconfirmed from the invariance of damping with Cu spacer layer thickness inserted between W and CoFeB. The observation of giant interfacial spin transparency and its strong dependence on crystal structures of W will be important for pure spin current based spin-orbitronic devices. 2 1. Introduction The rapid emergence of spintronics has promised a new paradigm of electronics based on the spin degree of freedom either associated with the charge or by itself.[1-3] This has potential advantages of non-volatility, reduced electrical power consumption, increased data processing speed, and increased integration densities as opposed to its semiconductor counterpart.[4] A major objective of modern spintronics is to harness pure spin current, which comprises of flow of spins without any net flow of charge current.[5, 6] This has the inherent benefit of reduced Joule heating and Oersted fields together with the ability to manipulate magnetization . Three major aspects of spin current are its generation, transport, and functionalization. Pure spin current can be generated by spin-Hall effect,[7,8] Rashba-Edelstein effect,[9,10] spin pumping,[11- 13] electrical injection in a lateral spin valve using a non-local geometry,[14,15] and spin caloritronic effects.[16,17] Among these, spin pumping is an efficient and extensively used method of spin injection from ferromagnet (FM) into normal metal (NM) where the precessing spins from FM transfer spin angular momentum to the conduction electrons of adjacent NM layer in NM/FM heterostructure, which gets dissipated by spin-flip scattering. The efficiency of spin pumping is characterized by spin-mixing conductance and spin diffusion length. The dissipation of spin current into the NM layer results in loss of spin angular momentum in the FM layer leading to an increase in its effective Gilbert damping parameter ( αeff). Thus, spin pumping controls the magnetization dynamics in NM/FM heterostructures, which is crucial for determining the switching efficiency of spin-torque based spintronic devices. The enhancement in αeff is more prominent in heavy metals (HM) with high spin-orbit coupling (SOC) due to stronger interaction between electron spin and lattice. Intense research in the field of spin- orbitronics has revealed that interface dependent spin transport is highly influenced by the spin transparency, which essentially determines the extent of spin current diffused through the NM/FM interface.[18,19] 3 The highly resistive β-W, which shows a distorted tetragonal phase commonly referred to as A15 structure, is well known for exhibiting large spin Hall angle (SHA) (up to ~0.50) [20] as compared to other transition metal elements such as Pt (0.08) [21] and β-Ta (0.12).[7] Besides, in W/FM heterostructures, W leads to highly stable perpendicular magnetic anisotropy[22] and interfacial Dzyaloshinskii-Moriya interaction.[23] Another important characteristic associated with W is that it shows a thickness-dependent phase transition in the sub-10 nm thickness regime.[24,25] In general, sputter-deposited W films with thickness well below 10 nm are found to have β phase with high resistivity, whereas the films with thickness above 10 nm possess predominantly α phase (bcc structure) with low resistivity. A small to moderate SHA has been reported for the α and mixed (α + β) phase (<0.2) of W.[24] As SHA and effective spin-mixing conductance ( Geff) are correlated, one would expect that interfacial spin transparency ( T), which is also a function of Geff, should depend on the structural phase of W thin films. Furthermore, the magnitude of the spin-orbit torque (SOT) depends on the efficiency of spin current transmission (i.e. T) across the NM/FM interface. It is worth mentioning that due to high SOC strength, W is a good spin-sink material and also cost-effective in comparison with the widely used NM like Pt. On the other hand, CoFeB due to its notable properties like high spin polarization, large tunnel magnetoresistance, and low intrinsic Gilbert damping, is used as FM electrode in magnetic tunnel junctions. The presence of Boron at the NM/CoFeB interface makes this system intriguing as some recent studies suggest that a small amount of boron helps in achieving a sharp interface and increases the spin polarization, although an excess of it causes contamination of the interface. To this end, determination of T of the technologically important W/CoFeB interface and its dependence on the W-crystal phase are extremely important but still absent in the literature. Besides spin pumping, there are different mechanisms like spin memory loss (SML),[26] Rashba effect,[10] two-magnon scattering (TMS),[27] and interfacial band hybridization[28] which may also cause loss of spin angular momentum at NM/FM interface, resulting in increase of αeff and 4 decrease of the spin transmission probability. However, for improved energy efficiency, the NM/FM interface in such engineered heterostructures must possess high spin transmission probability. Consequently, it is imperative to get a deeper insight into all the mechanisms involved in generation and transfer of spin current for optimizing its efficiency. Here, we investigate the effects of spin pumping on the Gilbert damping in W/CoFeB bilayer system as a function of W-layer thickness using recently developed all-optical technique, which is free from delicate micro-fabrication and electrical excitation and detection.[29] This is a local and non-invasive method based on time-resolved magneto-optical Kerr effect (TR-MOKE) magnetometry. Here, the damping is directly extracted from the decaying amplitude of time- resolved magnetization precession, which is free from experimental artifacts stemming from multimodal oscillation, sample inhomogeneity, and defects. From the modulation of damping with W layer thickness, we have extracted the intrinsic spin-mixing conductance ( G↑↓) of the W/CoFeB interface which excludes the backflow of spin angular momentum and spin diffusion length(𝜆௦ௗ) of W. Furthermore, we have modeled the spin transport using both the ballistic transport model[30, 31] and the model based on spin diffusion theory[32,33]. Subsequently, Geff, which includes the backflow of spin angular momentum, is estimated from the dependence of damping on the CoFeB layer thicknesses. By using both the spin Hall magnetoresistance model[34] and spin transfer torque based model utilizing the drift-diffusion approximation[35], we have calculated the T of W/CoFeB interface. The spin Hall magnetoresistance model gives lower value of T than the drift-diffusion model, but the former is considered more reliable as the latter ignores the spin backflow. We found a giant value of T exceeding 0.8 in the β phase of W, which exhibits a sharp decrease to about 0.6 in the mixed (α+β) phase using spin Hall magnetoresistance model. We have further investigated the other possible interface effects in our W/CoFeB system, by incorporating a thin Cu spacer layer of varying thickness between the W and CoFeB layers. Negligible modulation of damping with Cu thickness confirms the 5 dominance of spin pumping generated pure spin current and its transport in the modulation of damping in our system. 2. Results and Discussion Figure 1 (a) shows the grazing incidence x-ray diffraction (GIXRD) patterns of Sub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures at the glancing angle of 2o. In these plots, the peaks corresponding to α and β phase of W are marked. The high-intensity GIXRD peak at ∼44.5° and low intensity peak at ∼64° correspond primarily to the β phase (A15 structure) of W (211) and W(222) orientation, respectively. Interestingly, we find these peaks to be present for all thicknesses of W, but when t > 5 nm, then an additional peak at ∼40.1° corresponding to α-W with (110) crystal orientation appears. Consequently, we understand that for t ≤ 5 nm, W is primarily in β-phase, while for t > 5 nm a fraction of the α phase appears, which we refer to as the mixed (α+β) phase of W. These findings are consistent with some existing literature.[24,25] Some other studies claimed that this transition thickness can be tuned by carefully tuning the deposition conditions of the W thin films.[36] The average lattice constants obtained from the β-W peak at 44.5o and α-W peak at 40.1o correspond to about 4.93 and 3.15 Å, respectively. By using the Debye-Scherrer formula, we find the average crystallite size in β and α phase of W to be about 14 and 7 nm, respectively. It is well known that the formation of β-W films is characterized by large resistivity due to its A-15 structure which is associated with strong electron-phonon scattering, while the α-W exhibits comparatively lower resistivity due to weak electron-phonon scattering. We measured the variation of resistivity of W with its thickness across the two different phases, using the four-probe method. The inverse of sheet resistance ( Rs) of the film stack as a function of W thickness is plotted in Figure 1(b). A change of the slope is observed beyond 5 nm, which indicates a change in the W resistivity. The data have been fitted using the parallel resistors model[24] (shown in Figure S1 of the Supporting Information). [37] We estimate the average 6 resistivity of W ( ρW) in β and mixed (α+β) phase to be about 287 ± 19 and 112 ± 14 µΩ.cm, respectively, while the resistivity of CoFeB (ρCoFeB) is found to be 139 ± 16 μΩ.cm. Thus, the resistivity results corroborate well with those of the XRD measurement. The AFM image of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) (t = 1, 5 and 10 nm) samples in Figure 1(c) revealed the surface topography. We have used WSxM software to process the images.[38] The variation in the average surface roughness of the films with W thickness is listed in Table 1. The roughness varies very little when measured at various regions of space of the same sample. The surface roughness in all samples is found to be small irrespective of the crystal phase of W. Due to the small thicknesses of various layers in the heterostructures, the interfacial roughness is expected to show its imprint on the measured topographical roughness. We thus understand that the interfacial roughness in these heterostructures is very small and similar in all studied samples. Details of AFM characterization is shown in Figure S2 of the Supporting Information.[37] 2.1. Principles behind the modulation of Gilbert damping with layer thickness: In an NM/FM bilayer magnetic damping can have various additional contributions, namely two-magnon scattering, eddy current, and spin pumping in addition to intrinsic Gilbert damping. Among these, the spin pumping effect is a non-local effect, in which an external excitation induces magnetization precession in the FM layer. The magnetization precession causes a spin accumulation at the NM/FM interface. These accumulated spins carry angular momentum to the adjacent NM layer, which acts as a spin sink by absorbing the spin current by spin-flip scattering, leading to an enhancement of the Gilbert damping parameter of FM. In 2002, Tserkovnyak and Brataas theoretically demonstrated the spin pumping induced enhancement in Gilbert damping in NM/FM heterostructures using time-dependent adiabatic scattering theory where magnetization dynamics in the presence of spin pumping can be described by a modified Landau-Lifshitz-Gilbert (LLG) equation as: [11-13] 7 ௗ𝒎 ௗ௧= −𝛾(𝒎×𝑯eff)+𝛼0(𝒎×ௗ𝒎 ௗ௧)+ఊ VMೞ𝑰௦ (1) where γ is the gyromagnetic ratio, Is is the total spin current, Heff is the effective magnetic field, α0 is intrinsic Gilbert damping constant, V is the volume of ferromagnet and Ms is saturation magnetization of the ferromagnet. As shown in equation (2), Is generally consists of a direct current contribution 𝑰𝒔𝟎 which is nonexistent in our case as we do not apply any charge current, 𝑰𝒔𝒑𝒖𝒎𝒑, i.e. spin current due to pumped spins from the FM to NM and 𝑰𝒔𝒃𝒂𝒄𝒌, i.e. a spin current backflow to the FM reflecting from the NM/substrate interface which is assumed to be a perfect reflector. 𝑰𝒔=𝑰𝒔𝟎+𝑰𝒔pump+𝑰𝒔back (2) Here, 𝑰𝒔𝒃𝒂𝒄𝒌 is determined by the spin diffusion length of the NM layer. Its contribution to Gilbert damping for most metals with a low impurity concentration is parametrized by a backflow factor β which can be expressed as:[39] 𝛽=൭2𝜋𝐺↑↓ටఌ ଷtanhቀ௧ ఒೞቁ൱ିଵ (3) where ε is the material-dependent spin-flip probability, which is the ratio of the spin-conserved to spin-flip scattering time. It can be expressed as: [40] 𝜀= (𝜆𝜆௦ௗ⁄)ଶ3⁄ (4) where λel and λsd are the electronic mean free path and spin diffusion length of NM, respectively. The spin transport through NM/FM interface directly depends on the spin-mixing conductance, which is of two types: (a) G↑↓, which ignores the contribution of backflow of spin angular momentum, and (b) Geff, which includes the backflow contribution. Spin-mixing conductance describes the conductance property of spin channels at the interface between NM and FM. Also, spin transport across the interface affects the damping parameter giving rise to αeff of the system 8 that can be modeled by both ballistic and diffusive transport theory. In the ballistic transport model, the αeff is fitted with the following simple exponential function:[30,31,39] 𝐺eff=𝐺↑↓൬1−𝑒ିమ ഊೞ൰=ସగdM ఓಳ(𝛼eff−𝛼) (5) 𝛥𝛼=𝛼eff−𝛼=ఓಳீ↑↓൭ଵିషమ ഊೞ൱ ସdMeff (6) Here, the exponential term signifies backflow spin current contribution and a factor of 2 in the exponent signifies the distance traversed by the spins inside the NM layer due to reflection from the NM/substrate interface. In the ballistic approach, the resistivity of NM is not considered while the NM thickness is assumed to be less than the mean free path. To include the effect of the charge properties of NM on spin transport, the model based on spin diffusion theory is used to describe αeff (t). Within this model, the additional damping due to spin pumping is described as:[32,33,36] 𝐺eff=ீ↑↓ ቆଵାమഐഊೞಸ↑↓ ୡ୭୲୦ቀ௧ఒೞൗቁቇ=ସగdM ఓಳ(𝛼eff−𝛼) (7) ∆𝛼=𝛼eff−𝛼=ఓಳீ↑↓ ସగdMቆଵା మഐഊೞಸ↑↓ ୡ୭୲୦ቀ௧ఒೞൗቁቇ (8) where ρ is the electrical resistivity of the W layer. Here the term మఘఒೞீ↑↓ cothቀ𝑡𝜆௦ௗൗቁ account for the back-flow of pumped spin current into the ferromagnetic layer. The reduction of spin transmission probability implies a lack of electronic band matching, intermixing, and disorder at the interface. The spin transparency, T of an NM/FM interface takes into account all such effects that lead to the electrons being reflected from the interface instead of being transmitted during transport. Further, T depends on both intrinsic and extrinsic interfacial factors, such as band-structure mismatch, Fermi velocity, interface imperfections, etc.[19,39] According to the spin Hall magnetoresistance model, the spin current density that 9 diffuses into the NM layer is smaller than the actual spin current density generated via the spin pumping in the FM layer. This model linked T with 𝐺eff by the following relation:[34,39] 𝑇=ீeff tanh൬ మഊೞ൰ ீeff coth൬ ഊೞ൰ା మഊೞమഐ (9) The interfacial spin transparency was also calculated by Pai et al. in the light of damping-like and field-like torques utilizing the drift-diffusion approximation. Here, the effects of spin backflow are neglected as it causes a reduction in the spin torque efficiencies. Assuming t ≫ λ and a very high value of d, T can be expressed as:[35] 𝑇=ଶீ↑↓ீಿಾ⁄ ଵାଶீ↑↓ீಿಾ⁄ (10) where, 𝐺ேெ= ఘఒೞమ is the spin conductance of the NM layer. In an NM/FM heterostructure, other than spin pumping, there is a finite probability to have some losses of spin angular momentum due to interfacial depolarization and surface inhomogeneities, known as SML and TMS, respectively. In SML, loss of spin angular momentum occurs when the atomic lattice at the interface acts as a spin sink due to the magnetic proximity effect or due to the interfacial spin-orbit scattering which could transfer spin polarization to the atomic lattice.[26] The TMS arises when a uniform FMR mode is destroyed and a degenerate magnon of different wave vector is created.[27] The momentum non- conservation is accounted for by considering a pseudo-momentum derived from internal field inhomogeneities or secondary scattering. SML and TMS may contribute to the enhancement of the Gilbert damping parameter considerably. Recently TMS is found to be the dominant contribution to damping for Pt-FM heterostructures.[41] In the presence of TMS and SML effective Gilbert damping can be approximated as:[41] αeff = α0 + αSP + αSML + αTMS ∆𝛼=𝛼eff−𝛼= 𝑔𝜇ீeff ା ீೄಾಽ ସdM+𝛽்ெௌ𝑑ିଶ (11) 10 where 𝐺ௌெ is the “effective SML conductance”, and βTMS is a “coefficient of TMS” that depends on both interfacial perpendicular magnetic anisotropy field and the density of magnetic defects at the FM surfaces. 2.2. All-optical measurement of magnetization dynamics: A schematic of the spin pumping mechanism along with the experimental geometry is shown in Figure 2(a). A typical time-resolved Kerr rotation data for the Sub/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample at a bias magnetic field, H = 2.30 kOe is shown in Figure 2(b) which consists of three different temporal regimes. The first regime is called ultrafast demagnetization, where a sharp drop in the Kerr rotation (magnetization) of the sample is observed immediately after femtosecond laser excitation. The second regime corresponds to the fast remagnetization where magnetization recovers to equilibrium by spin-lattice interaction. The last regime consists of slower relaxation due to heat diffusion from the lattice to the surrounding (substrate) superposed with damped magnetization precession. The red line in Figure 2(b) denotes the bi-exponential background present in the precessional data. We are mainly interested here in the extraction of decay time from the damped sinusoidal oscillation about an effective magnetic field and its modulation with the thickness of FM and NM layers. We fit the time-resolved precessional data using a damped sinusoidal function given by: 𝑀(𝑡)=𝑀(0)𝑒ିቀ ഓቁsin(2π𝑓𝑡+𝜑) (12) where τ is the decay time, φ is the initial phase of oscillation and f is the precessional frequency. The bias field dependence of precessional frequency can be fitted using the Kittel formula given below to find the effective saturation magnetization ( Meff): 𝑓=ఊ ଶ(𝐻(𝐻+4π𝑀eff))ଵ/ଶ (13) where γ = gµB/ħ, g is the Landé g-factor and ћ is the reduced Planck’s constant. From the fit, Meff and g are determined as fitting parameters. For these film stacks, we obtained effective 11 magnetization, Meff ≈ 1200 ± 100 emu/cc, and g = 2.0 ± 0.1. The comparison between Meff obtained from the magnetization dynamics measurement and Ms from VSM measurement for various thickness series are presented systematically in Figures S3-S5 of the Supporting Information.[37] For almost all the film stacks investigated in this work, Meff is found to be close to Ms, which indicates that the interface anisotropy is small in these heterostructures. We estimate αeff using the expression: [42] 𝛼eff=1 γτ(𝐻+2π𝑀eff) (14) where τ is the decay time obtained from the fit of the precessional oscillation with equation (12). We have plotted the variation of time-resolved precessional oscillation with the bias magnetic field and the corresponding fast Fourier transform (FFT) power spectra in Figure S6 of the Supporting Information.[37] The extracted values of αeff are found to be independent of the precession frequency f. Recent studies show that in presence of extrinsic damping contributions like TMS, αeff should increase with f, while in presence of inhomogeneous anisotropy in the system αeff should decrease with f.[43] Thus, frequency-independent αeff rules out any such extrinsic contributions to damping in our system. 2.3. Modulation of the Gilbert damping parameter: In Figure 3 (a) we have presented time-resolved precessional dynamics for Sub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) samples with 0 ≤ t ≤ 15 nm at H = 2.30 kOe. The value of α0 for the 3-nm-thick CoFeB layer without the W underlayer is found to be 0.006 ± 0.0005. The presence of W underlayer causes αeff to vary non monotonically over the whole thickness regime as shown by the αeff vs. t plot in Figure 3(b). In the lower thickness regime, i.e. 0 ≤ t ≤ 3 nm, Δ α increases sharply by about 90% due to spin pumping but it saturates for t ≥ 3 nm. However, for t > 5 nm, Δ α drops by about 30% which is most likely related to due to the thickness-dependent phase transition of W. At first, we have fitted our result for t ≤ 5 nm with equation (6) of the ballistic transport model and determined G↑↓ = (1.46 ± 0.01) × 1015 cm- 12 2 and λsd = 1.71 ± 0.10 nm as fitting parameters. Next, we have also fitted our results with equation (8) based on spin diffusion theory, where we have obtained G↑↓ = (2.19 ± 0.02) × 1015 cm-2 and λsd = 1.78 ± 0.10 nm. The value of G↑↓ using spin diffusion theory is about 28% higher than that of ballistic model while the value of λsd is nearly same in both models. Using values for λel (about 0.45 nm for W) from the literature[44] and λsd derived from our experimental data, we have determined the spin-flip probability parameter, ε = 2.30 × 10−2 from equation (4). To be considered as an efficient spin sink, a nonmagnetic metal must have ε ≥ 1.0 × 10-2 and hence we can infer that the W layer acts as an efficient spin sink here.[13] The backflow factor β can be extracted from equation (3). We have quantified the modulation of the backflow factor (Δ β) to be about 68% within the experimental thickness regime. To determine the value of 𝐺eff directly from the experiment, we have measured the time- resolved precessional dynamics for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) samples with 1 nm ≤ d ≤ 10 nm at H = 2.30 kOe as shown in Figure 4(a). The αeff is found to increase with the inverse of FM layer thickness ( Figure 4(b)). We have fitted our results first with equation (5), from which we have obtained 𝐺eff and 𝛼 to be (1.44 ± 0.01) × 1015 cm-2 and 0.006 ± 0.0005, respectively. By modelling the W thickness dependent modulation of damping of Figure 3(b) using equation (5), we have obtained 𝐺eff of W/CoFeB in β-phase (where ∆𝛼 ≈ 0.006) and α+β-mixed phase (where ∆𝛼 ≈ 0.004) of W to be (1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2, respectively. From these, we conclude that β-phase of W has higher conductance of spin channels in comparison to the α+β-mixed phase. The variation of 𝐺eff with W layer thickness is presented in Figure 5(a), which shows that 𝐺eff increases non monotonically and nearly saturates for t ≥ 3 nm. For t > 5 nm, 𝐺eff shows a sharp decrease in consonance with the variation of αeff. We have further fitted the variation of αeff with the inverse of FM layer thickness ( Figure 4(b)) using with equation (11) to isolate the contributions from SML, TMS and spin pumping (SP). 13 The values of 𝐺ௌெ , and βTMS are found to be (2.45 ± 0.05) × 1013 cm-2 and (1.09 ± 0.02) × 10- 18 cm2, respectively. 𝐺ௌெ is negligible in comparison with 𝐺eff which confirms the absence of SML contribution in damping. Contribution of TMS to damping modulation ( 𝛽்ெௌ𝑑ଶ) is also below 2% for all the FM thicknesses. The relative contributions are plotted in Figure 5(b). It is clear that spin pumping contribution is highly dominant over the SML and TMS for our studied samples. The value of our 𝐺eff in β-W/CoFeB is found to be much higher than that obtained for β-Ta/CoFeB[39] measured by all-optical TRMOKE technique as well as various other NM/FM heterostructures measured by conventional techniques as listed in Table 2. This provides another confirmation of W being a good spin sink material giving rise to strong spin pumping effect. We subsequently investigate the value of T for W/CoFeB interface, which is associated with the spin-mixing conductances of interface, spin diffusion length, and resistivity of NM as denoted in equations (9) and (10). T is an electronic property of a material that depends upon electronic band matching of the two materials on either side of the interface. After determining the resistivity, spin diffusion length and spin-mixing conductances experimentally, we have determined the value of T which depends strongly on the structural phase of W. Using equation (9) based on the spin-Hall magnetoresistance model, Tβ-W and T(α+β)-W are found to be 0.81 ± 0.03 and 0.60 ± 0.02, respectively. On the other hand, equation (10) of spin transfer torque based model utilizing the drift-diffusion approximation gives Tβ-W and T(α+β)-W to be 0.85 ± 0.03 and 0.63 ± 0.02, respectively, which are slightly higher than the values obtained from spin-Hall magnetoresistance model. However, we consider the values of T obtained from the spin-Hall magnetoresistance model to be more accurate as it includes the mandatory contribution of spin current backflow from W layer into the CoFeB layer. Nevertheless, our study clearly demonstrates that the value of spin transparency of the W/CoFeB interface is the highest reported among the NM/FM heterostructures as listed in Table 2. This high value of T, combined with the high spin Hall angle of β-W makes it an extremely useful material for pure 14 spin current based spintronic and spin-orbitronic devices. The structural phase dependence of T for W also provides a particularly important guideline for choosing the correct thickness and phase of W for application in the above devices. Finally, to directly examine the additional possible interfacial effects present in the W/CoFeB system, we have introduced a copper spacer layer of a few different thicknesses between the W and CoFeB layers. Copper has very small SOC and spin-flip scattering parameters and it shows a very high spin diffusion length. Thus, a thin copper spacer layer should not affect the damping of the FM layer due to the spin pumping effect but can influence the other possible interface effects. Thus, if other interface effects are substantial in our samples, the introduction of the copper spacer layer would cause a notable modulation of damping with the increase of copper spacer layer thickness ( c).[19,39] The time-resolved Kerr rotation data for the Sub/W(4 nm)/Cu(c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures with 0 ≤ c ≤ 1 nm are presented in Figure 6(a) at H = 2.30 kOe and Figure 6(b) shows the plot of αeff as a function of c. The invariance of αeff with c confirms that the interface of Cu/CoFeB is transparent for spin transport and possible additional interfacial contribution to damping is negligible, which is in agreement with our modelling as shown in Figure 5(b). 3. Conclusion In summary, we have systematically investigated the effects of thickness-dependent structural phase transition of W in W( t)/CoFeB( d) thin film heterostructures and spin pumping induced modulation of Gilbert damping by using an all-optical time-resolved magneto-optical Kerr effect magnetometer. The W film has exhibited structural phase transition from a pure β phase to a mixed (α + β) phase for t > 5 nm. Subsequently, β-W phase leads to larger modulation in effective damping ( αeff) than (α+β)-W. The spin diffusion length of W is found to be 1.71 ± 0.10 nm, while the spin pumping induced effective spin-mixing conductance 𝐺eff is found to be (1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2 for β and mixed (α+β) phase of W, 15 respectively. This large difference in 𝐺eff is attributed to different interface qualities leading towards different interfacial spin-orbit coupling. Furthermore, by analyzing the variation of αeff with CoFeB thickness in W (4 nm)/CoFeB (d)/SiO2 (2 nm), we have isolated the contributions of spin memory loss and two-magnon scattering from spin pumping, which divulges that spin pumping is the dominant contributor to damping. By modeling our results with the spin Hall magnetoresistance model, we have extracted the interfacial spin transparency ( T) of β- W/CoFeB and (α + β)-W/CoFeB as 0.81 ± 0.03 and 0.60 ± 0.02, respectively. This structural phase-dependent T value will offer important guidelines for the selection of material phase for spintronic applications. Within the framework of ballistic and diffusive spin transport models, the intrinsic spin-mixing conductance ( G↑↓) and spin-diffusion length ( λsd) of β-W are also calculated by studying the enhancement of αeff as a function of β-W thickness. Irrespective of the used model, the value of T for W/CoFeB interface is found to be highest among the NM/FM interfaces, including the popularly used Pt/FM heterostructures. The other possible interface effects on the modulation of Gilbert damping are found to be negligible as compared to the spin pumping effect. Thus, our study helps in developing a deep understanding of the role of W thin films in NM/FM heterostructures and the ensuing spin-orbit effects. The low intrinsic Gilbert damping parameter, high effective spin-mixing conductance combined with very high interface spin transparency and spin Hall angle can make the W/CoFeB system a key material for spin- orbit torque-based magnetization switching, spin logic and spin-wave devices. 4. Experimental Section/Methods 4.1. Sample Preparation Thin films of Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) were deposited by using RF/DC magnetron sputtering system on Si (100) wafers coated with 285 nm-thick SiO 2. We varied the W layer thickness as t = 0, 0.5, 1, 1.5, 2, 3, 4, 5, 8, 10 and 15 nm and CoFeB layer thickness as d = 1, 2, 3, 5 and 10 nm. The depositions were performed at an average base pressure of 1.8 × 10-7 Torr 16 and argon pressure of about 0.5 mTorr at a deposition rate of 0.2 Å/s. Very slow deposition rates were chosen for achieving a uniform thickness of the films even at a very thin regime down to sub-nm. The W and CoFeB layers were deposited using average DC voltages of 320 and 370 V, respectively, while SiO 2 was deposited using average RF power of 55 watts. All other deposition conditions were carefully optimized and kept almost identical for all samples. In another set of samples, we introduced a thin Cu spacer layer in between the CoFeB and W layers and varied its thickness from 0 nm to 1 nm. The Cu layer was deposited at a DC voltage of 350 V, argon pressure of 0.5 mTorr and deposition rate of 0.2 Ǻ/s. 4.2. Characterization Atomic force microscopy (AFM) was used to investigate the surface topography and vibrating sample magnetometry (VSM) was used to characterize the static magnetic properties of these heterostructures. Using a standard four-probe technique the resistivity of the W films was determined and grazing incidence x-ray diffraction (GIXRD) was used for investigating the structural phase of W. To study the magnetization dynamics, we used a custom-built TR- MOKE magnetometer based on a two-color, collinear optical pump-probe technique. Here, the second harmonic laser pulse (λ = 400 nm, repetition rate = 1 kHz, pulse width >40 fs) of an amplified femtosecond laser, obtained using a regenerative amplifier system (Libra, Coherent) was used to excite the magnetization dynamics, while the fundamental laser pulse (λ = 800 nm, repetition rate = 1 kHz, pulse width ~40 fs) was used to probe the time-varying polar Kerr rotation from the samples. The pump laser beam was slightly defocused to a spot size of about 300 µm and was obliquely (approximately 30° to the normal on the sample plane) incident on the sample. The probe beam having a spot size of about 100 µm was normally incident on the sample, maintaining an excellent spatial overlap with the pump spot to avoid any spurious contribution to the Gilbert damping due to the dissipation of energy of uniform precessional mode flowing out of the probed area. A large enough magnetic field was first applied at an angle of about 25° to the sample plane to saturate its magnetization. This was followed by a 17 reduction of the magnetic field to the bias field value ( H = in-plane component of the bias field) to ensure that the magnetization remained saturated along the bias field direction. The tilt of magnetization from the sample plane ensured a finite demagnetizing field along the direction of the pump pulse, which was modified by the pump pulse to induce a precessional magnetization dynamics in the sample. The pump beam was chopped at 373 Hz frequency and the dynamic Kerr signal in the probe pulse was detected using a lock-in amplifier in a phase- sensitive manner. The pump and probe fluences were kept constant at 10 mJ/cm2 and 2 mJ/cm2, respectively, during the measurement. All the experiments were performed under ambient conditions at room temperature. Acknowledgements AB gratefully acknowledges the financial assistance from the S. N. Bose National Centre for Basic Sciences (SNBNCBS), India under Project No. SNB/AB/18-19/211. SNP, SM and SC acknowledge SNBNCBS for senior research fellowship. ArB acknowledges SNBNCBS for postdoctoral research associateship. SD acknowledges UGC, Govt of India for junior research fellowship. 18 References: [1] C. Chappert, A. Fert, F. 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[37] See Supporting Information for 1) determination of resistivity of W and Co 20Fe60B20 layers, 2) measurement of surface roughness of the sample using AFM, 3) determination of saturation magnetization of the samples from static and dynamic measurements and 4) variation of effective damping with precessional frequency. [38] I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, A. M. Baro, Rev. Sci. Instrum ., 2007, 78, 013705. [39] S. N. Panda, S. Mondal, J. Sinha, S. Choudhury, A. Barman, Sci. Adv., 2019, 5, eaav7200. [40] S. Akansel, A. Kumar, N. Behera, S. Husain, R. Brucas, S. Chaudhary, P. Svedlindh, Phys. Rev. B , 2018, 97, 134421. [41] L. Zhu, D. C. Ralph, R. A. Buhrman, Phys. Rev. Lett., 2019, 123, 057203. [42] J. Walowski, M. Djordjevic Kaufmann, B. Lenk, C. Hamann, J. McCord, M. Munzenberg, J. Phys. D: Appl. Phys., 2008, 41, 164016. [43] G. Woltersdorf, M. Buess, B. Heinrich, C. H. Back, Phys. Rev. Lett., 2005, 95, 037401. [44] Q. Hao, W. Chen, G. Xiao, Appl. Phys. Lett ., 2015, 106, 182403. [45] A. K. Patra, S. Singh, B. Barin, Y. Lee, J.-H. Ahn, E. D. Barco, E. R. Mucciolo, B. Özyilmaz, Appl. Phys. Lett ., 2012, 101, 162407. [46] N. Behera, M. S. Singh, S. Chaudhary, D. K. Pandya, P. K. Muduli, J. Appl. Phys ., 2015, 117, 17A714. [47] M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Geprägs, H. Huebl, S. T. B. Goennenwein, G. Woltersdorf, Phys. Rev. B , 2015, 92, 054437. 21 [48] S. Husain, A. Kumar, P. Kumar, A. Kumar, V. Barwal, N. Behera, S. Choudhary, P. Svedlindh, S. Chaudhary, Phys. Rev. B , 2018, 98, 180404(R). [49] A. Kumar, S. Akansel, H. Stopfel, M. Fazlali, J. Åkerman, R. Brucas, P. Svedlindh, Phys. Rev. B , 2017, 95, 064406. [50] M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rogalev, W. E. Bailey, Phys. Rev. B , 2016, 94, 014414. [51] S. Husain, A. Kumar, V. Barwal, N. Behera, S. Akansel, P. Svedlindh, S. Chaudhary, Phys. Rev. B , 2018, 97, 064420. [52] G. Wu, Y. Ren, X. He, Y. Zhang, H. Xue, Z. Ji, Q. Y. Jin, Z. Zhang, Phys. Rev. Appl ., 2020, 13, 024027. [53] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, F. Y. Yang, Phys. Rev. Lett ., 2014, 112, 197201. 22 Figure 1. (a) X-ray diffraction patterns measured at 2° grazing angle incidence for different W thickness. (b) Variation of inverse sheet resistance with W thickness. (c) AFM images of the samples showing the surface topography. 23 Figure 2. (a) Schematic of experimental geometry and (b) typical TR-MOKE data from Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructure at an applied bias magnetic field of 2.30 kOe. The three important temporal regimes are indicated in the graph. The solid red line shows a biexponential fit to the decaying background of the time-resolved Kerr rotation data. 24 Figure 3. (a) Background subtracted time-resolved Kerr rotation data showing precessional oscillation for Sub/W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of W thickness at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation damping with t (symbol) fitted with theoretical models (solid and dashed lines) of spin pumping. Two different regions corresponding to W crystal phase, namely β and α+β are shown. 25 Figure 4. (a) Background subtracted time-resolved Kerr rotation data showing precessional oscillation for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) as function of Co 20Fe60B20 thickness d at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation of damping vs 1/d (symbol) fitted with theoretical models (solid and dashed lines). 26 Figure 5. (a) Variation of effective spin-mixing conductance( 𝐺eff ) with W layer thickness t (symbol). The solid line is guide to the eye. (b) Contributions of SP, SML and TMS to the modulation of damping for different Co 20Fe60B20 layer thickness d (symbol). The solid line is guide to the eye. 0 2 4 6 8 10039095100 SP TMS SML Damping (%) d (nm) 0 2 4 8 12 160.00.51.01.5 Geff (1015 cm-2) t (nm)(a) (b) 27 Figure 6. (a) Background subtracted time-resolved Kerr rotation data showing precessional oscillation for Sub/W(4 nm)/Cu( c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of Cu layer thickness c at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation of damping vs c. The dotted line is guide to the eye, showing very little dependence of damping on Cu layer thickness. 28 Table 1. The average surface roughness values of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) samples obtained using AFM. Table 2. Comparison of the effective spin-mixing conductance and interfacial spin transparency of the W/CoFeB samples studied here with the important NM/FM interfaces taken from the literature. Material Interface Effective Spin-Mixing Conductance (×1015 cm-2) Interfacial Spin Transparency Pt/Py 1.52 [19] 0.25 [19] Pt/Co 3.96 [19] 0.65 [19] Pd/CoFe 1.07 [31] N.A. Pt/FM 0.6-1.2 [35] 0.34-0.67 [35] β-Ta/CoFeB 0.69 [39] 0.50 [39] β-Ta/ CFA 2.90 [40] 0.68 [40] Pd0.25Pt0.75/Co 9.11 [41] N.A. Au0.25Pt0.75/Co 10.73 [41] N.A. Pd/Co 4.03 [41] N.A. Pd0.25Pt0.75/FeCoB 3.35 [41] N.A. Au0.25Pt0.75/ FeCoB 3.64 [41] N.A. Gr/Py 5.26 [45] N.A. Ru/Py 0.24 [46] N.A. Pt/YIG 0.3-1.2 [47] N.A. MoS2/CFA 1.49 [48] 0.46 [48] Pd/Fe 0.49-1.17 [49] 0.04-0.33 [49] Pd/Py 1.40 [50] N.A. Mo/CFA 1.56 [51] N.A. MoS2/CoFeB 16.11 [52] N.A. Ta/YIG 0.54 [53] N.A. W/YIG 0.45 [53] N.A. Cu/YIG 0.16 [53] N.A. Ag/YIG 0.05 [53] N.A. Au/YIG 0.27 [53] N.A. β-W/CoFeB 1.44 (This work) 0.81 (This work) Mixed(α+β)-W/CoFeB 1.07 (This work) 0.60 (This work) ((N.A. = Not available)) t (nm) 0 0.5 1.0 1.5 2 3 5 8 10 15 Roughness (nm) 0.23 0.21 0.32 0.28 0.25 0.21 0.19 0.29 0.28 0.22 29 Supporting Information Structural Phase Dependent Giant Interfacial Spin Transparency in W/CoFeB Thin Film Heterostructure Surya Narayan Panda, Sudip Majumder, Arpan Bhattacharyya, Soma Dutta, Samiran Choudhury and Anjan Barman* Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India E-mail: abarman@bose.res.in This file includes: 1. Determination of resistivity of W and Co 20Fe60B20 layers. 2. Measurement of surface roughness of the sample using AFM. 3. Determination of saturation magnetization of the samples from static and dynamic measurements. 4. Variation of effective damping with precessional frequency. 1. Determination of resistivity of W and CoFeB layers : The variation of sheet resistance ( Rs) of the W( t)/Co20Fe60B20(3 nm) film stack with W layer thickness, t is shown in Figure S1 . The data is fitted with a parallel resistor model (Ref. 24 of the article) by the formula given in the inset of the figure. This yields the resistivity of W in its β and (α+β) phase as: 287 ± 19 µΩ.cm and 112 ± 14 µΩ.cm, respectively. On the other hand, the resistivity of Co 20Fe60B20 is found to be 139 ± 16 µΩ.cm. 30 Figure S1. Variation of sheet resistance ( Rs) of the W ( t)/ Co20Fe60B20(3 nm) film stack vs. W thickness t used for the determination of resistivity of the W and Co 20Fe60B20 layers. 2. Measurement of surface roughness of the sample using AFM: We have measured the surface topography of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) thin films by atomic force microscopy (AFM) in dynamic tapping mode by taking scan over 10 μm × 10 μm area. We have analyzed the AFM images using WSxM software. Figures S2 (a) and S2(d) show two-dimensional planar AFM images for t = 1 nm and 10 nm, respectively. Figures S2(b) and S2(e) show the corresponding three-dimensional AFM images for t = 1 nm and 10 nm, respectively. The dotted black lines on both images show the position of the line scans to obtain the height variation. Figures S2 (c) and S2(f) show the surface roughness profile along that dotted lines, from which the average roughness ( Ra) is measured as 0.32 ± 0.10 nm and 0.28 ± 0.12 nm for t = 1 nm and 10 nm, respectively. Topographical roughness is small and constant within the error bar in all samples irrespective of the crystal phase of W. Furthermore, surface roughness varies very little when measured at different regions of same sample. The interfacial roughness is expected to show its imprint on the measured topographical roughness 0 3 10 150200400 Rs(Ω) t(nm)(ρW)β= 287 µΩ.cm (ρW)α+β= 112 µΩ.cm = 139 µΩ.cm 31 due to the small thickness of our thin films. Small and constant surface roughness in these heterostructures proves the high quality of the thin films. Figure S2. (a) The two-dimensional AFM image, (b) the three-dimensional AFM image, and (c) the line scan profile along the black dotted line for W(1 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 nm) sample. (d) The two-dimensional AFM image, (e) the three-dimensional AFM image, and (f) the line scan profile along the black dotted line for W(10 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 nm) sample. 3. Determination of saturation magnetization of the samples from static and dynamic magnetic measurements : We have measured the in-plane saturation magnetization ( Ms) of all the W( t)/ Co20Fe60B20(d)/SiO2(2 nm) samples using vibrating sample magnetometry (VSM). Typical magnetic hysteresis loops (magnetization vs. magnetic field) for W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm), W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm) and W(4 nm)/Cu( c)/ Co20Fe60B20(3 32 nm)/SiO 2(2 nm) series are plotted in Figures S3 (a), S4(a) and S5(a), respectively. Here, Ms is calculated from the measured magnetic moment divided by the total volume of the Co 20Fe60B20 layer. These films have very small coercive field (~5 Oe). The effective magnetization Meff of the samples are obtained by fitting the bias magnetic field ( H) dependent precessional frequency (f) obtained from the TR-MOKE measurements, with the Kittel formula (equation (13) of the article) (see Figures S3 (b), S4(b) and S5(b)). We have finally plotted the variation of Meff and Ms with W, Co 20Fe60B20, and Cu thickness in Figures S3 (c), S4(c), and S5(c), respectively. The Meff and Ms values are found to be in close proximity with each other, indicating that the interfacial anisotropy is small for all these samples. Since these films were not annealed post- deposition, the interfacial anisotropy stays small and plays only a minor role in modifying the magnetization dynamics for these heterostructures. Figure S3. (a) VSM loops for W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit (solid line) to experimental data (symbol) of precessional frequency vs. magnetic field for W( t)/ Co20Fe60B20(3 nm)/SiO 2( 2 nm) samples. (c) Comparison of variation of Ms from VSM and Meff from TR-MOKE as a function of W layer thickness. 0 4 8 12 16500100015002000 t (nm)Ms (emu/cc) 500100015002000 Meff (emu/cc)-100001000 -100001000 -0.4 0.0 0.4-100001000 t =1 nm t = 8 nm H (kOe)t = 15 nm M (emu/cc)141618 141618 1.5 2.0 2.5141618 t = 1 nm t = 8 nm t = 15 nm f (GHz) H (kOe) (a) (b) (c) 33 Figure S4. (a) VSM loops for W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm). (b) Kittel fit (solid line) to experimental data (symbol) of precessional frequency vs. magnetic field for W(4 nm)/ Co20Fe60B20(d)/SiO2(2 nm) samples. (c) Comparison between variation of Ms from VSM and Meff from TR-MOKE as a function of Co 20Fe60B20 layer thickness. Figure S5. (a) VSM loops for W(4 nm)/Cu( c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit (solid line) to experimental data (symbol) of precessional frequency vs. magnetic field for W(4 nm)/ Cu(c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) samples. (c) Comparison between variation of Ms from VSM and Meff from TR-MOKE as a function of Cu layer thickness. 0 3 6 9500100015002000 d (nm)Ms (emu/cc) 500100015002000 Meff (emu/cc)141618 141618 1.5 2.0 2.514161820 f (GHz) H (kOe)d = 10 nmd = 5 nmd = 2 nm -100001000 -100001000 -0.3 0.0 0.3-100001000 d = 5 nmd = 2 nm d = 10 nmM (emu/cc) H (kOe) (a) (b) (c) 0.00 0.25 0.50 0.75 1.00500100015002000 c (nm)Ms (emu/cc) 500100015002000 Meff (emu/cc)141618 141618 1.5 2.0 2.5141618 f (GHz) H (kOe)c = 1 nmc = 0.5 nmc = 0 nm -100001000 -100001000 -0.4 0.0 0.4-100001000 H (kOe)M (emu/cc) c = 0.5 nmc = 0 nm c = 1 nm (a) (b) (c) 34 4. Variation of effective damping with precessional frequency: For all the sample series the time-resolved precessional oscillations have been recorded at different bias magnetic field strength. The precessional frequency has been extracted by taking the fast Fourier transform (FFT) of the background-subtracted time-resolved Kerr rotation. Subsequently, the time-resolved precessional oscillations have also been fitted with a damped sinusoidal function given by equation (12) of the article to extract the decay time τ. The value of effective Gilbert damping parameter ( αeff) have then been extracted using equation (14). Variation of this αeff with precessional frequency ( f) is plotted to examine the nature of the damping. Here, we have plotted the time-resolved precessional oscillations ( Figure S6( a)), FFT power spectra ( Figure S6( b)) and αeff vs. f (Figure S6( c)) for Sub/W(0.5 nm)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample. It is clear from this data that damping is frequency independent, which rules out the contribution of various extrinsic factors such as two-magnon scattering, inhomogeneous anisotropy, eddy current in the damping for our samples. Figure S6. (a) Background subtracted time-resolved precessional oscillations at different bias magnetic fields for Sub/W(0.5 nm)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample, where symbols represent the experimental data points and solid lines represent fits using equation (12) of the article. (b) The FFT power spectra of the time-resolved precessional oscillations showing the 0.0 0.3 0.6 0.9 1.2 1.5 H = 1.50 kOeH = 1.80 kOeH = 2.10 kOe Kerr Rotation (arb. units) Time (ns)H = 2.30 kOe 0 10 20 30 Power (arb. units) f (GHz)12 14 16 180.0000.0080.016 f (GHz)eff (c) (b) (a) 35 precessional frequency. (c) Variation of effective damping with precessional frequency is shown by symbol and the dotted line is guide to the eye.

1107.0753v2.Minimization_of_the_Switching_Time_of_a_Synthetic_Free_Layer_in_Thermally_Assisted_Spin_Torque_Switching.pdf

arXiv:1107.0753v2 [cond-mat.mes-hall] 16 Sep 2011Applied Physics Express Minimizationof theSwitchingTime of aSyntheticFreeLayer inThermallyAssistedSpin TorqueSwitching TomohiroTaniguchiandHiroshi Imamura Nanosystem Research Institute, AIST, 1-1-1 Umezono, Tsuku ba 305-8568, Japan Wetheoreticallystudiedthethermallyassistedspintorqu eswitchingofasyntheticfreelayerandshowedthattheswit ching timeisminimizedifthecondition HJ=|Hs|/(2α)issatisfied,where HJ,Hs,andαarethecouplingfieldoftwoferromagnetic layers,theamplitudeofthespintorque,andtheGilbertdam pingconstant, respectively. Wealsoshowed thatthecoupli ng fieldof the synthetic freelayer can be determined from there sonance frequencies of thespin-torque diode effect. Spin random access memory (Spin RAM) using the tun- neling magnetoresistance (TMR) e ffect1,2)and spin torque switching3,4)is one of the important spin-electronicsdevices for future nanotechnology. For Spin RAM application, it is highly desired to realize the magnetic tunnel junction (MTJ ) with high thermal stability ∆0, a low spin-torque switching currentIc, and a fast switching time. Recently, large ther- malstabilitieshavebeenobservedinanti-ferromagnetica lly5) and ferromagnetically6)coupled synthetic free (SyF) layers inMgO-basedMTJs.Inparticular,theferromagneticallyco u- pledSyFlayerisaremarkablestructurebecauseitshowsthe r- malstabilityofmorethan100withalowswitchingcurrent.6) Sincethecouplingbetweenthe ferromagneticlayersinthe SyF layer is indirect exchange coupling, we can systemati- callyvarythesignandstrengthofthecouplingfieldbychang - ing the spacer thickness between the two ferromagnetic lay- ers. As shown in ref.7), the thermal switching probability of theSyFlayerisadoubleexponentialfunctionofthecouplin g field, anda tinychangein the couplingfield cansignificantly increaseordecreasetheswitchingtime.Therefore,itisof in- teresttophysicalsciencetostudythedependenceofthethe r- malswitchingtimeonthecouplingfield. In this paper, we theoretically studied the spin-current- induced dynamics of magnetizations in an SyF layer of an MTJ. We found the optimum condition of the coupling field, whichminimizesthethermallyassistedspintorqueswitchi ng time. We showedthat the couplingfield of the two ferromag- netic layers in the SyF layer can be determined by using the spintorquediodee ffect. Let us first briefly describe the thermal switching of the SyF layer in the weak coupling limit, KV≫JS, where K,J,V, andSare the uniaxial anisotropy energy per unit volume, the coupling energy per unit area, and the volume andcross-sectionalarea of the single ferromagneticlayer ,re- spectively.For simplicity,we assume that all the material pa- rameters of the two ferromagnetic layers (F 1and F2) in the SyF layer are identical. A typical MTJ with an SyF layer is structured as a pinned layer /MgO barrier/ferromagnetic (F 1) layer/nonmagnetic spacer /ferromagnetic (F 2) layer (see Fig. 1), where the F 1and F2layers are ferromagneticallycoupled dueto the interlayerexchangecoupling.6)The F1and F2lay- ers have uniaxial anisotropy along the zaxis and two energy minima at mk=±ez, wheremkis the unit vector pointing in the direction of the magnetization of the F klayer. The spin current injected from the pinned layer to the F 1layer exerts spin torque on the magnetization of the F 1layer.8)Then, the magnetization of the F 1layer switches its direction due to the spin torque,after which the magnetizationof the F 2layerelectron (positive current)p m1 m2Hz xy F1 layer F2 layer spacer MgO pinned layer Fig. 1. Schematic view of the SyF layer. mkandpare the unit vectors pointing in the directions of the magnetizations of the F kand pinned layers, respectively. The positive current is defined as the electro n flow from the pinned layer to the free layer. Hrepresents the applied field. switchesits directiondueto coupling.By increasingthe co u- pling field, the potential height of the F 1(F2) layer for the switching becomes high (low), which makes the switching time of the F 1(F2) layer long (short). Then, a minimum of the totalswitchingtime appearsat a certaincouplingfield, as we shallshowbelow. Theswitchingprobabilityfromtheparallel(P)toantipara l- lel(AP)alignmentofthepinnedandfreelayermagnetizatio ns isgivenby7) P=1−(νF1e−νF2t−νF2e−νF1t)/(νF1−νF2),(1) whereνFk=fFkexp(−∆Fk)istheswitchingrateoftheF klayer. The attempt frequency is given by fFk=f0δk, wheref0= [αγHan/(1+α2)]√∆0/π,δ1=[1−(H+HJ+Hs/α)2/H2 an][1+(H+ HJ+Hs/α)/Han],andδ2=[1−(H−HJ)2/H2 an][1+(H−HJ)/Han].α, γ,H,Han=2K/M,HJ=J/(Md), and∆0=KV/(kBT) are the Gilbert damping constant, gyromagnetic ratio, applied fiel d, uniaxialanisotropyfield,couplingfield,andthermalstabi lity, respectively,and distheferromagneticlayerthickness. ∆Fkis givenby7,9) ∆F1=∆0[1+(H+HJ+Hs/α)/Han]2,(2) ∆F2=∆0[1+(H−HJ)/Han]2. (3) ∆F1is the potential height of the F 1layer before the F 2layer switches its magnetization while ∆F2is the potential height of the F 2layer after the F 1layer switches its magnetization. Hs=/planckover2pi1ηI/(2eMSd) is the amplitude of the spin torque in the unit of the magnetic field, where ηis the spin polariza- tion of the current I. The positive current corresponds to the electron flow from the pinned to the F 1layer; i.e., the nega- tive current I(Hs<0) induces the switching of the F 1layer. The field strengthsshouldsatisfy |H+HJ+Hs/α|/Han<1and |H−HJ|/Han<1becauseeq.(1)isvalidinthethermalswitch- ing region. In particular, |H+HJ+Hs/α|/Han<1 means that |I|<|Ic|. Theeffect of thefield like torqueis neglectedin Eq. (2) because its magnitude, βHswhere the beta term satisfies β<1, is less than 1 Oe in the thermal switching region and 12 Applied Physics Express coupling field, H J (Oe)I=-8, -9, and -10 (μA)solid : P=0.50 dotted : P=0.95 10 (μs)100 (μs)1 (ms)10 (ms)100 (ms)1 (s)switching time 20 40 60 80 100 Fig. 2. Dependences of the switching time at P=0.50 (solid lines) and P=0.95 (dotted lines) on the coupling field HJwith currents I=−8 (yel- low),−9 (blue), and−10 (red)µA. thus,negligible. Figure 2 shows the dependences of the switching times at P=0.50andP=0.95onthecouplingfieldwiththecurrents (a)−8, (b)−9, and (c)−10µA. The valuesof the parameters are taken to beα=0.007,γ=17.32 MHz/Oe,Han=200 Oe,M=995 emu/c.c.,S=π×80×35 nm2,d=2 nm, and T=300 K.6)The values of Handηare taken to be−65 Oe and 0.5,respectively.The value of His chosen so as to make thepotentialheightsfortheswitchinglowasmuchaspossib le (|H+HJ+Hs/α|/Han/lessorsimilar1 and|H−HJ|/Han/lessorsimilar1). As shown in Fig. 2, the switching time is minimized at a certain coupling field. We call this HJas the optimum coupling field for the fast thermallyassisted spintorqueswitching. Letusestimatetheoptimumcouplingfield.Forasmall HJ, the switching time of the F 2layer is the main determinant of the total switching time; thus, eq. (1) canbe approximateda s P≃1−e−νF2t. By increasing HJ,νF2increases and the switch- ing time (∼1/νF2) decreases. Fast switching is achieved for νF2∼νF1in this region. On the other hand, for a large HJ, the switching time of the F 1layer dominates, and eq. (1) is approximated as P≃1−e−νF1t. The switching time ( ∼1/νF1) decreaseswithdecreasing HJ.Fastswitchinginthisregionis also achieved forνF1∼νF2. The switching rate νFkis mainly determined by∆Fk. By putting∆F1=∆F2, the optimum cou- plingfield isobtainedas HJ=|Hs|/(2α). (4) Thisisthemainresultofthispaper.Thevaluesobtainedwit h eq.(4)for I=−8,−9and−10µAare53.7,60.5,and67.2Oe, respectively,whichshowgoodagreementwithFig.2. TheconditionνF1≃νF2meansthatthemoste fficientswitch- ing can be realized when two switching processes of the F 1 and F2layers occur with the same rate. νF1>νF2means that the magnetization of the F 1layer can easily switch due to a largespintorque.However,thesystemshouldstayinthisst ate for a long time because of a small switching rate of the F 2 layer. On the otherhand, when νF1<νF2, it takes a longtime to switch the magnetization of the F 1layer. Thus, when νF1 andνF2are different, the system stays in an unswitched state of the F 1or F2layer for a long time, and the total switching timebecomeslong.Forthermallyassistedfieldswitching,w e cannot find the optimum condition of the switching time be- cause the switching probabilities of the F 1and F2layers are the same. Factor 2 in eq. (4) arises from the fact that HJaf- fectstheswitchingsofboththeF 1andF2layers,while Hsas- siststhatofonlytheF 1layer.When HJ≪|Hs|/(2α),thetotalswitchingtimeisindependentofthecurrentstrength,beca use thetotalswitchingtimeinthisregionismainlydetermined by theswitchingtimeoftheF 2layer,whichisindependentofthe current. In the strong coupling limit, KV≪JS, two magne- tizations switch simultaneously,7)and the switching time is independentofthecouplingfield. For the AP-to-P switching, the factors δkand∆Fkare given byδ1=[1−(H−HJ+Hs/α)2/H2 an][1−(H− HJ+Hs/α)/Han],δ2=[1−(H+HJ)2/H2 an][1−(H+ HJ)/Han],∆F1= ∆0[1−(H−HJ+Hs/α)/Han]2, and∆F2= ∆0[1−(H+HJ)/Han]2.Inthiscase,apositivecurrent( Hs>0) inducestheswitching.Bysetting ∆F1=∆F2,theoptimumcou- pling field is obtained as HJ=Hs/(2α). Thus, for both P- to-AP and AP-to-Pswitchings, the optimumcouplingfield is expressedas HJ=|Hs|/(2α). Inthecaseoftheanti-ferromagneticallycoupledSyFlayer , H+HJandH−HJineqs.(2)and(3)shouldbereplacedby H+ |HJ|and−H−|HJ|,respectively,wherethesignofthecoupling fieldisnegative( HJ<0).Theoptimumconditionisgivenby |HJ|=−H+|Hs|/(2α),wherethenegativecurrentisassumedto enhancethe switching of the F 1layer. For a sufficiently large positive field H>|Hs|/(2α),this conditioncannot be satisfied becauseνF1isalwayssmallerthan νF2. One might notice that the condition ∆F1= ∆F2for the ferromagnetically coupled SyF layer has another solution |Hs|/(2α)=H+Han, which is independent of the coupling field. We exclude this solution because such HandHscan- not satisfy the conditions for the thermal switching region s |H+HJ+Hs/α|<Hanand|H−HJ|<Hansimultaneously.Sim- ilarly, for the anti-ferromagnetically coupled SyF layer, we excludethesolution |Hs|/(2α)=Hanobtainedfrom∆F1=∆F2. Thenaturalquestionfromtheabovediscussionishowlarge the coupling field is. The coupling field of a large plane film can be determinedfromtwo ferromagneticresonance(FMR) frequencies10,11)corresponding to the acoustic and optical modes,whichdependon HJ. Theantiferromagneticcoupling field can also be determined by the magnetization curve,5) in which finite magnetization appears when the applied field exceeds the saturation field Hs=−2HJ. These methods are, however,not applicable to nanostructuredferromagnetssu ch as the Spin RAM cells becausethe signal intensity is propor- tional to the volume of the ferromagnet, and thus, the inten- sity from the Spin RAM cell is negligibly small. It is desir- able to measure the coupling field of each cell because HJ strongly depends on the surface state and may di ffer signifi- cantlyamongthecellsobtainedfromasinglefilm plane. Here,weproposethatthecouplingfieldcanbedetermined by using the spin torque diode e ffect12–14)of the SyF layer. This method is applicable to a nanostructured ferromagnet, although the basic idea is similar to that of FMR measure- ment. The spin torque diode e ffect is measured by applying an alternating current Ia.c.cos(2πft) to an MTJ, which induces oscillating spin torque on the magnetization of the F 1layer. The free layer magnetizations oscillate due to the oscillat ing spin torque and the coupling, which lead to the oscillation o f the TMR RTMR=RP+(1−p·m1)∆R/2 and the d.c. voltage Vd.c.. Here,∆R=RAP−RP, andRPandRAPcorrespond to the resistancesattheparallelandantiparallelalignmentsof pand m1, respectively. pis the unit vector pointingin the directionApplied Physics Express 3 -0.080.04 0d.c. voltage (mV) current frequency (GHz)0 2 4 6 8 10 12 14 single free F coupled SyF AF coupled SyF-0.04 Fig. 3. Dependences of the spin torque diode voltage of the single fr ee layer (solid), the ferromagnetically (F) coupled SyF layer (dotted), and the anti-ferromagnetically (AF) coupled SyF layer (dashed) on the applied cur- rent frequency. ofthepinnedlayermagnetization. Vd.c.isgivenby Vd.c.=1 T/integraldisplayT 0dtIa.c.cos(2πft)−∆R 2p·m1,(5) whereT=1/f. The SyF layer shows large peaksof d.c. volt- age at the FMR frequencies of the acoustic facousticand opti- calfopticalmodes. The couplingfield can be determined from thesefrequencies. The resonancefrequencyof the ferromagneticallycoupled system is obtained as follows. The free energy of the SyF layerisgivenby F MV=−H·(m1+m2)−Han 2/bracketleftBig (m1·ez)2+(m2·ez)2/bracketrightBig +2πM/bracketleftBig (m1·ey)2+(m2·ey)2/bracketrightBig −HJm1·m2,(6) where the first, second, third, and fourth terms are the Zee- manenergy,uniaxialanisotropyenergy,demagnetizationfi eld energy, and coupling energy, respectively. The yandzaxes are normal to the plane and parallel to the easy axis, re- spectively. The applied field, H=H(sinθHex+cosθHez), lies in thexzplane with angleθHfrom the zaxis. The equilib- rium point is located at m1=m2=m(0)=(sinθ0,0,cosθ0), whereθ0satisfiesHsin(θ0−θH)+Hansinθ0cosθ0=0. We employ a new XYZcoordinate in which the YandZaxes are parallel to the yaxis andm(0), respectively, and denote a small component of the magnetization around m(0)asδmk= (mkX,mkY,0).The magnetizationdynamicsis desribedbyus- ing the Landau-Lifshitz-Gilbert (LLG) equation d mk/dt= −γmk×Hk+αmk×(dmk/dt), whereHk=−(MV)−1∂F/∂mk is the field acting on mk. By assuming the oscillating solu- tion (∝e2πi˜ft) ofmkXandmkY, keeping the first-order terms ofmkXandmkY, and neglecting the damping term, the LLG equations can be linearized as M(m1X,m1Y,m2X,m2Y)t=0. The nonzero components of the coe fficient matrix are M11= M22=M33=M44=2πi˜f/γ,M12=M34=[Hcos(θH−θ0)+ Hancos2θ0+HJ+4πM],M21=M43=−[Hcos(θH−θ0)+ Hancos2θ0+HJ], andM14=−M23=M32=−M41=−HJ. The FMR resonance frequencies are obtained under the con- ditiondet[ M]=0,andaregivenby facoustic=γ√h1h2/(2π)and foptical=γ√(h1+2HJ)(h2+2HJ)/(2π), whereh1=Hcos(θH− θ0)+Hancos2θ0andh2=Hcos(θH−θ0)+Hancos2θ0+4πM. HJcan be determined from these frequencies. For the anti- ferromagneticallycoupledsystem, m1/nequalm2inequilibriumin general, and the resonance frequencies are obtained by solv -ingthe4×4matrixequation. Figure 3 showsthe dependencesof the d.c. voltage Vd.c.of the single free layer (solid) and the ferromagnetically (do t- ted) and anti-ferromagnetically (dashed) coupled SyF laye rs on the applied current frequency calculated by solving the LLG equations of the F 1, F2, and pinned layers. The spin torque term,γHsm1×(p×m1)+γβHsp×m1, is added to the LLG equation of the F 1layer. Here the field like torque is taken into account because it a ffects the shape of Vd.c. significantly.12)The magnetic field acting on pis given by Hpin=H−4πMpyey+(Hanpz+Hp)ez, whereHpis the pinning field due to the bottom anti-ferromagnetic layer.6)In Fig. 3, Ia.c.=0.1 mA,∆R=400Ω,H=200 Oe,|HJ|=100 Oe, Hp=2 kOe,θH=30◦andβ=0.3.12)Thefacousticandfoptical oftheferromagneticallycoupledSyFlayerareestimatedto be 5.98and7.50GHz,respectively,whichshowgoodagreement with the peak points in Fig. 3. These results indicate that th e spin torquediode e ffect is useful in determiningthe coupling field. Insummary,wetheoreticallystudiedthedependenceofthe thermally assisted spin torque switching time of a SyF layer onthecouplingfield.Wefoundthattheswitchingtimeismin- imized if the condition of HJ=|Hs|/(2α) is satisfied. We showed that the coupling field can be determined from the resonancefrequencyofthespin torquediodee ffect. The authors would like to acknowledge H. Kubota, T. Saruya, D. Bang, T. Yorozu, H. Maehara, and S. Yuasa of AISTfortheirsupportandthe discussionstheyhadwithus. 1) S.Yuasa,T.Nagahama,A.Fukushima,Y.Suzuki,andK.Ando :Nature Materials 3(2004) 868. 2) S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughe s, M. Samant, and S.H. Yang: Nature Materials 3(2004) 862. 3) J.C.Slonczewski:Phys.Rev.B 39,6995(1989);J.Magn.Magn.Mater. 159(1996) L1. 4) L.Berger: Phys.Rev. B 54(1996) 9353. 5) J.Hayakawa, S.Ikeda,K.Miura,M.Yamanouchi, Y.M.Lee,R .Sasaki, M. Ichimura, K. Ito, T. Kawahara, R. Takemura, T. Meguro, F. M at- sukura, H. Takahashi, H.Matsuoka, and H.Ohno: IEEE.Trans. Magn. 44(2008) 1962. 6) S. Yakata, H.Kubota, T.Sugano, T.Seki, K. Yakushiji, A.F ukushima, S.Yuasa, and K.Ando: Appl. Phys.Lett. 95(2009) 242504. 7) T.Taniguchi and H.Imamura: Phys.Rev. B 83(2011) 054432. 8) Private communication with Hitoshi Kubota. It was experi mentally shown that the critical current in the CoFeB /Ru/CoFeB spin valve is one order of magnitude larger than that in CoFeB /MgO/CoFB MTJs (unpublished). This result means that the spin torque arisi ng between the free layers is negligible compared with that arising fro m the spin current injected from thepinned layer. 9) In ref.7),∆F1is expressed as∆0[1+(Happl+HJ)/Han]2(1−I/Ic)2, which is equivalent to eq. (2). 10) Z.Zhang, L.Zhou,and P.E.Wigen: Phys.Rev. B 50(1994) 6094. 11) J.Lindner,U.Wiedwald,K.Baberschke,andM.Farle:J.V ac.Sci.Tech- nol. A23(2005) 796. 12) A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. M aehara, K.Tsunekawa,D.D.Djayaprawira, N.Watanabe, andS.Yuasa: Nature 438(2005) 339. 13) H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yu asa, K. Ando, H. Maehara, Y. Nagamine, K. Tsunekawa, D. D. Djayapraw ira, N.Watanabe, and Y. Suzuki: Nature Physics 4(2008) 37. 14) J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A. Buhrman, and D.C. Ralph: Nature Physics 4,(2008) 67.

1507.03075v1.Realization_of_the_thermal_equilibrium_in_inhomogeneous_magnetic_systems_by_the_Landau_Lifshitz_Gilbert_equation_with_stochastic_noise__and_its_dynamical_aspects.pdf

Realization of the thermal equilibrium in inhomogeneous magnetic systems by the Landau-Lifshitz-Gilbert equation with stochastic noise, and its dynamical aspects Masamichi Nishino1and Seiji Miyashita2;3 1Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan 2Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-Ku, Tokyo, Japan 3CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan (Dated: July 14, 2015) 1arXiv:1507.03075v1 [cond-mat.mtrl-sci] 11 Jul 2015Abstract It is crucially important to investigate eects of temperature on magnetic properties such as critical phenomena, nucleation, pinning, domain wall motion, coercivity, etc. The Landau-Lifshitz- Gilbert (LLG) equation has been applied extensively to study dynamics of magnetic properties. Approaches of Langevin noises have been developed to introduce the temperature eect into the LLG equation. To have the thermal equilibrium state (canonical distribution) as the steady state, the system parameters must satisfy some condition known as the uctuation-dissipation relation. In inhomogeneous magnetic systems in which spin magnitudes are dierent at sites, the condition requires that the ratio between the amplitude of the random noise and the damping parameter depends on the magnitude of the magnetic moment at each site. Focused on inhomogeneous mag- netic systems, we systematically showed agreement between the stationary state of the stochastic LLG equation and the corresponding equilibrium state obtained by Monte Carlo simulations in various magnetic systems including dipole-dipole interactions. We demonstrated how violations of the condition result in deviations from the true equilibrium state. We also studied the characteris- tic features of the dynamics depending on the choice of the parameter set. All the parameter sets satisfying the condition realize the same stationary state (equilibrium state). In contrast, dierent choices of parameter set cause seriously dierent relaxation processes. We show two relaxation types, i.e., magnetization reversals with uniform rotation and with nucleation. PACS numbers: 75.78.-n 05.10.Gg 75.10.Hk 75.60.Ej 2|||||||||||||||||||||||||- I. INTRODUCTION The Landau-Lifshitz-Gilbert (LLG) equation1has been widely used in the study of dy- namical properties of magnetic systems, especially in micromagnetics. It contains a relax- ation mechanism by a phenomenological longitudinal damping term. The Landau-Lifshitz- Bloch (LLB) equation2contains, besides the longitudinal damping, a phenomenological transverse damping and the temperature dependence of the magnetic moment are taken into account with the aid of the mean-eld approximation. Those equations work well in the region of saturated magnetization at low temperatures. Thermal eects are very important to study properties of magnets, e.g., the amount of spontaneous magnetization, hysteresis nature, relaxation dynamics, and the coercive force in permanent magnets. Therefore, how to control temperature in the LLG and LLB equations has been studied extensively. To introduce temperature in equations of motion, a coupling with a thermal reservoir is required. For dynamics of particle systems which is naturally expressed by the canonical conjugated variables, i.e., ( q;p), molecular dynamics is performed with a Nose-Hoover (NH) type reservoir3{5or a Langevin type reservoir6. However, in the case of systems of magnetic moments, in which dynamics of angular momenta is studied, NH type reservoirs are hardly used due to complexity7. On the other hand, the Langevin type reservoirs have been rather naturally applied2,8{18although multiplicative noise19requires the numerical integration of equations depending on the interpretation, i.e., Ito or Stratonovich type. To introduce temperature into a LLG approach by a Langevin noise, a uctuation- dissipation relation is used, where the temperature is proportional to the ratio between the strength of the uctuation (amplitude of noise) and the damping parameter of the LLG equation. For magnetic systems consisting of uniform magnetic moments, the ratio is uniquely given at a temperature and it has been often employed to study dynamical prop- erties, e.g., trajectories of magnetic moments of nano-particles8, relaxation dynamics in a spin-glass system20or in a semiconductor21. The realization of the equilibrium state by stochastic LLG approaches by numerical simulations is an important issue, and it has been conrmed in some cases of the Heisenberg model for uniform magnetic moments.22,23 3In general cases, however, magnetic moments in atomic scale have various magnitudes of spins. This inhomogeneity of magnetization is important to understand the mechanisms of nucleation or pinning.24{28To control the temperature of such systems, the ratio between the amplitude of noise and the damping parameter depends on the magnetic moment at each site. In order to make clear the condition for the realization of the canonical distribution as the stationary state in inhomogeneous magnetic systems, we review the guideline of the derivation of the condition in the Fokker-Planck equation formalism in the Appendix A. Such a generalization of the LLG equation with a stochastic noise was performed to study properties of the alloy magnet GdFeCo29, in which two kinds of moments exist. They ex- ploited a formula for the noise amplitude, which is equivalent to the formula of our condition A (see Sec II). They found surprisingly good agreements of the results between the stochas- tic LLG equation and a mean-eld approximation. However, the properties in the true canonical distribution is generally dierent from those obtained by the mean-eld analysis. The LLG and LLB equations have been often applied for continuous magnetic systems or assemblies of block spins in the aim of simulation of bulk systems, but such treatment of the bulk magnets tend to overestimate the Curie temperature11, and it is still under develop- ment to obtain properly magnetization curves in the whole temperature region2,11,17,18. The in uence of coarse graining of block spin systems on the thermal properties is a signicant theme, which should be claried in the future. To avoid such a diculty, we adopt a lattice model, in which the magnitude of the moment is given at each magnetic site. Within the condition there is some freedom of the choice of parameter set. In the present paper, in particular, we investigate the following two cases of parameter sets, i.e., case A, in which the LLG damping constant is the same in all the sites and the amplitude of the noise depends on the magnitude of the magnetic moment at each site, and case B, in which the amplitude of the noise is the same in all the sites and the damping constant depends on the magnitude of the moment. (see Sec II.). We conrm the realization of the equilibrium state, i.e., the canonical distribution in various magnetic systems including critical region by comparison of magnetizations obtained by the LLG stochastic approach with those obtained by standard Monte Carlo simulations, not by the mean-eld analysis. We study systems with not only short range interactions but also dipole-dipole interactions, which causes the demagnetizing eld statically. We nd that dierent choices of the parameter set which satises the uctuation-dissipation relation give the same stationary state (equilibrium state) 4even near the critical temperature. We also demonstrate that deviations from the relation cause systematic and signicant deviations of the results. In contrast to the static properties, we nd that dierent choices of parameter set cause serious dierence in the dynamics of the relaxation. In particular, in the rotation type relaxation in isotropic spin systems, we nd that the dependences of the relaxation time on the temperature in cases A and B show opposite correlations as well as the dependences of the relaxation time on the magnitude of the magnetic moment. That is, the relaxation time of magnetization reversal under an unfavorable external eld is shorter at a higher temperature in case A, while it is longer in case B. On the other hand, the relaxation time is longer for a larger magnetic moment in case A, while it is shorter in case B. We also investigate the relaxation of anisotropic spin systems and nd that the metastability strongly aects the relaxation at low temperatures in both cases. The system relaxes to the equilibrium state from the metastable state by the nucleation type of dynamics. The relaxation time to the metastable state and the decay time of the metastable state are aected by the choice of the parameter set. The outline of this paper is as follows. The model and the method in this study are ex- plained in Sec II. Magnetization processes as a function of temperature in uniform magnetic systems are studied in Sec III. Magnetizations as a function of temperature for inhomoge- neous magnetic systems are investigated in Sec. IV, in which not only exchange interactions (short-range) but also dipole interactions (long-range) are taken into account. In Sec. V dynamical aspects with the choice of the parameter set are considered, and the dependences of the relaxation process on the temperature and on the magnitude of magnetic moments are also discussed. The relaxation dynamics via a metastable state is studied in Sec. VI. Sec. VII is devoted to summary and discussion. In Appendix A the Fokker-Planck equation for inhomogeneous magnetic systems is given both in Stratonovich and Ito interpretations, and Appendix B presents the numerical integration scheme in this study. II. MODEL AND METHOD As a microscopic spin model, the following Hamiltonian is adopted, H=X hi;jiJi;jSi
SjX iDA i(Sz i)2X ihi(t)Sz i+X i6=kC r3 ik Si
Sk3(rik
Si)(rik
Sk) r2 ik :(1) 5Here we only consider a spin angular momentum Sifor a magnetic moment Miat each site (iis the site index) and regard Mi=Siignoring the dierence of the sign between them and setting a unit: gB= 1 for simplicity, where gis the g-factor and Bis the Bohr magneton30. Interaction Ji;jbetween the ith andjth magnetic sites indicates an exchange coupling,hi;jidenotes a nearest neighbor pair, DA iis an anisotropy constant for the ith site,hiis a magnetic eld applied to the ith site, and the nal term gives dipole interactions between the ith andkth sites whose distance is ri;k, whereC=1 40is dened using the permeability of vacuum 0. The magnitude of the moment Miis dened as MijMij, which is not necessarily uniform but may vary from site to site. In general, the damping parameter may also have site dependence, i.e., i, and thus the LLG equation at the ith site is given by d dtMi= MiHe i+i MiMidMi dt; (2) or in an equivalent formula: d dtMi= 1 +2 iMiHe ii (1 +2 i)MiMi(MiHe i); (3) where is the gyromagnetic constant. Here He iis the eective eld at the ith site and described by He i=@ @MiH(M1;


;MN;t) (4) , which contains elds from the exchange and the dipole interactions, the anisotropy, and the external eld. We introduce a Langevin-noise formalism for the thermal eect. There have been several ways for the formulation to introduce a stochastic term into the LLG equation. The stochas- tic eld can be introduced into the precession term and/or damping term8,9,11. Furthermore, an additional noise term may be introduced10,12. In the present study we add the random noise to the eective eld He i!He i+iand we have d dtMi= 1 +2 iMi(He i+i)i (1 +2 i)MiMi(Mi(He i+i)); (5) where iis the(=1,2 or 3 for x,yorz) component of the white Gaussian noise applied at theith site and the following properties are assumed: h k(t)i= 0;h k(t) l(s)i= 2Dkkl(ts): (6) 6We call Eq. (5) stochastic LLG equation. We derive a Fokker-Planck equation6,8for the stochastic equation of motion in Eq. (5) in Stratonovich interpretation, as given in appendix A, @ @tP(M1;


;MN;t) =X i 1 +2 i@ @Mi
i MiMi(MiHe i) (7) DiMi(Mi@ @Mi) P(M1;


;MN;t) : Here we demand that the distribution function at the stationary state ( t!1 ) of the equation of motion (Eq. (7)) agrees with the canonical distribution of the system (Eq. (1)) at temperature T, i.e., Peq(M1;


;MN)/exp H(M1;


;MN) ; (8) where=1 kBT. Considering the relation @ @MiPeq(M1;


;MN) =He iPeq(M1;


;MN); (9) we nd that if the following relation i Mi Di= 0 (10) is satised at each site i, the canonical distribution in the equilibrium state is assured. When the magnetic moments are uniform, i.e., the magnitude of each magnetic moment is the same and Mi=jMij=M, the parameters iandDiare also uniform i=and Di=Dfor a given T. However, when Miare dierent at sites, the relation (10) must be satised at each site independently. There are several ways of the choice of the parameters iandDito satisfy this relation. Here we consider the following two cases: A and B. A: we take the damping parameter ito be the same at all sites, i.e., 1=2=


=N . In this case the amplitude of the random eld at the ith site should be Di= MikBT /1 Mi: (11) B: we take the amplitude of the random eld to be the same at all sites, i.e., D1=D2=


=DND. In this case the damping parameter at the ith site should be i=D Mi kBT/Mi: (12) 700.20.40.60.81 0123456m TFIG. 1: (color online) Comparison of the temperature dependence of min the stationary state between the stochastic LLG method and the Langevin function (green circles). Crosses and boxes denotemin case A ( = 0:05) and case B ( D= 1:0), respectively. In the stochastic LLG simulation
t= 0:005 was set and 80000 time steps (40,000 steps for equilibration and 40,000 steps for measurement) were employed. The system size N=L3= 103was adopted. We study whether the canonical distribution is realized in both cases by comparing data obtained by the stochastic LLG method with the exact results or with corresponding data obtained by Monte Carlo simulations. We set the parameters = 1 andkB= 1 hereafter. III. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN HOMOGE- NEOUS MAGNETIC SYSTEMS A. Non-interacting magnetic moments As a rst step, we check the temperature eect in the simplest case of non-interacting uniform magnetic moments, i.e., Ji;j= 0,DA i= 0,C= 0 in Eq. (1) and Mi=M(or Si=S), whereandDhave no site i-dependence. In this case the magnetization in a magnetic eld ( h) at a temperature ( T) is given by the Langevin function: m=1 NhNX i=1Sz ii=M cothhM kBT kBT hM! : (13) We compare the stationary state obtained by the stochastic LLG method and Eq. (13). 8We investigate m(T) ath= 2 forM= 1. Figure 1 shows m(T) when= 0:05 is xed (case A) and when D= 1:0 is xed (case B). We nd a good agreement between the results of the stochastic LLG method and the Langevin function in the whole temperature region as long as the relation (10) is satised. Numerical integration scheme is given in Appendix B. The time step of
t= 0:005 and total 80000 time steps (40000 steps for equilibration and 40000 steps for measurement) were adopted. B. Homogeneous magnetic moments with exchange interactions Next, we investigate homogenous magnetic moments ( Mi=jMij=M) in three di- mensions. The following Hamiltonian ( C= 0,Ji;j=J,DA i=DA, andh(t) =hin Eq. (1)): H=X hi;jiJSi
SjX iDA(Sz i)2X ihSz i (14) is adopted. There is no exact formula for magnetization ( m) as a function of temperature for this system, and thus a Monte Carlo (MC) method is applied to obtain reference magnetization curves for the canonical distribution because MC methods have been established to obtain nite temperature properties for this kind of systems in the equilibrium state. Here we employ a MC method with the Metropolis algorithm to obtain the temperature dependence of magnetization. In order to check the validity of our MC procedure, we investigated magnetization curves as functions of temperature (not shown) with system-size dependence for the three- dimensional classical Heisenberg model ( DA= 0 andh= 0 in Eq. (14)), and conrmed that the critical temperature agreed with past studies31, wherekBTc= 1:443Jfor the innite system size with M= 1. We givem(T) for a system of M= 2 with the parameters J= 1,h= 2 andDA= 1:0 for cases A and B in Fig 2. The system size was set N=L3= 103and periodic boundary conditions (PBC) were used. Green circles denote mobtained by the Monte Carlo method. At each temperature ( T) 10,000 MC steps (MCS) were applied for the equilibration and following 10,00050,000 MCS were used for measurement to obtain m. Crosses and boxes denotemin the stationary state of the stochastic LLG equation in case A ( = 0:05) and in case B ( D= 1:0), respectively. Here
t= 0:005 was set and 80000 steps (40000 for 900.511.52 0 5 10 15 20 25m TFIG. 2: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method in the homogeneous magnetic system with M= 2. Crosses and boxes denote case A with = 0:05 and case B with D= 1:0, respectively. transient and 40000 for measurement) were used to obtain the stationary state of m. The m(T) curves show good agreement between the MC method and the stochastic LLG method in both cases. We checked that the choice of the initial state for the MC and the stochastic LLG method does not aect the results. The dynamics of the stochastic LLG method leads to the equilibrium state at temperature T. IV. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN INHOMO- GENEOUS MAGNETIC SYSTEMS A. Inhomogeneous magnetic moments with exchange interactions Here we study a system which consists of two kinds of magnitudes of magnetic moments. The Hamiltonian (14) is adopted but the moment Mi=jMijhasi-dependence. We investi- gate a simple cubic lattice composed of alternating M= 2 andM= 1 planes (see Fig. 3 (a)), whereJ= 1,h= 2 andDA= 1:0 are applied. We consider two cases A and B mentioned in Sec. II. The reference of m(T) curve was obtained by the MC method and is given by green circles in Figs. 3 (b) and (c). In the simulation, at each temperature ( T) 10,000 MCS were applied for the equilibration and following 10,000 50,000 MCS were used for measurement. 10(a) 00.511.5 05 1 0 1 5 2 0m T(b) 00.511.5 05 1 0 1 5 2 0m T(c)FIG. 3: (color online) (a) A part of the system composed of alternating M= 2 (red long arrows) and M= 1 (short blue arrows) layers. (b) Comparison of temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method for = 0:05.
t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were employed. Crosses denote mwhenDi=D(Mi) MikBT was used. Triangles and Diamonds are mfor Di=D(1) =kBT for alliandDi=D(2) = 2kBT for alli, respectively. (c) Comparison of temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method for D= 1:0.
t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were employed. Crosses denote mwheni=(Mi)D Mi kBTwas used. Triangles are mfori=(Mi= 1) =D 1 kBTfor alliand Diamonds are mfori=(2) =D 2 kBT for alli. 11The system size N=L3= 103was adopted with PBC. In case A, (= 0:05) is common for all magnetic moments in the stochastic LLG method and Mi(orSi) dependence is imposed onDiasDi=D(Mi) MikBT . In case B, D= 1:0 is common for all magnetic moments in the stochastic LLG method and i=(Mi)D Mi kBT. Crosses in Figs. 3 (b) and (c) denote mby the stochastic LLG method for cases A and B, respectively. For those simulations
t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were employed at each temperature. In both Figs. 3 (b) and (c), we nd good agreement between m(T) by the stochastic LLG method (crosses) and m(T) by the MC method (green circles). Next, we investigate how the results change if we take wrong choices of parameters. We studym(T) when a uniform value Di=Dfor case A ( i=for case B) is used for all spins, i.e., for both Mi= 1 andMi= 2. IfD(Mi= 2) = 2kBT is used for all spins, m(T) is shown by Diamonds in Fig. 3 (b), while if D(Mi= 1) =kBT is applied for all spins, m(T) is given by triangles in Fig. 3 (b). In the same way, we study m(T) for a uniform value of. In Fig. 3 (c) triangles and diamonds denote m(T) wheni=(Mi= 1) and i=(Mi= 2) are used, respectively. We nd serious dierence in m(T) when we do not use correct Mi-dependent choices of the parameters. The locations of triangle (diamond) at each temperature Tare the same in Figs. 3 (a) and (b), which indicates that if the ratio =D is the same in dierent choices, the same steady state is realized although this state is not the true equilibrium state for the inhomogeneous magnetic system. Thus we conclude that to use proper relations of Mi-dependence of Dioriis important for m(T) curves of inhomogeneous magnetic systems and wrong choices cause signicant deviations. B. Critical behavior of Inhomogeneous magnetic moments In this subsection, we examine properties near the critical temperature. Here we adopt the case ofh= 0 andDA= 0 in the same type of lattice with M= 1 and 2 as Sec. IV A. We investigate both cases of the temperature control (A and B). The Hamiltonian here has O(3) symmetry and mis not a suitable order parameter. Thus we dene the following quantity as the order parameter31: ma=q m2 x+m2 y+m2 z; (15) 1200.511.5 0123456ma TFIG. 4: (color online) Comparison of temperature ( T) dependence of mabetween the MC method (green circles) and the stochastic LLG method for the system of inhomogeneous magnetic moments. N=L3= 203. PBC were used. In the MC method 10,000 MCS and following 50,000 MCS were used for equilibration and measurement at each temperature, respectively. The stochastic LLG method was performed in case A with = 0:05 (croses) and in case B with D= 1:0 (diamonds). Here
t= 0:005 was applied and 240,000 steps were used (40,000 for transient and 200,000 for measurement). where mx=1 NhNX i=1Sx ii; my=1 NhNX i=1Sy ii;andmz=m=1 NhNX i=1Sz ii: (16) In Fig. 4, green circles denote temperature ( T) dependence of magiven by the MC method. The system size N=L3= 203with PBC was adopted and in MC simulations 10,000 MCS and following 50,000 MCS were employed for equilibration and measurement, respectively at each temperature. The magnetizations of maobtained by the stochastic LLG method for case A (crosses) and case B (diamonds) are given in Fig. 4. Here = 0:05 and D= 1:0 were used for (a) and (b), respectively.
t= 0:005 was set and 240,000 steps (40,000 for transient and 200,000 for measurement) were applied. In both cases ma(T) curve given by the stochastic LLG method shows good agreement with that obtained by the MC method. Thus, we conclude that as long as the relation (10) is satised, the temperature dependence of the magnetization is reproduced very accurately even around the Curie temperature, regardless of the choice of the parameter set. 1300.511.5 0123456m TFIG. 5: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method. Crosses and diamonds denote case A with= 0:05 and case B with D= 1:0, respectively. A reduction of mfrom fully saturated magnetization is observed at around T= 0 due to the dipole interactions. As a reference, mby the MC method without the dipole interactions ( C= 0) is given by open circles. C. Inhomogeneous magnetic moments with exchange and dipole interactions We also study thermal eects in a system with dipole interactions. We use the same lattice as in the previous subsections. The system is ( Ji;j=J,DA i=DA, andhi(t) =hin Eq. (1)) given by H=X hi;jiJSi
SjX iDA(Sz i)2X ihSz i+X i6=kC r3 ik Si
Sk3(rik
Si)(rik
Sk) r2 ik :(17) Here a cubic lattice with open boundary conditions (OBC) is used. Since Jis much larger thanC=a3(JC=a3) for ferromagnets, where ais a lattice constant between magnetic sites. However, we enlarge dipole interaction as C= 0:2 witha= 1 forJ= 1 to highlight the eect of the noise on dipole interactions. We set other parameters as h= 0:1,DA= 0:1. Studies with realistic situations will be given separately. We study cases A ( = 0:05) and B ( D= 1:0) for this system. We depict in Fig. 5 the temperature ( T) dependences of mwith comparison between the MC (green circles) and stochastic LLG methods. Crosses and diamonds denote m(T) for cases A and B, respectively. Dipole interactions are long-range interactions and we need longer equilibration steps, and 14we investigate only a small system with N=L3= 63. In the MC method 200,000 MCS were used for equilibration and 600,000 steps were used for measurement of m, and for the stochastic LLG method
t= 0:005 was set and 960,000 steps (160,000 and 800,000 time steps for equilibration and measurement, respectively) were consumed. A reduction of mfrom fully saturated magnetization is observed. As a reference, mby the MC method without the dipole interactions ( C= 0) is given by open circles in Fig. 5. This reduction of mis caused by the dipole interactions. We nd that even when dipole interactions are taken into account in inhomogeneous magnetic moments, suitable choices of the parameter set leads to the equilibrium state. Finally, we comment on the comparison between the LLG method and the Monte Carlo method. To obtain equilibrium properties of spin systems, the Monte Carlo method is more ecient and powerful in terms of computational cost. It is much faster than the stochastic LLG method to obtain the equilibrium m(T) curves, etc. For example, it needs more than 10 times of CPU time of the MC method to obtain the data for Fig. 5. However, the MC method has little information on the dynamics and the stochastic LLG method is used to obtain dynamical properties because it is based on an equation of motion of spins. Thus, it is important to clarify the nature of stochastic LLG methods including the static properties. For static properties, as we saw above, the choice of the parameter set, e.g., cases A and B, did not give dierence. However, the choice gives signicant dierence in dynamical properties, which is studied in the following sections. V. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER SET IN ISOTROPIC SPIN SYSTEMS ( DA=0) Now we study the dependence of dynamics on the choice of parameter set. The temper- ature is given by kBT= DiMi i; (18) which should be the same for all the sites. In general, if the parameter D(amplitude of the noise) is large, the system is strongly disturbed, while if the parameter (damping parameter) is large, the system tends to relax fast. Therefore, even if the temperature is the same, the dynamics changes with the values of Dand. When the anisotropy term exists, i.e.,DA6= 0, in homogeneous systems ( Mi=M) given by Eq. (14), the Stoner- 15Wohlfarth critical eld is hc= 2MDAatT= 0. If the temperature is low enough, the metastable nature appears in relaxation. On the other hand, if Tis rather high orDA= 0, the metastable nature is not observed. In this section we focus on dynamics of isotropic spin systems, i.e.,DA=0. A. Relaxation with temperature dependence In this subsection we investigate the temperature dependence of magnetization relaxation in cases A and B. We adopt a homogeneous system ( Mi=M= 2) withDA= 0 in Eq. (14). Initially all spins are in the spin down state and they relax under a unfavorable external eldh= 2. The parameter set M= 2,= 0:05,D= 0:05 givesT= 2 by the condition (Eq. (10)). Here we study the system at T= 0:2;1;2, and 10. We set = 0:05 in case A and the control of the temperature is performed by D, i.e.D= 0:005;0:025;0:05, and 0:25, respectively. In case B we set D= 0:05, and the control of the temperature is realized by , i.e.,= 0:5;0:1;0:05, and 0:01, respectively. We depict the temperature dependence of m(t) for cases A and B in Figs. 6 (a) and (b), respectively. Here the same random number sequence was used for each relaxation curve. Red dash dotted line, blue dotted line, green solid line, and black dashed line denote T= 0:2, T= 1,T= 2 andT= 10, respectively. Relaxation curves in initial short time are given in the insets. In case A, as the temperature is raised, the initial relaxation speed of mbecomes faster and the relaxation time to the equilibrium state also becomes shorter. This dependence is ascribed to the strength of the noise with the dependence D/T, and a noise with a larger amplitude disturbs more the precession of each moment, which causes faster relaxation. On the other hand, in case B, the relaxation time to the equilibrium state is longer at higher temperatures although the temperature dependence of the initial relaxation speed of mis similar to the case A. In the initial relaxation process all the magnetic moments are in spin-down state ( Sz i'2). There the direction of the local eld at each site is given byHe i'JP jSz j+h=26 + 2 =10, which is downward and the damping term tends to x moments to this direction. Thus, a large value of the damping parameter at a low temperature T(/1 T) suppresses the change of the direction of each moment and the initial relaxation speed is smaller. However, in the relaxation process thermal uctuation 16-2-1012 0 50 100 150 200m time(a) -2.2-2-1.8-1.6-1.4-1.2-1 012345678 -2-1012 0 50 100 150 200m time-2.2-2-1.8-1.6-1.4-1.2-1 012345678(b)FIG. 6: (color online) (a) Time dependence of the magnetization ( m(t)) in case A, where = 0:05 for a homogeneous system with M= 2. Red dash dotted line, blue dotted line, green solid line, and black dashed line denote T= 0:2,T= 1,T= 2 andT= 10, respectively. Inset shows the time dependence of m(t) in the initial relaxation process. (b) Time dependence of the magnetization (m(t)) in case B, where D= 0:05 for a homogeneous system with M= 2. Correspondence between lines and temperatures is the same as (a). causes a deviation of the local eld and then a rotation of magnetic moments from z tozdirection advances (see also Fig. 11 ). Once the rotation begins, the large damping parameter accelerates the relaxation and nally the relaxation time is shorter. B. Relaxation with spin-magnitude dependence Next we study the dependence of relaxation on the magnitude of magnetic moments in cases A and B. Here we adopt a homogeneous system ( Mi=M) without anisotropy( DA= 0) atT= 2 andh= 2. The initial spin conguration is the same as the previous subsection. Because D/T M;and/M T; (19) raising the value of Mis equivalent to lowering temperature in both cases A and B and it causes suppression of relaxation in case A, while it leads to acceleration of relaxation in case B. Because Maects the local eld from the exchange energy at each site, changing the value ofMunder a constant external eld his not the same as changing Tand it may show 17-1.5-1-0.500.511.5 01 0 2 0 3 0 4 0 5 0m time(a) -1.5-1-0.500.511.5 02468 1 0m time(b)FIG. 7: (color online) Comparison of the time dependence of mbetween cases A and B by the stochastic LLG method. Red and blue lines denote cases A and B, respectively. (a) = 0:05 for case A and D= 1:0 for case B, (b) = 0:2 for case A and D= 1:0 for case B. some modied features. In the relation (19), T= 0:2, 1, 2, 10 at M= 2 (Fig.6 (a) and (b)) are the same as M= 20, 4, 2, 0.4 at T= 2, respectively. We studied the relaxation ratio dened as m(t)=M withMdependence at T= 2 for these four values of M, and compared with the relaxation curves of Fig.6 (a) and (b). We found qualitatively the same tendency between relaxation curves with Mdependence and those with 1 =Tdependence in both cases. A dierence was found in the initial relaxation speed (not shown). When M > 2, the initial relaxation at T= 2 is slower than that of the corresponding TatM= 2. The downward initial local eld at each site is stronger for larger Mdue to a stronger exchange coupling, which also assist the suppression of the initial relaxation. It is found that the relaxation time under a constant external led becomes longer as the value of Mis raised in case A, while it becomes shorter in case B. This suggests that dierent choices of the parameter set lead to serious dierence in the relaxation dynamics withMdependence. 18VI. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER SET IN ANISOTROPIC SPIN SYSTEMS ( DA6= 0) A. Dierent relaxation paths to the equilibrium in magnetic inhomgeneity If the anisotropy term exists DA6= 0 but the temperature is relatively high, metastable nature is not observed in relaxation. We consider the relaxation dynamics when Mihas idependence in this case. We study the system (alternating M= 2 andM= 1 planes) treated in Sec. IV A. We set a conguration of all spins down as the initial state and observe relaxation of min cases A and B. In Sec. IV A we studied cases A ( =0.05) and B ( D=1.0) for the equilibrium state and the equilibrium magnetization is m'0:95 atT= 5. We give comparison of the time dependence of mbetween the two cases in Fig. 7 (a), with the use of the same random number sequence. The red and blue curves denote cases A and B, respectively. We nd a big dierence in the relaxation time of mand features of the relaxation between the two cases. The parameter values of andDare not so close between the two cases at this tempera- ture (T= 5), i.e.,D(M= 1) = 0:25 andD(M= 2) = 0:125 for case A and (M= 1) = 0:2 and(M= 2) = 0:4 for case B. Thus, to study if there is a dierence of dynamics even in close parameter values of andDbetween cases A and B at T= 5, we adopt common = 0:2, whereD(M= 1) = 1 and D(M= 2) = 0:5, as case A and common D= 1:0, where (M= 1) = 0:2 and(M= 2) = 0:4, as case B. We checked that this case A also gives the equilibrium state. In Fig. 7 (b), the time dependence of mfor both cases is given. The red and blue curves denote cases A and B, respectively. There is also a dierence (almost twice) of the relaxation time of mbetween cases A and B. Thus, even in close parameter region of andD, dynamical properties vary depending on the choice of the parameters. B. Relaxation with nucleation mechanism In this subsection we study a system with metastability. We adopt a homogeneous system (M= 2) withJ= 1,DA= 1 andh= 2. Here the Stoner-Wohlfarth critical eld ishc= 2MDA=4, and if the temperature is low enough, the system has a metastable state underh= 2. At a high temperature, e.g., T= 10 (= 0:05,D= 0:25), the magnetization relaxes 19(a) -2-1012 0 80 160 240 320m time -2-1012 0 50 100 150 200 250 300 350m time(b) -2-1012 0 50 100 150 200 250 300 350m time(c)FIG. 8: (color online) (a) Dashed line shows m(t) for= 0:05,D= 0:25, andT= 10. Blue and green solid lines give m(t) for= 0:05 atT= 3:5 (case A) and D= 0:25 atT= 3:5 (case B), respectively. These two lines were obtained by taking average over 20 trials with dierent random number sequences. The 20 relaxation curves for cases A and B are given in (b) and (c), respectively. without being trapped as depicted in Fig 8(a) with a black dotted line. When the tempera- ture is lowered, the magnetization is trapped at a metastable state. We observe relaxations in cases A and B, where = 0:05 for case A and D= 0:25 for case B are used. In Figs. 8(b) and (c), we show 20 samples (with dierent random number sequences) of relaxation pro- cesses atT= 3:5 for case A ( = 0:05,D= 0:0875) and case B ( D= 0:25,= 0:143), respectively. The average lines of the 20 samples are depicted in Fig 8(a) by blue and green solid lines for cases A and B, respectively. In both cases, magnetizations are trapped at a metastable state with the same value of m(m'1:55). This means that the metastabil- ity is independent of the choice of parameter set. Relaxation from the metastable state to the equilibrium is the so-called stochastic process and the relaxation time distributes. The relaxation time in case A is longer. If the temperature is further lowered, the escape time from the metastable state becomes longer. In Figs. 9 (a) and (b), we show 20 samples of relaxation at T= 3:1 for cases A and B, respectively. There we nd the metastable state more clearly. Here we investigate the initial relaxation to the metastable state at a relatively low temperature. In Figs. 10 (a) and (b), we depict the initial short time relaxation of 20 samples at T= 2 in cases A ( = 0:05,D= 0:05) and B(D= 0:25,= 0:25), respectively. The insets show the time dependence of the magnetization in the whole measurement time. 20-2-1012 0 200 400 600 800m time(a) -2-1012 0 200 400 600 800m time(b)FIG. 9: (a) and (b) illustrate 20 relaxation curves for = 0:05 atT= 3:1 (case A) and D= 0:25 atT= 3:1 (case B), respectively. Metastability becomes stronger than T= 3:5. No relaxation occurs in all 20 trials in (a), while ve relaxations take place in 20 trials in (b). We nd that the relaxation is again faster in case B. The metastability also depends on Mas well asDAand largeMgives a strong metastabil- ity. Here we conclude that regardless of the choice of the parameter set, as the temperature is lowered, the relaxation time becomes longer due to the stronger metastability, in which largerD(larger) gives faster relaxation from the initial to the metastable state and faster decay from the metastable state. Finally we show typical congurations in the relaxation process. When the anisotropy DAis zero or weak, the magnetization relaxation occurs with uniform rotation from z tozdirection, while when the anisotropy is strong, the magnetization reversal starts by a nucleation and inhomogeneous congurations appear with domain wall motion. In Figs. 11 we give an example of the magnetization reversal of (a) the uniform rotation type (magneti- zation reversal for DA= 0 withD= 0:05,T= 2,= 0:1,M= 4) and of (b) the nucleation type (magnetization reversal for DA= 1 withD= 0:25,T= 3:1,= 0:161,M= 2 ). VII. SUMMARY AND DISCUSSION We studied the realization of the canonical distribution in magnetic systems with the short-range (exchange) and long-range (dipole) interactions, anisotropy terms, and magnetic elds by the Langevin method of the LLG equation. Especially we investigated in detail the 21-2.2-2-1.8-1.6-1.4-1.2-1.0 012345678m time(a) -2-1012 0 200 400 600 800 time -2.2-2-1.8-1.6-1.4-1.2-1 012345678m time(b) -2-1012 0 200 400 600 800 timeFIG. 10: Initial relaxation curves of magnetization. Insets show m(t) in the whole measurement time. (a) and (b) illustrate 20 relaxation curves for = 0:05 atT= 2 (case A) and D= 0:25 at T= 2 (case B), respectively. (b)(a) FIG. 11: (a) Typical uniform rotation type relaxation observed in the isotropic spin system. (b) Typical nucleation type relaxation observed in the anisotropic spin system. thermal equilibration of inhomogeneous magnetic systems. We pointed out that the spin- magnitude dependent ratio between the strength of the random eld and the coecient of the damping term must be adequately chosen for all magnetic moments satisfying the condition (10). We compared the stationary state obtained by the present Langevin method of the 22LLG equation with the equilibrium state obtained by the standard Monte Carlo simulation for given temperatures. There are several choices for the parameter set, e.g., A and B. We found that as long as the parameters are suitably chosen, the equilibrium state is realized as the stationary state of the stochastic LLG method regardless of the choice of the parameter set, and the temperature dependence of the magnetization is accurately produced in the whole region, including the region around the Curie temperature. We also studied dynamical properties which depend on the choice of the parameters. We showed that the choice of the parameter values seriously aects the relaxation process to the equilibrium state. In the rotation type relaxation in isotropic spin systems under an unfavorable external eld, the dependences of the relaxation time on the temperature in cases A and B exhibited opposite correlations as well as the dependences of the relaxation time on the magnitude of the magnetic moment. The strength of the local eld in the initial state strongly aects the speed of the initial relaxation in both cases. We also found that even if close parameter values are chosen in dierent parameter sets for inhomogeneous magnetic systems, these parameter sets cause a signicant dierence of relaxation time to the equilibrium state. In the nucleation type relaxation, the metastability, which depends on DAandM, strongly aects the relaxation in both cases A and B. Lowering temperature reinforces the metastability of the system and causes slower relaxation. The relaxation to the metastable state and the decay to the metastable state are aected by the choice of the parameter set, in which larger Dcauses fast relaxation at a xed T. In this study we adopted two cases, i.e., A and B in the choice of the parameter set. Generally more complicated dependence of MiorTon the parameters is considered. How to chose the parameter set is related to the quest for the origin of these parameters. It is very important for clarication of relaxation dynamics but also for realization of a high speed and a low power consumption, which is required to development of magnetic devices. Studies of the origin of have been intensively performed32{41. To control magnetization relaxation at nite temperatures, investigations of the origin of Das well aswill become more and more important. We hope that the present work gives some useful insight into studies of spin dynamics and encourages discussions for future developments in this eld. 23Acknowledgments The authors thank Professor S. Hirosawa and Dr. S. Mohakud for useful discussions. The present work was supported by the Elements Strategy Initiative Center for Magnetic Materials under the outsourcing project of MEXT and Grant-in-Aid for Scientic Research on Priority Areas, KAKENHI (C) 26400324. 24Appendix A: Fokker-Planck equation The LLG equation with a Langevin noise (Eq. (5)) is rewritten in the following form for component ( = 1;2 or 3 forx;yorz) of theith magnetic moment, dM i dt=f i(M1;


;MN;t) +g i(Mi) i(t): (A1) Heref iandg iare given by f i= 1 +2 i M iHe; i+i MiM iM iHe; i (A2) and g i= 1 +2 i M i+i Mi(M2 i +M iM i) ; (A3) whereHe; ican have an explicit time ( t) dependence, and denotes the Levi-Civita symbol. We employ the Einstein summation convention for Greek indices ( ,


). We consider the distribution function FF(M1;


;MN;t) in the 3N-dimensional phase space ( M1 1;M2 1;M3 1;


;M1 N;M2 N;M3 N). The distribution function F(M1;


;MN;t) satises the continuity equation of the distribution: @ @tF(M1;


;MN;t) +NX i=1@ @M id dtM i
F = 0: (A4) Substituting the relation (A1), the following dierential equation for the distribution func- tionFis obtained. @ @tF(M1;


;MN;t) =NX i=1@ @M in fi+g i i
Fo : (A5) Regarding the stochastic equation (A1) as the Stratonovich interpretation, making use of the stochastic Liouville approach42, and taking average for the noise statistics (Eq. (6)), we have a Fokker-Planck equation. @ @tP(M1;


;MN;t) =NX i=1@ @M i f iPDig i@ @M i(g iP) ; (A6) wherePP(M1;


;MN;t) is the averaged distribution function hFi. Substituting the relation @ @M ig i= i Mi(1 +2 i)4M i (A7) 25and Eq. (A3) into g i(@ @M ig i), we nd g i(@ @M ig i) = 0: (A8) Thus Eq.(A6) is simplied to @ @tP(M1;


;MN;t) =NX i=1@ @M i f iDig ig i@ @M i
P : (A9) Substituting Eqs. (A2) and (A3), we have a formula in the vector representation. @ @tP(M1;


;MN;t) = (A10) X i 1 +2 i@ @Mi
 MiHe i+i MiMi(MiHe i) DiMi(Mi@ @Mi) P(M1;


;MN;t) : Since@ @Mi
(MiHe i) = 0, it is written as @ @tP(M1;


;MN;t) =X i 1 +2 i@ @Mi
i MiMi(MiHe i) (A11) DiMi(Mi@ @Mi) P(M1;


;MN;t) : In the case that Eq. (A1) is given under Ito denition, we need Ito-Stratonovich trans- formation, and the corresponding equation of motion in Stratonovich interpretation is dM i dt=f i(M1;


;MN;t)Dig i(Mi)@g i(Mi) @M i+g i(Mi) i(t): (A12) Then the Fokker-Planck equation in Ito interpretation is @ @tP(M1;


;MN;t) =NX i=1@ @M i f iDig i@g i @M iDig ig i@ @M i
P : Sinceg i@g i @M i=2 2 1+2 iM i, the vector representation is given by @ @tP(M1;


;MN;t) =X i 1 +2 i@ @Mi
i MiMi(MiHe i) 2 DiMi DiMi(Mi@ @Mi) P(M1;


;MN;t) : (A13) 26Appendix B: Numerical integration for stochastic dierential equations In stochastic dierential equations, we have to be careful to treat the indierentiability of the white noise. In the present paper we regard the stochastic equation, e.g., Eq. (5), as a stochastic dierential equation in Stratonovich interpretation: dM i=f i(M1;


;MN;t)dt+g i1 2 Mi(t) +Mi(t+dt)
 dW i(t); (B1) wheredW i(t) =Rt+dt tds i(s), which is the Wiener process. This equation is expressed by dM i=f i(M1;


;MN;t)dt+g i(Mi(t))dW i(t); (B2) whereindicates the usage of the Stratonovich denition. A simple predictor-corrector method called the Heun method8,19, superior to the Euler method, is given by M i(t+
t) =M i(t) +1 2[f i(^M1(t+
t);


;^MN(t+
t);t+
t) +f i(M1(t);


;MN(t);t)]
t +1 2[g i(^Mi(t+
t)) +g i(Mi(t))]
W i; (B3) where
W iW i(t+
t)W(t) and ^M i(t+
t) is chosen in the Euler scheme: ^M i(t+
t) =M i(t) +f i(M1(t);


;MN(t);t)
t+g i(Mi(t))
W i: (B4) This scheme assures an approximation accuracy up to the second order of
Wand
t. Sev- eral numerical dierence methods19for higher-order approximation, which are often compli- cated, have been proposed. Here we adopt a kind of middle point method equivalent to the Heun method. M i(t+
t) =M i(t) +f i(M1(t+
t=2);


;MN(t+
t=2);t+
t=2)
t +g i(Mi(t+
t=2))
W i; (B5) whereM i(t+
t=2) is chosen in the Euler scheme: M i(t+
t=2) =M i(t) +f i(M1(t);


;MN(t);t)
t=2 +g i(Mi(t))
~Wi; (B6) 27where
~WiW i(t+
t=2)W i(t). Considering the following relations, h
~Wi
W ii= [W i(t+
t=2)W i(t)][W i(t+
t)W i(t)] =Di
t; (B7) h
W ii= 0 andh
~Wii= 0, this method is found equivalent to the Heun method. We can formally replace
~Wiby
W i=2 in Eq. (B6) in numerical simulations. Corresponding author. Email address: nishino.masamichi@nims.go.jp 1H. Kronm ullar and M. F ahnle, \Micromagnetism and the Microstructure of Ferromagnetic Solids" Cambridge University Press, (2003). 2D. A. Garanin, Phys. Rev. B 55, 3050 (1997). 3M. Nishino, K. Boukheddaden, Y. Konishi, and S. Miyashita, Phys. Rev. Lett. 98, 247203 (2007). 4S. Nos e, J. Chem. Phys. 81, 511 (1984). 5W. G. Hoover, Phys. Rev. A 31, 1695 (1985). 6H. Risken, The Fokker-Planck Equation, 2nd ed. Springer, Berlin (1989). 7A. Bulgac and D. Kusnezov, Phys. Rev. A 42, 5045 (1990). 8J. L. Garc a-Palacios and F. J. L azaro, Phys. Rev. B 58, 14937 (1998) 9Y. Gaididei, T. Kamppeter, F. G. Mertens, and A. Bishop, Phys. Rev. B 59, 7010 (1999). 10T. Kamppeter, F. G. Mertens, E. Moro, A. S anchez, A. R. Bishop, Phys. Rev. B 59, 11349 (1999). 11G. Grinstein and R. H. Koch, R.W. Chantrell, U. Nowak, Phys. Rev. Lett. 90, 207201 (2003). 12O. Chubykalo, R. Smirnov-Rueda, J.M. Gonzalez, M.A. Wongsam, R.W. Chantrell, U. Nowak, J. Magn. Magn. Mater. 266, 28 (2003). 13A. Rebei and M. Simionato, Phys. Rev. B 71, 174415 (2005). 14U. Atxitia, O. Chubykalo-Fesenko, R.W. Chantrell, U. Nowak, and A. Rebei, Phys. Rev. Lett. 102, 057203 (2009). 15K. Vahaplar, A. M. Kalashnikova, A.V. Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk and Th. Rasing, Phys. Rev. Lett. 103, 117201 (2009) 16D. A. Garanin and O. Chubykalo-Fesenko, Phys. Rev. B 70, 212409 (2004). 2817O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and D. Garanin, Phys. Rev. B 74, 094436 (2006). 18R. F. L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R. W. Chantrell and O. Chubykalo-Fesenko, Phys. Rev. B 85, 014433 (2012) . 19P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Dierential Equations, 3rd corr. printing., Springer (1999). 20B. Skubic, O. E. Peil, J. Hellsvik, P. Nordblad, L. Nordstr om, and O. Eriksson, Phys. Rev. B 79, 024411 (2009). 21J. Hellsvik, B. Skubic, L. L Nordstr om, B. Sanyal, and O. Eriksson, Phys. Rev. B 78, 144419 (2008). 22B Skubic, J Hellsvik, L Nordstr om, and O Eriksson, J. Phys. Condens. Matter 20315203 (2008). 23R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M.O.A. Ellis and R. W. Chantrell, J. Phys. Condens. Matter 26103202 (2014). 24K.-D. Durst and H. Kronm uller, J. Magn. Magn. Mater. 68, 63 (1987). 25H. Kronm uller and K.-D. Durst, J. Magn. Magn. Mater. 74, 291 (1988). 26B. Barbara and M. Uehara, Inst. Phys. Conf. Ser. No. 37 Chapter8 (1978). 27A. Sakuma, S. Tanigawa and M. Tokunaga, J. Magn. Magn. Mater. 84, 52 (1990). 28A. Sakuma, J. Magn. Magn. Mater. 88, 369 (1990). 29A. Ostler et al., Phys. Rev. B 84, 024407 (2011). 30In realistic materials, in particular rare earth systems, the total angular momentum at each magnetic site is given by J=S+L, whereSis a spin angular momentum and Lis an orbital angular momentum, and the relation between MandJshould be treated carefully. 31P. Peczak, A. M. Ferrenberg, and D. P. Landau, Phys. Rev. B 43, 6087 (1991) and references therein. 32H. J. Skadsem, Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 75094416 (2007). 33E. Simanek and B. Heinrich, Phys. Rev. B 67, 144418 (2003). 34V. Kambersk y, Phys. Rev. B 76134416 (2007). 35K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). 36K. Gilmore, Y. U. Idzerda, and M. D. Stiles, J. Appl. Phys. 10307D303 (2008). 37A. Brataas, Y. Tserkovnyak, and G. E. Bauer: Phys. Rev. Lett. 101037207 (2008). 2938A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010). 39H. Ebert, S. Mankovsky, D. Kodderitzsch, and P. J. Kelly: Phys. Rev. Lett. 107066603 (2011). 40A. Sakuma, J. Phys. Soc. Jpn. 4, 084701 (2012). 41A. Sakuma, J. Appl. Phys. 117, 013912 (2015). 42R. Kubo, J. Math. Phys. 4, 174 (1963) 30

0802.4455v3.Heat_conduction_and_Fourier_s_law_in_a_class_of_many_particle_dispersing_billiards.pdf

arXiv:0802.4455v3 [nlin.CD] 30 Aug 2008Heat conduction and Fourier’s law in a class of many particle dispersing billiards Pierre Gaspard †, Thomas Gilbert ‡ Center for Nonlinear Phenomena and Complex Systems, Universit´ e Libre de Bruxelles, C. P. 231, Campus Plaine, B-1050 Brussels, Belgium Abstract. We consider the motion of many confined billiard balls in interaction and discusstheirtransportandchaoticproperties. Inspiteoftheab senceofmasstransport, due to confinement, energy transport can take place through bin ary collisions between neighbouring particles. We explore the conditions under which relaxa tion to local equilibrium occurs on time scales much shorter than that of binary co llisions, which characterize the transport of energy, and subsequent relaxat ion to local thermal equilibrium. Starting from the pseudo-Liouville equation for the time e volution of phase-space distributions, we derive a master equation which gove rns the energy exchange between the system constituents. We thus obtain analy tical results relating the transport coefficient of thermal conductivity to the frequen cy of collision events and compute these quantities. We also provide estimates of the Lya punov exponents and Kolmogorov-Sinai entropy under the assumption of scale sepa ration. The validity of our results is confirmed by extensive numerical studies. Submitted to: New J. Phys. PACS numbers: 05.20.Dd,05.45.-a,05.60.-k,05.70.Ln E-mail:†gaspard@ulb.ac.be, ‡thomas.gilbert@ulb.ac.beFourier’s law in many particle dispersing billiards 2 1. Introduction Understanding the dynamical origin of the mechanisms which underly the phenomenol- ogy of heat conduction has remained one of the major open problem s of statistical mechanics ever since Fourier’s seminal work [1]. Fourier himself actua lly warned his reader that the effects of heat conduction “make up a special ran ge of phenomena which cannot be explained by the principles of motion and equilibrium,” th us seemingly rejecting the possibility of a fundamental level of description. Of c ourse, with the sub- sequent developments of the molecular kinetic theory of heat, sta rting with the works of the founding-fathers of statistical mechanics, Boltzmann, Gib bs and Maxwell, and later with the definite triumph of atomism, thanks to Perrin’s 1908 me asurement of the Avogadro number in relation to Einstein’s work on Brownian motion, Fo urier’s earlier perception was soon discarded as it became clear that heat transp ort was indeed the effect of mechanical causes. Yet, after well over a century of hard labor, the community is still a ctively hunting for a first principles based derivation of Fourier’s law, some authors going so far as to promise “a bottle of very good wine to anyone who provides [a satisfa ctory answer to this challenge]” [2]. Thus the challenge is, starting from the Hamiltonian time evolution of a system of interacting particles which models a fluid or a crystal, t o derive the conditions under which the heat flux and temperature gradients ar e linearly related by the coefficient of heat conductivity. This is the embodiment of Fourie r’s law. As outlined in [2], it is necessary, in order to achieve this, that the mod el statistical properties be fully determined in terms of the local temperature, a notionwhich involves that of local thermalization. To visualize this, we imagine that the sys tem is divided up into a large number of small volume elements, each large enough to contain a number of particles whose statistics is accurately described by the equilibrium statistics at the local temperature associated to the volume element under c onsideration. To establish this property, one must ensure that time scales separat e, which implies that the volume elements settle to local thermal equilibrium on time scales m uch shorter than the ones which characterize the transport of heat at the ma croscopic scale. Local thermal equilibrium therefore relies on very strong ergodic proper ties of the model. The natural framework to apply this programme is that of chaotic b illiards. Indeed non-interacting particle billiards, where individual tracers move and make specular collisions among a periodic array of fixed convex scatterers (the pe riodic Lorentz gases) aretheonlyknownHamiltoniansystems forwhichmasstransport[3 ]andshearandbulk viscosities [4] have been established in a rigorous way. Here, rather than local thermal equilibrium, itisarelaxationtolocalequilibrium, occurringontheconst antenergysheet of the individual tracer particles, which allows to model the mass tra nsport by a random walk and yields an analytic estimate of the diffusion coefficient [5]. Argua bly Lorentz gases, whether periodic or disordered, in or out of equilibrium, have played, over more thanacentury, aprivilegedandmostimportantroleinthedevelopme nt oftransportand kinetic theories [6]. However, in the absence of interaction among th e tracer particles,Fourier’s law in many particle dispersing billiards 3 there is no mechanism for energy exchange and therefore no proc ess of thermalization before heat is conducted. If, on the other hand, one adds some in teraction between the tracers, say, as suggested in [2], by assigning them a size, the r esulting system will typically have transport equations for the diffusion of both mass an d energy. Though these systems may be conceptually simple, they are usually mathema tically too difficult to handle with the appropriate level or rigor. An example of billiard with interacting tracer particles is the modified Lo rentz gas withrotatingdiscsproposedbyMej´ ıa–Monasterio etal.[7]. Thissystemexhibitsnormal transport with non-trivial coupled equations for heat and mass tr ansport. Though this model has attracted much attention among mathematicians [8, 9, 10, 11, 12, 13], the rigorous proof of Fourier’s law for such a system of arbitrary size a nd number of tracers relies on assumptions which, though they are plausible, are themselv es not resolved. Moreover the equations which govern the collisions and the discs rot ations do not, as far as we know, derive from a Hamiltonian description. Yet a remedy to such limitations with respect to the number of partic les in interaction that a rigorous treatment will handle may have been jus t around the corner [14], and, this, within a Hamiltonian framework. Indeed in [15], Bunimov ichet al. introduced a class of dispersing billiard tables with particles that are in geometric confinement – i. e.trapped within cells– but that can nevertheless interact among particles belonging to neighbouring cells. The authors proved ergod icity and strong chaotic properties of such systems with arbitrary number of part icles. More recently, the idea that energy transport can be modeled as a slow diffusion process resulting from the coupling of fast energy-conserving dy namics has led to proofs of central limit theorems in the context of models of random walks an d coupled maps which describe the diffusion of energy in a strongly chaotic, fast cha nging environment [16, 17]. Although the extension of these results to symplectic coup led maps, let alone Hamiltonian flows, is not yet on the horizon, it is our belief that such sy stems as the dispersing billiardtablesintroducedin[15]will, ifany, lendthemselves to afullyrigorous treatment of heat transport within a Hamiltonian framework. As we announced in [18], the reason why we should be so hopeful is that the particles con finement has two important consequences : first, relaxation to local thermal equilib rium is preceded by a relaxation of individual particles to local equilibrium ‡, which occurs at constant energy within each cell, and has strong ergodic properties that guarantee the rapid decay of statistical correlations; and, second, heat transport, unlike in t he rotating discs model, can be controlled by the mere geometry of the billiard, which also cont rols the absence of mass transport. As of the first property, the relaxation to a lo cal equilibrium before energy exchanges take place is characterized by a fast time scale, much faster than that of relaxation to local thermal equilibrium among neighbouring cells, wh ich is itself much faster than the hydrodynamic relaxation scale. There is therefor e a hierarchy of three ‡Let us underline the distinction we make here and in the sequel betwe en relaxation to local equilibrium, which precedes energy exchanges, and relaxation to loc al thermal equilibrium, which involves energy exchanges among particles belonging to neighbourin g cells.Fourier’s law in many particle dispersing billiards 4 separate time scales in this system, the first accounting for relaxa tion at the microscopic scale of individual cells, the second one at the mesoscopic scale of ne ighbouring cells, and the third one at the macroscopic scale of the whole system. In this paper, we achieve two important milestones towards a comple te first principles derivation of the transport properties of such models. H aving defined the model, we establish the conditions for separation of time scales and r elaxation to local equilibrium, identifying a critical geometry where binary collisions beco me impossible. Assuming relaxation to local equilibrium holds, we go on to considering t he time evolution of phase-space densities and derive, from it, a master eq uation which governs theexchangeofenergyinthesystem[19], thusgoingfromamicrosc opicscaledescription of the Liouville equation to the mesoscopic scale at which energy tran sport takes place. We regard this as the first milestone, namely identifying the condition s under which one can rigorously reduce the level of descrition from the determin istic dynamics at the microscopic level to a stochastic process described by a maste r equation at the mesoscopic level of energy exchanges. This master equation is then used to compute the frequency of bina ry collisions and to derive Fourier’s law and the macroscopic heat equation, which results from the application of a small temperature gradient between neighbouring c ells. This is our second milestone : an analytic formula for heat conductivity, exact for the stochastic system, and thus valid for the determinisitc system at the critical g eometry limit. These results are then checked against numerical computations o f these quantities, with outstanding agreement, and shown to extend beyond the critical geometry, with very good accuracy, to a wide range of parameter values. We further characterize the chaotic properties of the model and offer arguments to account for the spectrum of Lyapunov exponents of the syst em, as well as the Kolmogorov-Sinai entropy, expressions which are exact at the cr itical geometry. Again, these results are very nicely confirmed by our numerical computat ions. The paper is organized as follows. The models, which we coin lattice billiar ds, are introduced in section 2. Their main geometric properties are establis hed, distinguishing transitions between insulating and conducting regimes under the tu ning of a single parameter. The same parameter controls the time scales separat ion responsible for local equilibrium. Section 3 provides the derivation of the master equation which, under the assumption of local equilibrium, governs energy transport. The ma in observables are computed and their scaling properties discussed. In section 4 we re view the properties of the model and assess the validity of the results of section 3 unde r the scope of our numerical computations. The chaotic properties of the models are discussed in section 5. We use simple theoretical arguments to predict some of these pr operties and compare them to numerical computations. Finally, conclusions are drawn in se ction 6.Fourier’s law in many particle dispersing billiards 5 2. Lattice billiards To introduce our model, we start by considering the uniform motion o f a point particle about a dispersing billiard table, Bρ, defined by the domain exterior to four overlapping discs of radii ρ, centered at the four corners of a square of sides l. The radius is thus restricted to the interval l/2≤ρ < l/√ 2, where the lower bound is the overlap (or bounded horizon) condition, and the upper bound is reached when Bρis empty. Ρl ΡfΡml Figure 1. Two equivalent representations of a dispersing billiard table : (left) a point particle moves uniformly inside the domain Bρand performs specular collisions with its boundary; (right) a disc of radius ρmmoves uniformly inside the domain Bρand performs specular collisions with fixed discs of radii ρf=ρ−ρm. The radius ρmis a free parameter which is allowed to take any value between 0 and ρ. As illustrated in figure 1, the motion of a point particle in this environme nt is a limiting case of a class of equivalent dispersing billiards, whereby the p oint particle becomes a moving disc with radius ρm, 0≤ρm≤ρ, and bounces off fixed discs of radii ρf=ρ−ρm. In all these cases, the motion of the center of the moving disc is eq uivalent to that of the point particle in Bρ. The border of the domain Bρconstitutes the walls of the billiards. In the absence of cusps (which occur at ρ=l/2), the ergodic and hyperbolic properties of these billiards are well established [20]. In particular, t he long term statistics of the billiard map, which takes the particle from one collision event to the next, preserves the measure cos φdrdφ, whereφdenotes the angle that the particle post-collisional velocity makes with respect to the normal vector t o the boundary. A direct consequence of this invariance is a general formula which rela tes the mean free path,ℓ, to the billiard table area, |Bρ|, and perimeter |∂Bρ|,ℓ=π|Bρ|/|∂Bρ|. The ratio between the speed of the particle, which we denote v, and the mean free path gives the wall collision frequency §,νc=v/ℓ, whose computation is shown in figure 2. With this quantity, one can relate the billiard map iterations to the t ime-continuous dynamics of the flow. In particular, the billiard map has two Lyapunov exponents, opposite in signs and equal in magnitudes, which, multiplied by the wall c ollision §Anticipating the more general definition of the wall collision frequenc y for interacting particle billiard cells, we adopt the subscript “c” in reference to “critical” for reas ons to be clarified below.Fourier’s law in many particle dispersing billiards 6 frequency, correspond to the two non-zero Lyapunov exponen ts of the flow (there are two additional zero exponents related to the direction of the flow a nd conservation of energy). The results of numerical computations of the positive on e, denoted λ+, are shown in figure 2 for different values of the parameters ρ. 0.360.380.400.420.440.460.480.501020304050 ΡΝc,Λ/PΛus Figure 2. Collision frequency νc(solid line) and numerical computation of the corresponding positive Lyapunov exponent of the flow (dots) of d ispersing billiards such as shown in figure 1. Here we took v= 1 and the square side to be l= 1/√ 2. Thus let Q(i,j) denote the rhombus of sides lcentered at point (cij,dij) =/braceleftBigg (√ 2li,lj/√ 2), j even, (√ 2li+l/√ 2,lj/√ 2), jodd.(1) The rhombic billiard cell illustrated in figure 1 becomes a domain centere d at point (cij,dij), defined according to Bρ(i,j) =/braceleftBig (x,y)∈Q(i,j)/vextendsingle/vextendsingle/vextendsingleδ[(x,y),(cij+pk,dij+qk)]≥ρ,k= 1,...,d/bracerightBig ,(2) whereδ[.,.] denotes the usual Euclidean distance between two points, and ( pk,qk) = (±l/√ 2,0),(0,±l/√ 2) are the coordinates of the d≡4 discs at the corners of the rhombus. Now consider a number of copies {Bρ(i,j)}i,jwhich tessellate a two-dimensional domain. We define a lattice billiard as a collection of billiard cells Lρ,ρm(n1,n2)≡/braceleftBig (xij,yij)∈ Bρ(i,j)/vextendsingle/vextendsingle/vextendsingleδ[(xij,yij),(xi′j′,yi′j′)]≥2ρm ∀1≤i,i′≤n1,1≤j,j′≤n2,i∝ne}ationslash=i′,j∝ne}ationslash=j′/bracerightBig . (3) Each individual cell of this billiard table possesses a single moving partic le of radius ρm, 0≤ρm≤ρand unit mass. All the moving particles are assumed to have independ ent initial coordinates within their respective cells, with the proviso that no overlap can occur between any pair of moving particles. The system energy, E=NkBT(with N=n1×n2, the number of moving particles, Tthe system temperature, and kB Boltzmann’s constant) is constant and assumed to be initially random ly divided amongFourier’s law in many particle dispersing billiards 7 thekinetic energies ofthemoving particles, E=/summationtext i,jǫij,ǫij=mv2 ij/2, where vijdenotes the speed of particle ( i,j). Energy exchanges occur when two moving particles located in neighb ouring cells collide. Such events can take place provided the radii of the moving p articlesρmis large enough compared to ρ. Indeed the value of the critical radius, below which binary collisions do not occur, is determined by half the separation between the corners of two neighbouring cells, ρc=/radicalbigg ρ2−l2 4. (4) Figure 3. A binary collision event in the critical configuration where ρm=ρcwould occur only provided the colliding particles visit the corresponding cor ners of their cells simultaneously. The value of ρin this figure is the same as in figures 1 and 4. For the sake of illustration, the unlikely occurrence of a binary collisio n event at the critical radius ρm=ρcis shown in figure 3. All the collisions are elastic and conserve energy, so that the dynam ical system is Hamiltonian with 2 Ndegrees of freedom. Its phase space of positions and velocities is 4N-dimensional. Accordingly, the sensitivity to initial conditions of the d ynamics is characterizedby4 NLyapunovexponents, {λi}4N i=1,obeyingthepairingruleofsymplectic systems, λ4N−i+1=−λi,i= 1,...,2N. Collision events between two moving particles are referred to as binary collision eventsand will be distinguished from wall collision events , which occur between the moving particles and the walls of their respective confining cells. The o ccurrences of the former are characterized by a binary collision frequency ,νb, and the latter by a wall collision frequency ,νw. Both frequencies depend on the difference ρm−ρc, separating the moving particles radii from the critical radius, equation (4). By definition of ρc, the binary collision frequency vanishes at ρm=ρc,νb|ρm=ρc= 0, and, correspondingly, the wall collision frequency at the critical radius is the collision freque ncy of the single- cell billiard, νw|ρm=ρc=νc. We will assume from now, unless otherwise stated, that the system is globally isolated and apply periodic boundary conditions a t the borders, thereby identifying Bρ(i+kn1,j+ln2) withBρ(i,j) for any k,l∈Z, 1≤i≤n1, 1≤j≤n2. Examples of such billiards are displayed in figure 4. Obviously the quincu nx rhombic lattice structure, which is generated by the rhombic cells, is but one amongFourier’s law in many particle dispersing billiards 8 Figure 4. Examples of lattice billiards with triangular (top), rhombic (middle) and hexagonal (bottom) tilings. The coloured particles move among an a rray of fixed black discs. The radii of both fixed and mobile discs are chosen so tha t (i) every moving particle is geometrically confined to its own billiard cell (identified as the area delimited by the exterior intersection of the black circles around the fixed discs), but (ii) can nevertheless exchange energy with the moving particles in th e neighbouring cells through binary collisions. The solid broken lines show the traject ories of the moving particles centers about their respective cells. The colours a re coded according to the particles kinetic temperatures (from blue to red with increas ing temperature).Fourier’s law in many particle dispersing billiards 9 different possible structures. Triangular, upright square, or hex agonal cells can be used as alternative periodic structures. One might also cover the plane w ith random or quasi-crystalline tessellations. The only relevant assumptions in wha t follows is that the moving particles must be confined to their (dispersing) billiard cells and that binary interactions between neighbouring cells can be turned on and off by t uning the system parameters. The two important features of such lattice billiards is that (i) there is no mass transport across the billiard cells since the moving particles are confi ned to their respective cells, and (ii) energy transport can occur through bina ry collision events which take place when the particles of two neighbouring cells come into contact. In periodicstructures such asthequincunx rhombiclattice, theposs ibility ofsuch collisions is controlled by tuning the parameter ρmabout the critical radius ρc, keeping ρfixed. We can therefore distinguish two separate regimes : •Insulating billiard cells : 0≤ρm< ρc Absence of interaction between the moving discs. No transport pr ocess across the individual cells can happen; •Conducting billiard cells :ρc< ρm< ρ Binary collision events are possible. Energy transport across the in dividual cells takes place. The case ρm=ρcis singular. We will refer to the critical geometry as the limit ρm>→ρc. In the insulating regime, there is no interaction among moving particle s so that the billiard cells are decoupled. The moving particles are independent a nd their kinetic energies are individually conserved, resulting in 2 Nzero Lyapunov exponents. The equilibrium measure in turn has a product structure and phase-spa ce distributions are locally uniform with respect to the particles positions and velocity dire ctions. The Npositive Lyapunov exponents of the system are all equal to the po sitive Lyapunov exponent of the single-cell dispersing billiard, up to a factor corres ponding to the particles speeds, vij=/radicalbig 2ǫij/m:λij+=vijλ+, whereλ+is the Lyapunov exponent of the single-cell billiard measured per unit length. When particles are allowed to interact, on the other hand, local ene rgies are exchanged through collision events. Thus only the total energy is c onserved in the conducting regime. The ergodicity of such systems of geometrically confined particles in interaction was proven by Bunimovich et al.[15]. The resulting dynamical system, whose equilibrium measure is the microcanonical one (taking into cons ideration that particles are otherwise uniformly distributed within their respective cells), enjoys the K-property. This implies ergodicity, mixing, and strong chaotic prope rties, including the positivity of the Kolmogorov-Sinai entropy. Two Lyapunov exp onents are zero, one associated to the conservation of energy, the other to the direc tion of the flow. The 2(2N−1) remaining Lyapunov exponents form non-vanishing pairs of expo nents with opposite signs, λ1> ... > λ 2N−1>0,λ4N−i+1=−λi,i= 1,...,2N−1. The regime of interest to us is that corresponding to particles inter acting rarely,Fourier’s law in many particle dispersing billiards 10 which is to say, in analogy with a solid, that particles mostly vibrate insid e their cells, ignorant of each other, and only seldom making collisions with their neig hbours, thereby exchanging energy. As we turn on the interaction and let ρm/greaterorsimilarρc, binary collisions, though they can occur, will remain unlikely. This is to say that the bina ry collision frequency, νb, will, in this regime, remain small with respect to the wall collision frequency, νw, which in the absence of interaction and, in particular, at the critica l geometry, we recall is equal to the wall collision frequency of the sin gle-cell billiard, νw|ρm=ρc=νc. When ρm/greaterorsimilarρc, we therefore expect νw≫νb, as well as νw≃νc. In words :time scales separate . The consequence is that relaxation to local equilibrium –i. e.uniformization of the distribution of the particles positions and veloc ity directions at fixed speeds– occurs typically much faster than the energy exc hange which drives the relaxationto theglobal equilibrium. This mechanism justifies resortin g tokinetic theory in order to compute the transport properties of the model. 3. Kinetic theory 3.1. From Liouville’s equation to the master equation The phase-space probability density is specified by the N-particles distribution function pN(r1,v1,...,rN,vN,t), whereraandva,a= 1,...,N, denote the ath particle position and velocity vectors. The index astands for the label ( i,j) of the cells defined by equation(2). Foroursystem, asiscustomaryforhardspheredy namics, thisdistribution satisfies a pseudo-Liouville equation [21], which is well defined despite t he singularity of the hard-core interactions. This equation, which describes the time evolution of pN is composed of three types of terms: (i) the advection terms, whic h account for the displacement of the moving particles within their respective billiard cells ; (ii) the wall collision terms, which account for the wall collision events, between t he moving particles and thedfixed scattering discs which form the cells walls; and (iii) the binary collis ion terms, which account for binary collision events, between moving pa rticles belonging to neighbouring billiard cells : ∂tpN=N/summationdisplay a=1/bracketleftBigg −va·∂ra+d/summationdisplay k=1K(a,k)/bracketrightBigg pN+1 2N/summationdisplay a,b=1B(a,b)pN. (5) Each wall collision term involves a single moving particle with index aand one of the d fixed discs in the corresponding cell, with index kand position Rk. Letrak=ra−Rk denote their relative position. Following [22], we have K(a,k)pN(...,ra,va,...) = ρ/integraldisplay ˆe·va>0dˆe(ˆe·va)/bracketleftBig δ(rak−ρˆe)pN(...,ra,va−2ˆe(ˆe·va),...) −δ(rak−ρˆe)pN(...,ra,va,...)/bracketrightBig , (6) whereˆedenotes the normal unit vector to the fixed disc kin the cell of particle a.Fourier’s law in many particle dispersing billiards 11 Likewise the binary collision operator, written in terms of the relative positions rab and velocities vabof particles aandb, and the unit vector ˆeabthat connects them, is B(a,b)pN(...,ra,va,...,rb,vb,...) = 2ρm/integraldisplay ˆeab·vab>0dˆeab(ˆeab·vab)/bracketleftBig δ(rab−2ρmˆeab) ×pN(...,ra,va−ˆeab(ˆeab·vab),...,rb,vb+ˆeab(ˆeab·vab),...) −δ(rab+2ρmˆeab)pN(...,ra,va,...,rb,vb,...)/bracketrightBig .(7) We notice that only the terms B(a,b)corresponding to first neighbours are non-vanishing and contribute to the double sum on the RHS of equation (5). Provided we have a separation of time scales between wall and binary collisions, the advection and wall collision terms on the RHS of equation (5) will typica lly dominate the dynamics on the short time ∼1/νw, which follows every binary collision event, thus ensuring, thanks to the mixing within individual billiard cells, the relaxat ion of the phase-space distribution pNto local equilibrium well before the occurrence of the next binary event, whose time scale is ∼1/νb. In other words, pN(r1,v1,...,rN,vN,t) quickly relaxes to a locally uniform distribution, which depends only on t he local energies, justifying the introduction of P(leq) N(ǫ1,...,ǫ N,t)≡/integraldisplayN/productdisplay a=1dradvapN(r1,v1,...,rN,vN,t)N/productdisplay a=1δ(ǫa−mv2 a/2),(8) whereva≡ |va|. On the time scale of binary collision events, this distribution subsequently relaxes to the global microcanonical equilibrium distrib ution. This process accounts for the transport of energy, and can be characterize d by the master equation [19] ∂tP(leq) N(ǫ1,...,ǫ N,t) =1 2N/summationdisplay a,b=1/integraldisplay dη ×/bracketleftBig W(ǫa+η,ǫb−η|ǫa,ǫb)P(leq) N(...,ǫa+η,...,ǫ b−η,...,t) −W(ǫa,ǫb|ǫa−η,ǫb+η)P(leq) N(...,ǫa,...,ǫ b,...,t)/bracketrightBig , (9) whereW(ǫa,ǫb|ǫa−η,ǫb+η) denotes the probability that an energy ηbe transferred from particle ato particle bas the result of a binary collision event between them. This equation is a closure for the local equilibrium distribution P(leq) N, obtained from equation (5) under the assumption that νw≫νb. The first two terms on the RHS of equation (5) are eliminated because they leave invariant the local distribution/producttextN a=1δ(ǫa−mv2 a/2). There remain the contributions (7) from the binary collisions, which, under the assumption that the local distibutions are uniform with respect to the positions and velocity directions, yield the following expression of W: W(ǫa,ǫb|ǫa−η,ǫb+η) =2ρmm2 (2π)2|Lρ,ρm(2)|/integraldisplay dφdR/integraldisplay ˆeab·vab>0dvadvb (10) ׈eab·vabδ/parenleftBig ǫa−m 2v2 a/parenrightBig δ/parenleftBig ǫb−m 2v2 b/parenrightBig δ/parenleftBig η−m 2[(ˆeab·va)2−(ˆeab·vb)2]/parenrightBig ,Fourier’s law in many particle dispersing billiards 12 where the first integration is performed over the positions of the c enter of mass, R≡(ra+rb)/2, between the two particles aandb, given that they are in contact and both located in their respective cells, and over the angle φof the unit vector connecting aandb,ˆeab= (cosφ,sinφ). The normalizing factor |Lρ,ρm(2)|denotes the 4-volume of the billiard corresponding to two neighbouring cells aandb, which, with the assumption thatρm/greaterorsimilarρc, can be approximated by |Lρ,ρm(2)| ≃ |B ρ|2. This substitution amounts to neglecting the overlap between the two particles; see equation ( 23) for a refinement of that approximation. We point out that the position and velocity int egrations in equation (10) can be formally decoupled; in this way, we can prove th at the transition rateWis given in terms of Jacobian elliptic functions, see Appendix A. 3.2. Geometric factor As we show in Appendix A, an important property of the master equa tion (9) is that the factor which accounts for the geometry of collision events fac torizes from the part of the kernel that accounts for energy exchanges. Therefore, a sρm→ρc, the critical value of the radius at which binary collision events become impossible, which is the regime where the billiard properties are accurately described in terms of th e master equation above, the geometric factor/integraltext dφdRencloses the scaling properties of observables with respect to the billiard geometry. We now compute this quantity. A binary collision occurs when particles aandbcome to a distance 2 ρmof each other, with ra∈ Bρ(a) andrb∈ Bρ(b). LetR= (x,y) be the center of mass coordinates andφbe the angle between the particles relative position and the axis conn ecting the center of the cells. Taking a reference frame centered between t he cells, we may write ri=1 2(x,y)+σiρm(cosφ,sinφ), (11) whereσi=±1 andi=aorb. The integral to be evaluated is the volume of the triplets (x,y,φ) about the origin so that /parenleftBigx 2+σiρmcosφ/parenrightBig2 +/parenleftbiggy 2+σiρmsinφ±l 2/parenrightbigg2 ≥ρ2. (12) As illustrated in figure 5, for different orientations φof the vector connecting the two particles, this is a region bounded by four arc-circles, which we deno te by yσ,τ(x)≡ −2σρmsinφ−τl+τ/radicalBig 4ρ2−(x+2σρmcosφ)2, (13) whereσ,τ=±1. As seen from figure 5, the area is connected for −φT≤φ≤φT, whereφTis the angleφat which opposite arcs intersect, φT= arcsinρ2 m−ρ2 c lρm. (14) Beyond that value, the area splits into two triangular areas. These areas shrink to zero at the angle φgiven by φM= arccosρmρc+l/2/radicalbig ρ2−ρ2m ρ2. (15)Fourier’s law in many particle dispersing billiards 13 Φ/EquΑΛ0 Φ/EquΑΛ0.05 Φ/EquΑΛ0.10 Φ/EquΑΛ0.15 Φ/EquΑΛ0.20 Figure 5. Possible positions of the center of mass ( x,y) for different values of φ, see equation (11), at a binary collision event. Here ρ= 13/25landρm= 13/50l. Let−φM≤φ≤φM. Wedenoteby x1< x2< x3< x4thefourcornersoftherectangular domain, x1=−x4=−2(ρmcosφ−ρc), x2=−x3=−2/radicalbig ρ2−ρ2msinφ.(16) Forφ≥φT, the points at which the opposite arcs intersect are given by xi=±(l−2ρmsinφ)/bracketleftbigg−l2+4lρmsinφ+4(ρ2−ρ2 m) l2−4lρmsinφ+4ρ2m/bracketrightbigg1/2 . (17) Combining equations (13)-(17) together, we can make use of the s ymmetry φ→ −φ and write the integral to be computed as α(ρ,ρm)≡/integraldisplay dφdR= 2/bracketleftbigg/integraldisplayφM 0A1(φ)dφ+/integraldisplayφM φTA2(φ)dφ/bracketrightbigg , (18) where A1(φ) =/integraldisplayx3 x1y+1,−1(x)dx+/integraldisplayx4 x3y−1,−1(x)dx−/integraldisplayx2 x1y+1,+1(x)dx−/integraldisplayx4 x2y−1,+1(x)dx,(19) which is the area bounded by the four arcs yσ,τ, and A2(φ) =/integraldisplayxi −xi[y−1,+1(x)−y+1,−1(x)]dx, (20) is the area of the overlapping opposite arcs y−1,+1andy+1,−1, which occurs when φT≤φ≤φM[it gives a negative contribution to A1(φ)]. Thecomputationoftheseexpressions iseasilyperformednumerica lly, andtheresult shown in figure 6. Near the critical geometry, we expand the quantity (18) in powers of the difference ρm−ρc, α(ρ,ρm) =∞/summationdisplay n=1cn(ρm−ρc)n. (21) The first two coefficients vanish, so that the leading term correspo nds ton= 3. TheFourier’s law in many particle dispersing billiards 14 0.010.02 0.05 0.10.2/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1 100000/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1 10000/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1 1000/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1 100/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1 101 Ρm/MiΝusΡcΑ/LParen1Ρ,Ρm/RParen1 Figure 6. Area of binary collisions α(ρ,ρm) versus ρm−ρcfor seven values of ρ ranging from 9 /25 to 42/100 (l= 1/√ 2). first few coefficients are derived in Appendix B and given by : c1=c2= 0, c3=128ρc 3l2, c4=256ρ2 c 3l4, c5=16 l2/parenleftbigg1 3ρc+8ρc 5l2+16ρ3 c l4+16ρ4 5l4ρc/parenrightbigg .(22) We further note that the computation of the two-cell 4-volume, |Lρ,ρm(2)|, which, as noticed above, is approximated by the square of the single cell ar ea|Bρ|2, can be improved using equation (21). Indeed it is easily seen that d |Lρ,ρm(2)|/dρm= −2α(ρ,ρm), which implies |Lρ,ρm(2)|=|Bρ|2−2∞/summationdisplay n=4cn−1 n(ρm−ρc)n. (23) It is immediate to check that corrections |Lρ,ρm(2)|−|Bρ|2=O(ρm−ρc)4. 3.3. Binary collision frequency Having computed the transition rates of the master equation with r espect to the billiard geometry, we now turn to the computation of observables. The fir st quantity of interest, which can be readily computed from equation (10), is the binary collisio n frequency νb. This is an equilibrium quantity which, in the (global) microcanonical ense mble with energyE=ǫ1+...+ǫN, involves the two-particle energy distribution, P(eq) 2(ǫa,ǫb) =(N−1)(N−2) E2/parenleftbigg 1−ǫa+ǫb E/parenrightbiggN−3 , (24) and can be written as νb=/integraldisplay dǫadǫbdηW(ǫa,ǫb|ǫa−η,ǫb+η)P(eq) 2(ǫa,ǫb). (25)Fourier’s law in many particle dispersing billiards 15 Taking the large Nlimit and letting E=NkBT≡N/β, we can write P(eq) 2(ǫa,ǫb)≃ β2exp[−β(ǫa+ǫb)][1+O(N−1)]. Substituting this expression into the above equation andinserting theexpression of Wfromequation(10), weobtain, aftersomecalculations, νb≃/radicalbigg kBT πm2ρm |Bρ|2/parenleftbigg/integraldisplay dφdR/parenrightbigg [1+O(N−1)]. (26) This expression involves the geometric factor (18) in the first brac ket. The leading term in the second bracket is the canonical expression of the binary collis ion frequency. The second term is a positive finite Ncorrection, which is useful in that it shows that the binary collision frequency decreases to its asymptotic value as N→ ∞. 3.4. Rescaled master equation The binary collision frequency (26) defines a natural dimensionless t ime scale for the stochastic process described by the master equation (9) with tra nsition rates (10). The master equation can thus be converted to dimensionless form by re scaling the energies by a reference thermal energy and time by the corresponding asy mptotic ( N→ ∞) value of the binary collision frequency (26). Introducing the variables ea≡ǫa kBT, (27) h≡η kBT, (28) τ≡νbt, (29) equation (9) becomes ∂τp(leq) N(e1,...,e N,τ) =1 2N/summationdisplay a,b=1/integraldisplay dh ×/bracketleftBig w(ea+h,eb−h|ea,eb)p(leq) N(...,ea+h,...,e b−h,...,τ) −w(ea,eb|ea−h,eb+h)p(leq) N(...,ea,...,e b,...,τ)/bracketrightBig , (30) with the transition rates w(ea,eb|ea−h,eb+h) =/radicalbigg 2 π3/integraldisplay x1/bardbl−x2/bardbl>0dx1/bardbldx1⊥dx2/bardbldx2⊥(x1/bardbl−x2/bardbl) (31) ×δ(ea−x2 1/bardbl−x2 1⊥)δ(eb−x2 2/bardbl−x2 2⊥)δ(h−x2 1/bardbl+x2 2/bardbl) This master equation shows that all the properties of heat conduc tion are rescaled by the binary collision frequency and temperature in its limit of validity whe re the collision frequency vanishes. In particular, this shows that the coefficient of heat conduction is proportional to the binary collision frequency in this limit, as explaine d in the next subsection.Fourier’s law in many particle dispersing billiards 16 3.5. Thermal conductivity Starting from the master equation (9), we derive an equation for t he evolution of the kinetic energy of each moving particle, ∝an}bracketle{tǫa∝an}bracketri}ht ≡kBTa, which defines the local temperature, where∝an}bracketle{t.∝an}bracketri}htdenotes an average with respect to the energy distributions. By t he structure of equation (9), such an equation can be expressed in terms of the transfer of energy due to the binary collisions between neighbouring cells. The time evolution o f the average local kinetic energy is given by ∂t∝an}bracketle{tǫa∝an}bracketri}ht=−/summationdisplay b∝an}bracketle{tJa,b(ǫa,ǫb)∝an}bracketri}ht, (32) with the energy flux defined as Ja,b(ǫa,ǫb)≡/integraldisplay dηηW(ǫa,ǫb|ǫa−η,ǫb+η), =−/integraldisplay dηηW(ǫa+η,ǫb−η|ǫa,ǫb).(33) This expresses the local conservation of energy. Over long time scales, the probability distribution becomes controlled by this local conservation of energy, the slowest variables being the local kinet ic energies ∝an}bracketle{tǫa∝an}bracketri}htor, equivalently, the local temperatures Taas defined above. This holds even though statisticalcorrelationsdevelopbetweenthelocalenergiesinthep robabilitydistributions. These statistical correlations are well known for transport proc esses ruled by master equations such as Eq. (9) [19, 23] and are observed in the present system as well. To be specific, we consider a one-dimensional chain, extending along thex-axis, formedwithasuccession ofpairsofrhombicbilliardcells arrangedinqu incunx, similarly to the middle panel of figure 4, except the vertical height is here on ly one unit of length. The unit of horizontal and vertical lengths is thus l√ 2 and there are two cells per each unit of length. Similar results hold for different choices of ge ometry modulo straightforward adaptations. We imagine that the system is in a non-equilibrium state, with a small tem perature difference δTabout an average temperature Tbetween neighbouring cells, and consider the average heat transfered from cell aat inverse temperature βa=β+δβ/2 to cell b at inverse temperature βb=β−δβ/2,δβ=−δT/(kBT2), both cells being assumed to be in thermal equilibrium at their respective temperatures. The sta tistical correlations we observe in the present system are of the order of δβ2, as it is the case in other systems [19]. In the non-equilibrium state, these statistical corre lations are controlled in the long-time limit by the local temperatures. Since the process is h ere ruled by a Markovian master equation in the limit of small binary collision frequenc y, we get the equation of heat for the temperature ∂tT(x,t) =∂x[κ∂xT(x,t)], (34) where the local temperature is here written as T(x,t) =∝an}bracketle{tǫij∝an}bracketri}ht/kB,x=√ 2l(i+j/2), i= 1,...,N/2,j= 0,1 [see equation (1)].Fourier’s law in many particle dispersing billiards 17 According to the rescaling property of the master equation discus sed in section 3.4, the heat conductivity is proportional to the binary collision frequen cy∝bardbl: κ l2=Aνb, (35) with a dimensionless constant A. An analytical estimation can be obtained by transforming the maste r equation into a hierarchy of equations for all the moments of the probability distr ibution:∝an}bracketle{tǫa∝an}bracketri}ht,∝an}bracketle{tǫaǫb∝an}bracketri}ht, ∝an}bracketle{tǫaǫbǫc∝an}bracketri}ht,... The evolution equations of these moments are coupled. Truncating the hierarchy at the equations for the averages ∝an}bracketle{tǫa∝an}bracketri}ht, we get the approximate heat conductivity : κ l2≃β4 2/integraldisplay dǫadǫbdη η(ǫb−ǫa)W(ǫa,ǫb|ǫa−η,ǫb+η)exp[−β(ǫa+ǫb)], =/radicalbigg kBT πm2ρm |Bρ|2/parenleftbigg/integraldisplay dφdR/parenrightbigg . (36) So thatA= 1 in this approximation. The same result holds if we include the equatio ns for the moments ∝an}bracketle{tǫaǫb∝an}bracketri}htwith|a−b| ≤1. Though the approximate result (36) above does not rule out possib le corrections to A= 1, wearguethat A= 1isanexactpropertyofthemasterequation(9)intheinfinite system limit. This claim is borne out by extensive studies of the stocha stic process described by equation (9) that will be reported in a separate publica tion [24]. The focus is here on the billiard systems whose conductivity may however bear corrections to this identity, due we believe to lack of sufficient separation of time s cales between wall and binary collision events. In the following, the results of numer ical computations are presented which support these claims. 4. Numerical results The above formulae for the binary collision frequency, equation (26 ), and the thermal conductivity, equation (36), together with the expressions of th e geometric factors, equations (22) and (23), provide a detailed picture of the mechanis m which governs the transport of heat in our model. Numerical computations of these q uantities further add to this picture and provide strong evidence of the validity of our the oretical approach. 4.1. Binary and wall collision frequencies For the sake of computing the binary collision frequency, we simulate the quasi-one dimensional channel of Ncells with rhombic shapes and apply periodic boundary conditions at the horizontal ends of the channel. Thus each cell ha s two neighbouring ∝bardblWe notice that the heat capacityper particle isequal to cV= (1/N)∂E/∂T=kB, sothat the thermal conductivity is also equal to the thermal diffusivity in units where kB= 1.Fourier’s law in many particle dispersing billiards 18 cells, left and right, and interactions between any two neighbouring cells can occur through both top and bottom corners ¶. In figure 7, we compare the computations of νbtoνwfor a system of N= 10 cells at unit temperature. The parameters are taken to be ρ= 9/25 andρm= 3/25,...,17/50 by steps of 1 /50. Both collision frequencies are computed in the units of the microcanonical average velocity, vN≡2N/radicalbig N/2(N−1)!/(2N−1)!!, which, as N→ ∞, converges to the canonical average velocity v∞=/radicalbig π/2. The wall collision frequency is compared to the collision frequency of the isolated cells νc, itself measured in the units of the single particle velocity. As expected, νb/νw≪1 andνw≃νcfor ρm/greaterorsimilarρc. The crossover νb≃νwoccurs at ρm≃11/50 for this value of ρ. 0.05 0.1 0.15 0.2 0.25 0.300.20.40.60.811.21.41.61.82 ρm − ρcνb, νw νw/νc νb νb/νw Figure 7. Wall and binary collision frequencies νwandνbversusρm−ρcin a one- dimensional channel of N= 10 rhombic cells with ρ= 9/25 and lattice spacing l= 1/√ 2. 4.2. Thermal conductivity 4.2.1. Heat flux. The thermal conductivity can be obtained by computing the heat flux in a nonequilibrium stationary state. Such stationary states oc cur when the two ends of the channel are put in contact with heat baths at separat e temperatures, say T−andT+,T−< T+. Let a system of Ncells be in contact with two thermostated cells at respective temperatures T±, and let these cell indices be n=±(N+1)/2 (we take Nodd for the sake of definiteness). Provided the difference between the bath t emperatures is small, T+−T−≪(T++T−)/2, a linear gradient oftemperature establishes throughthesyste m, with local temperatures Tn=1 2(T++T−)+n N+1(T+−T−). (37) ¶We mention that this set-up allowsfor re-collisionbetween two partic les (under stringent conditions), due to the vertical periodic boundary conditions. This conflicts with the assumptions of [15], but does not seem to affect the results as far as numerics are concerned.Fourier’s law in many particle dispersing billiards 19 Under these conditions, the nonequilibrium stationary state is expe cted to be locally well approximated by a canonical equilibrium at temperature Tn. Thiscanbechecked numerically. Infactthelocalthermalequilibrium isverified(by comparing the moments of ǫnto their Gaussian expectation values) under the weaker property of small local temperature gradients, i. e.(T+−T−)/N≪(T++T−)/2, for which the temperature profile is generally not linear since the therma l conductivity depends on the temperature. Indeed, since κ∝T1/2, we expect in that case, according to Fourier’s law, the profile Tn=/bracketleftbigg1 2(T3/2 −+T3/2 +)+n N+1(T3/2 +−T3/2 −)/bracketrightbigg2/3 . (38) The thermal conductivity can therefore be computed from the he at exchanges of the chain in contact with the two cells at the ends of the chain, respe ctively thermalized at temperatures T±δT/2,δT≪T. The thermalization of the end cells is achieved by randomizing the velocities of the two particles at every collision they m ake with their cells walls, according to the usual thermalization procedure of part icles colliding with thermalized walls [28]. First, we consider a chain containing a single cell in order to test the v alidity of the master equation. In this case, the procedure amounts to simu lating a single particle confined to its cell and performing random collisions with stochastic p articles which penetrate the cell corners according to the statistics of binary c ollisions. For a chain with a single cell, the heat conductance is given by Eq. (36) with A= 1, as there are no correlations with the stochastic particles. Figure 8 shows the results of the computations of the heat conduc tance+with this method and provides a comparison with the binary collision frequency νbon the one hand (left panel), as well as with the results of our kinetic theory pr edictions on the other hand (right panel). Theagreementbetweenthedataandequations(26), (36),(35) andthecomputation of the integrals (18)-(20), especially as ρm→ρc, demonstrates the validity of the stochastic description of the billiard system, equation (9). Next, we increase the size of the chain in order to reach the therma l conductivity in the limit of an arbitrarily large chain. The results are that statistica l correlations appear between the kinetic energies along the chain. As we show belo w, their influence on the computed value of the conductivity diminishes as ρm→ρc. FixT−= 0.5 andT+= 1.5 to be the baths temperatures, and let the size of the system increase from N= 1 toN= 20asρmisprogressively decreased from ρm= 11/50 toρm= 13/100, with fixed ρ= 9/25. As one can see from figure 8, this range of values ofρmcrosses over from a regime where the separation of time scales is no t effective +Here and in the sequel, the thermal conductance or conductivity a re further divided by l2√ T, where l= 1/√ 2 is the rhombic cell size, so as to eliminate its length and temperature dependences, thus defining the reduced thermal conductivity κ∗=κ/(l2√ T). In these expressions and from here on, we further set kB≡1.Fourier’s law in many particle dispersing billiards 20 00.05 0.10.15 0.20.25 0.30.90.9511.051.11.151.21.251.3 ρm − ρcκ/(l2νb) 10−210−110−610−410−2100102 ρm − ρcκ/(l2 T1/2), νb//T1/2 κ νb K.T. Figure 8. Reduced thermal conductance κ, computed from the heat exchange in a chainwith asingle particlewith thermalizedneighbours, and binarycollis ionfrequency νb, as functions of ρm−ρc. (Left) ratio between κandνb; (Right) comparison with the results of section 3. The only relevant parameter is ρ= 9/25. For each value of ρm, several temperature differences δTwere taken, all giving consistent values of κ. The solid line shows the result of kinetic theory (K.T.). to one where it appears to be and where the stochastic model shou ld therefore be a reasonable approximation to the process of energy transport in t he billiard. Forallvaluesof ρm, wemeasuredthetemperatureprofileandheatfluxesthroughou t the system and inferred the value of the reduced thermal conduc tivity by linearly extrapolating the ratios between the average heat flux and local t emperature gradient divided by the square root of the local temperature as functions o f 1/Nto the vertical axis intercept, corresponding to N=∞. Given the parameter values, every realisation was carried out over a time corresponding to 1,000 interactions bet ween the system and baths and repeated over 104realisations. For Nup to 20, this time provided satisfying stationary statistics, with temperature profiles verifying equatio n (38) and statistically constant heat fluxes. The results of the linear regression used to compute the value of κ/νbfor selective values of ρmare shown in figures 9 and 10. Our results thus make it plausible that the ratio between the therma l conductivity and binary collision frequency approaches unity as the parameter ρmdecreases towards the critical value ρc. As will be show in a separate publication [24], this is indeed a property of the stochastic model described by equation (9) and has been verified numerically by direct simulation of the master equation within an accur acy of 4 digits. As of the billiard system, it is unfortunately difficult to improve the res ults beyond those presented here as the CPU times necessary to either increa seNor decrease ρm quickly become prohibitive. Nevertheless, the data displayed in figur es 9 and 10 offer convincing evidence that the thermal conductivity is well approxima ted by the binary collision frequency so long as the separation of time scales between w all and binary collision events is effective.Fourier’s law in many particle dispersing billiards 21 Κ/Slash1ΝB/TildeEqual1.0272Ρm/EquΑΛ0.13 0.00.20.40.60.81.01.01.11.21.31.41.5 1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝBΚ/Slash1ΝB/TildeEqual1.0426Ρm/EquΑΛ0.16 0.00.20.40.60.81.01.01.11.21.31.41.5 1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝB Κ/Slash1ΝB/TildeEqual1.0811Ρm/EquΑΛ0.19 0.00.20.40.60.81.01.01.11.21.31.41.5 1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝBΚ/Slash1ΝB/TildeEqual1.17911Ρm/EquΑΛ0.22 0.00.20.40.60.81.01.01.11.21.31.41.5 1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝB Figure 9. Ratio between thermal conductivity and binary collision frequency, κ/νb, extrapolated from the computation of the average ratio betwee n heat current and local temperature gradients, divided by the local binary collision frequency, 1/n/summationtext i[Ji,i+1/(Ti+1−Ti)]/νb(i). The system sizes are N= 1,2,5,10,15,20. The four pannels correspond to different values of ρm= 13/100,4/25,19/100,11/50, with ρ= 9/25 and thus ρc≃0.068. The red dots are the data points with corresponding error bars and the black solid line shows the result of a linear regress ion performed with data associated to systems of lengths N≥2. 0.060.080.100.120.140.161.001.051.101.151.20 Ρm/MiΝusΡcΚ/Slash1ΝB Figure 10. Ratio between thermal conductivity and binary collision frequency, κ/νb, computated as in figure 9, here collected for a largerset of values o fρm. The horizontal axis shows the difference ρm−ρc. The error bars are of the same order as those in figure 8, with deviations from unity of the data points of the same or der as those in the latter figure.Fourier’s law in many particle dispersing billiards 22 4.2.2. Helfand moment. The computation of the thermal conductivity can also be performed in the global equilibrium microcanonical ensemble using the method of Helfand moments [25, 26, 27]. The Helfand moment has expression H(t) =/summationtext axa(t)ǫa(t), wherexa(t) denotes the horizontal position of particle aat timetandǫa(t) =|va(t)|2/2 its kinetic energy (the masses are taken to be unity). The computation of the time evolutio n of this quantity proceeds by discrete steps, integrating the Helfand moment from one collision event to the next, whether between a particle and the walls of its cell, or betw een two particles. Let{τn}n∈Zdenote the times at successive collision events. In the absence of b inary collisions, the energies are locally conserved and the Helfand moment changes according toH(τn) =H(τn−1) +/summationtext a[xa(τn)−xa(τn−1)]ǫa(τn−1). If, on the other hand, a binary collision occurs between particles kandl, the Helfand moment changes by an additional term [xk(τn)−xl(τn)][ǫk(τn+0)−ǫk(τn−0)]. Computing the time average of the squared Helfand moment, we obtain an expression of the thermal conductiv ity according to κ= lim L→∞1 L(kBT)2lim n→∞1 2τn/angbracketleftBig [H(τn)−H(τ0)]2/angbracketrightBig (39) whereL=N/2 is the horizontal length of the system. Figure 11 shows the results of a computation of the thermal condu ctivity through equation (39) for different system sizes. Though the actual value s ofκvary wildly with N, it is clear that a finite asymptotic value is reached for N≃102. In this case, the constant of proportionality in equation (35) takes the value A= 0.98±0.08, close to 1. Similar results were obtained for other parameter values, and othe r cell geometries as well. 0 0.05 0.1 0.15 0.2 0.250.080.0850.090.0950.10.1050.110.1150.120.125 1/Nκ/(l2 T1/2) Figure 11. Reduced thermal conductivity, computed from the mean squared Helfand moment, versus 1 /N. The parameters are ρ= 9/25,ρm= 9/50. The system sizes vary from N= 4 toN= 100. The dashed line shows the binary collision frequency, νb≃0.1225. The solid line shows a linear fit of the data, with y-intercept 0 .12±0.01, in agreement with the prediction (35).Fourier’s law in many particle dispersing billiards 23 5. Lyapunov spectrum A key aspect of our model, which justifies the assumption of local eq uilibrium, is that it is strongly chaotic. This property can be illustrated through the c omputation of the Lyapunov spectrum and Kolmogorov-Sinai entropy in equilibrium con ditions. As mentioned earlier, in the absence of interaction between the cells ,ρm< ρc, The Lyapunov spectrum of a system of Ncells has Npositive and Nnegative Lyapunov exponents, which, if divided by the average speed of the particle to which they are attached, are all equal in absolute value. This reference value we d enote by λ+. The 2Nremaining Lyapunov exponents vanish. As we increase ρmand let the particles interact, we expect that, in the regime 0< ρm−ρc≪1, where binary collision events are rare, the Lyapunov exponents will essentially be determined by λ+multiplied by a factor which is specified by the particle velocities. The exchange of velocities thus produces an ordering of the exponents which can be computed as shown below. We note that the other half of the spectrum, which remains zero in this approximation, will only pick up positive values as a r esult of the interactions. Assume for the sake of the argument that Nis large. The probability that a given particle with velocity vhas exponent λ=vλ+less than a value λi=viλ+can be approximated by the probability that the particle velocity be less tha nvi, which, if we assume a canonical form of the equilibrium distribution, is Prob(λ < λi) = Prob( v < vi), =β/integraldisplaymv2 i/2 0dǫexp(−βǫ), = 1−exp(−βmv2 i/2). (40) But this probability is simply ( N−i+ 1/2)/N. Therefore the half of the positive Lyapunov exponent spectrum, which is associated to the isolated m otion of particles within their cells, becomes, in the presence of rare collision events, λi=λ+/radicalbigg2 mβ/bracketleftbigg lnN i−1/2/bracketrightbigg1/2 , i= 1,...,N, (41) with ordering λ1> λ2> ... > λ N. In particular, the largest exponent λ1grows like√ lnN. The Kolmogorov-Sinai entropy on the other hand is extensive : hKS= Nλ+/radicalbig π/(2mβ). Refined expressions can be computed by taking the microcanonical distribution associated to a finite N. In particular, the expressions of the Lyapunov exponents become λi=λ+/radicalBigg 2N mβ/bracketleftBigg 1−/parenleftbiggi−1/2 N/parenrightbigg1/(N−1)/bracketrightBigg1/2 . (42) We mention in passing that similar arguments are relevant and can be u sed to approximate the Lyapunov spectrum (actually half of it) of other m odels, such as a mixture of light and heavy particles [30].Fourier’s law in many particle dispersing billiards 24 /SoΛi∆me∆sqΡ /SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ/StΑΡ /StΑΡ 0 10 20 30 40/MiΝus20/MiΝus1001020 iΛi /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛeti/EquΑΛ1,...,10 0.000.050.100.150.200.250.3051015 Ρm/MiΝusΡcΛi/BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛet /BuΛΛeti/EquΑΛ11,...,19 0.10 0.200.150.010.1110 Ρm/MiΝusΡcΛi Figure 12. (Top) Lyapunov exponents λiversusicomputed with a rhombic channel of sizeN= 10 cells and parameter ρ= 9/25. The different curves correspond to different values of ρm= 0.11 (bottom curve) to 0 .34 (top curve) by steps of 0 .01. The lines of stars are obtained for ρm= 0.04< ρc, yielding the exponents λ+and λ−associated to isolated cells, with all the particles at the same speed. The squares correspond to the first half of the spectrum as predicted by equa tion (42). (Bottom) λiversusρm−ρc. The first of the two figures displays the first half of the positive part of the spectrum of exponents, λ1,...,λ Nand compares them to the asymptotic estimate equation (42) (straight lines). The second plot shows the second half of the positive part of the spectrum, λN+1,...,λ 2N−1, displaying their power-law scaling to zero asρm→ρc.Fourier’s law in many particle dispersing billiards 25 /Bullet /Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet /Bullet /Bullet /Bullet /Bullet/SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle/SoΛi∆me∆sqΡ /SoΛi∆me∆sqΡ /SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ /SoΛi∆me∆sqΡ /SoΛi∆me∆sqΡ /SoΛi∆me∆sqΡ/MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond /MedSolidDiamond/SolidDownTriangle /SolidDownTriangle /SolidDownTriangle /SolidDownTriangle /SolidDownTriangle /SolidDownTriangle /SolidDownTriangle /SolidDownTriangle /SolidDownTriangle /SolidDownTriangle 0.10 0.200.151.00 0.1010.00 0.01 Ρm/MiΝusΡchKS/Slash1N/MiΝusΛ/PΛus Figure 13. Extensivity of the Kolmogorov-Sinai entropy. The vertical axis sh ows the difference between the Kolmogorov Sinai entropy, here compu ted from the sum of the positive Lyapunov exponents for system sizes N= 3 (magenta circles), 5 (blue up triangles), 10 (green squares), 15 (cyan diamonds), 20 (red dow n triangles), divided by the corresponding microcanonical average velocity, and the Ly apunov exponent of the isolated billiard cell at unit velocity. The curves fall nicely upon eac h other and converge to zero as ρm→ρc. Let us again insist that equations (41) and (42) account for only Nof the 2N−1 positive Lyapunov exponents. N−1 are vanishing within this approximation. For all these Lyapunov exponents, we expect the corrections to vanish with the binary collision frequency as we go to the critical geometry. Figures 12 and 13 show the results of a numerical computation of th e whole spectrum of Lyapunov exponents and corresponding Kolmogorov -Sinai entropy for the one-dimensional channel of rhombic cells. The agreement with equa tion (42) as ρm→ρc isverygood,attheexceptionperhapsofthelastfewamongthefir stNexponents, whose convergence to the asymptotic value (42) appears to be slower. I nterestingly, the largest exponents have a minimum which occurs at about the value of ρmfor which the binary and wall collision frequencies have ratio unity, see figure 7. Indeed, for larger radii ρm, the spectrum is similar to that of a channel of hard discs (without ob stacles) [29]. Note that the same holds for the ratio between the thermal conductivit y and binary collision frequency, as seen from figure 8. Thus one can interpret the occ urrence of a minimum of the largest exponent as evidence of a crossover from a near clo se-packing solid-like phase (ρm/lessorsimilarρ) to a gaseous-like phase trapped in a rigid structure ( ρm/greaterorsimilarρc). 6. Conclusions To summarize, lattice billiards form the simplest class of Hamiltonian mod els for which one can observe normal transport of energy, consistent with Fo urier’s law. Geometric confinement restricts the transport properties of this system t o heat conductivity alone, thereby avoiding the complications of coupling mass and heat transp orts, which are common to other many particle billiards.Fourier’s law in many particle dispersing billiards 26 The strong chaotic properties of the isolated billiard cells warrant, in a parametric regime where interactions among moving particles are seldom, the pr operty of local equilibrium. This is to say, assuming that wall collision frequencies are o rder-of- magnitude larger than binary collision frequencies, that one is entitle d to making a Markovian approximation according to which phase-space distribu tions are spread over individual cells. We are thus allowed to ignore the details of the dis tribution at the level of individual cells and coarse-grain the phase-space distr ibutions to a many- particle energy distribution function, thereby going from the pseu do-Liouville equation, governing the microscopic statistical evolution, to the master equ ation, which accounts for the energy exchanges at a mesoscopic cell-size scale and local t hermalization. The energy exchange process further drives the relaxation of the wh ole system to global equilibrium. This separation of scales, from the cell scale dynamics, correspon ding to the microscopic level, to the energy exchange among neighbouring cells a t the mesoscopic level, and to the relaxation of the system to thermal equilibrium at th e macroscopic level, is characterized by three different rates. The process of re laxation to local equilibrium has a rate given by the wall collision frequency, much larger than the rate of binary collisions, which characterizes the rate of energy ex changes which accompany the relaxation to local thermal equilibrium, itself much lar ger than the hydrodynamic relaxation rate, given by the binary collision rate divide d by the square of the macroscopic length of the system. On this basis, having reduced the deterministic dynamics of the many particles motions to a stochastic process of energy exchanges between ne ighboring cells, we are able to derive Fourier’s law and the macroscopic heat equation. The energytransport master equation canbesolved withtheresu lt, tobepresented elsewhere [24], that the binary collision frequency and heat conduct ivity are equal. Under the assumption that the wall and binary collision time scales of t he billiard are well separated, the transposition of this result to the billiard dynam ics is that : (i) The heat conductivity of the mechanical model is proportional t o the binary collision frequency, i. e.the rate of collisions among neighbouring particles, κ l2νb=A, ν b≪νw, with a constant Athat is exactly 1 at the critical geometry, ρm→ρc, where the evolution of probability densities is rigorously described by the maste r equation, and remains close to unity over a large range of parameter values fo r which we conclude the time scale separation is effective and the master equat ion therefore gives a good approximation to the energy transport process of th e billiard ; (ii) The heat conductivity and the binary collision frequency both van ish in the limit of insulating system, ρm→ρc, with (ρm−ρc)3, lim ρm→ρcκ l2(ρm−ρc)3= lim ρm→ρcνb (ρm−ρc)3=2ρm |Bρ|2/radicalbigg kBT πmc3, where the coefficient c3depends on the specific geometry of the binary collisions.Fourier’s law in many particle dispersing billiards 27 Though both results are exact strictly speaking only in the limit ρm→ρc, the first one, according to our numerical computations, is robust and holds throughout the range ofparameters for which the binarycollision frequency ismuch less th anthe wall collision frequency, νb≪νw. The deviations from A= 1 which we observed at intermediary values of ρm, where the separation of time scales is less effective, are interpret ed as actual deviations of the energy transport process of the billiard f rom that described by the master equation, where correlations between the motions of n eighboring particles must be accounted for. Under the conditions of local equilibrium, the Lyapunov spectrum ha s a simple structure, half of it being determined according to random velocity distributions within the microcanonical ensemble, while the other half remains close to ze ro. The analytic expression of the Lyapunov spectrum that we obtained is thus exa ct at the critical geometry. The Kolmogorov-Sinai entropy is equal to the sum of th e positive Lyapunov exponents andthusdetermined byhalfofthem. Itisextensive inth enumber ofparticles in the system, whereas the largest Lyapunov exponent grows like t he square root of the logarithm of that number. As we mentioned earlier, we believe our met hod is relevant to the computation of the Lyapunov spectrum of other models of in teracting particles [30]. The computation of the full spectrum, particularly regarding t he effect of binary collisions on the exponents, remains an open problem. Acknowledgments The authors wish to thank D. Alonso, J. Bricmont, J. R. Dorfman, M . D. Jara Valenzuela, A. Kupiainen, R. Lefevere, C. Liverani, S. Olla and C. Mej ´ ıa-Monasterio for fruitful discussions and comments at different stages of this w ork. This research is financially supported by the Belgian Federal Government under th e Interuniversity AttractionPoleprojectNOSYP06/02andtheCommunaut´ efran¸ caisedeBelgiqueunder contract ARC 04/09-312. TG is financially supported by the Fonds d e la Recherche Scientifique F.R.S.-FNRS. Appendix A. Computation of the Kernel Weprovideinthisappendix theexplicit formofthetransitionrate W, givenbyequation (10). We first substitute the two velocity integrals by two angle integrals, eliminating the two delta functions which involve only the local energies. /integraldisplay ˆeab·vab>0dvadvbˆeab·vabδ/parenleftbigg ǫa−mv2 a 2/parenrightbigg δ/parenleftbigg ǫb−mv2 b 2/parenrightbigg δ/parenleftBig η−m 2[(ˆeab·va)2−(ˆeab·vb)2]/parenrightBig =√ 2 m5/2/integraldisplay D+dθadθb(√ǫacosθa−√ǫbcosθb)δ(η−ǫacos2θa+ǫbcos2θb),(A.1) whereθa/bdenote the angles of the velocity vectors va/bwith respect to the direction φof the relative position vector joining particles aandb,ˆeab= (cosφ,sinφ), and theFourier’s law in many particle dispersing billiards 28 angle integration is performed over the domain D+such that√ǫacosθa>√ǫbcosθb. With the above expression (A.1), the explicit φdependence has disappeared so that we have effectively decoupled the velocity integration from the integration over the direction of the relative position between the two colliding particles. W e can further transform this expression in terms of Jacobian elliptic functions as f ollows. Letxi= cosθi,i=a,b, in equation (A.1), which becomes 4√ 2 m5/2/integraldisplay1 −1dxa/radicalbig 1−x2 a/integraldisplay1 −1dxb/radicalbig 1−x2 bθ(√ǫaxa−√ǫbxb)(√ǫaxa−√ǫbxb)δ(η−ǫax2 a+ǫbx2 b),(A.2) whereθ(.) is the Heaviside step function. We thus have to perform the xaandxb integrations along the line defined by the argument of the delta func tion, η=ǫax2 a−ǫbx2 b, (A.3) and that satisfies the condition √ǫaxa>√ǫbxb. (A.4) To carry out this computation, we have to consider the following alte rnatives : (i)ǫa< ǫb, 0< η < ǫ a The solution of equation (A.3) which is compatible with equation (A.4) is xa=/parenleftbiggη+ǫbx2 b ǫa/parenrightbigg1/2 . (A.5) Plugging this solution into equation (A.2) and setting the bounds of th exb-integral to±/radicalbig (ǫa−η)/ǫb, the expression (A.2) reduces to (omitting the prefactors) /integraldisplay√ (ǫa−η)/ǫb 0dxb1/radicalbig ǫa−η−ǫbx2 b/radicalbig 1−x2 b=1√ǫbK/parenleftbiggǫa−η ǫb/parenrightbigg ,(A.6) whereKdenotes the Jabobian elliptic function of the first kind, K(m) =/integraldisplayπ/2 0(1−msin2θ)−1/2dθ(m <1). (A.7) Thus the kernel is, in this case, W(ǫa,ǫb|ǫa−η,ǫb+η) =2ρm π2|Lρ,ρm(2)|/radicalbigg 2 mǫbK/parenleftbiggǫa−η ǫb/parenrightbigg/integraldisplay dφdR. (A.8) (ii)ǫa< ǫb,ǫa−ǫb< η <0 This case is similar to case (i), with equation (A.5) replaced by xb=−/parenleftbigg−η+ǫax2 a ǫb/parenrightbigg1/2 (A.9) and−1< xa<+1. The expression of the kernel corresponding to this case is therefore W(ǫa,ǫb|ǫa−η,ǫb+η) =2ρm π2|Lρ,ρm(2)|/radicalBigg 2 m(ǫb+η)K/parenleftbiggǫa ǫb+η/parenrightbigg/integraldisplay dφdR.(A.10)Fourier’s law in many particle dispersing billiards 29 (iii)ǫa< ǫb,−ǫb< η < ǫ a−ǫb<0 This case is similar to case (ii), with −/radicalbig (ǫb+η)/ǫa< xa<+/radicalbig (ǫb+η)/ǫa. In this case, the expression of the kernel is given by W(ǫa,ǫb|ǫa−η,ǫb+η) =2ρm π2|Lρ,ρm(2)|/radicalbigg 2 mǫaK/parenleftbiggǫb+η ǫa/parenrightbigg/integraldisplay dφdR. (A.11) The cases with ǫa> ǫbare obtained from the cases above with the roles of aandb interchanged and η→ −η. Appendix B. Collision area near the critical geometry Thereasonwhy c1andc2inequation(22)vanishisthattheangledifferenceis O(ρm−ρc) and the area A1(φ) =O[(ρm−ρc)2]. These quantities are easily computed. Letρm= (1+ε)ρc,ε≪1. We have φT=2ρc lε−ρc lε2+O(ε3), (B.1) φM=2ρc lε+4ρ3 c l3ε2+O(ε3), (B.2) which indicates that the bounds of the angle integrals appearing in eq uation (18) are O(ε), withφMandφTdiffering only to O(ε2). The leading contribution to the integral α(ρ,ρm) therefore stems only from the integration of A1(φ), which we can compute explicitly by expanding ρmaboutρcand taking into consideration that φisO(ε). The result is /integraldisplayφM 0A1(φ)dφ≃/integraldisplayφT 0A1(φ)dφ, ≃/integraldisplayφT 0/parenleftBig16ρ3 c lε2−4lρcφ2/parenrightBig dφ, =64ρ4 c 3l2ε3, (B.3) which yields the leading coefficient c3in equation (22). We can compute the coefficients of the next few powers in the expan sion (21) in a similar fashion. First we notice that A2(φ) isO(ε3) so that its integral between φTand φMisO(ε5). Therefore only the integral of A1(φ) contributes to c4in equation (22). The computation of the next terms in the expansion is more involved s ince it requires the integration of A2(φ), whose expression is : A2(φ) = 8ρ2arcsin/bracketleftbiggρmlsinφ−(ρ2 m−ρ2 c) ρ2/bracketrightbigg1/2 (B.4) −4[ρmlsinφ−(ρ2 m−ρ2 c)]1/2(4ρ2 m+l2−4ρmlsinφ)1/2. Expanding this expression for ρmε-close to ρcandφ ε2-close to φT, we get, to leading order, /integraldisplayφM φTA2(φ)dφ=1024ρ4ρ4 c 15l6ε5. (B.5)Fourier’s law in many particle dispersing billiards 30 Combining this expression with the 5 thorder contribution to the integral of A1(φ), we obtainc5, equation (22). We point out that this coefficient is actually much larg er than c3andc4, a reason being that negative powers of ρcappear in its expression. The same holds of the next few coefficients. [1] Fourier J 1822 Th´ eorie analytique de la chaleur (Didot, Paris); Reprinted 1988 (J Gabay, Paris); Facsimile available online at http://gallica.bnf.fr . [2] Bonetto F, Lebowitz J L, and Rey-Bellet L 2000 Fourier Law: A Challenge To Theorists in Fokas A, Grigoryan A, Kibble T, Zegarlinski B (Eds.) Mathematical Physics 2000 (Imperial College, London). [3] BunimovichLAandSinaiYaG1980 Statistical properties of lorentz gas with periodic configu ration of scatterers Commun. Math. Phys. 78479. [4] Bunimovich L A and Spohn K 1996 Viscosity for a periodic two disk fluid: An existence proof Commun. Math. Phys. 176661. [5] Machta J and Zwanzig R 1983 Diffusion in a Periodic Lorentz Gas Phys. Rev. Lett. 501959. [6] Gaspard P, Nicolis G, and Dorfman J R 2003 Diffusive Lorentz gases and multibaker maps are compatible with irreversible thermodynamics Physica A 323294. [7] Mej´ ıa-Monasterio C, Larralde H, and Leyvraz F 2001 Coupled Normal Heat and Matter Transport in a Simple Model System Phys. Rev. Lett. 865417. [8] Eckmann J P and Young L S 2004 Temperature profiles in Hamiltonian heat conduction Europhys. Lett.68790. [9] Eckmann J P and Young L S 2006 Nonequilibrium energy profiles for a class of 1-D models Comm. Math. Phys. 262237. [10] EckmannJP,Mej´ ıa-MonasterioC,andZabeyE2006 Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems J. Stat. Phys. 1231339. [11] Lin K K and Young L S 2007 Correlations in nonequilibrium steady states of random hal ves models J. Stat. Phys. 128607. [12] Ravishankar K and Young L S 2007 Local thermodynamic equilibrium for some stochastic model s of Hamiltonian origin J. Stat. Phys. 128641. [13] Eckmann J P and Jacquet P 2007 Controllability for chains of dynamical scatterers Nonlinearity 201601. [14] Kupiainen A 2007 On Fouriers law in coupled lattice dynamics oral communication (Institut Henri Poincar´ e, Paris, France). [15] Bunimovich L, Liverani C, Pellegrinotti A, and Suhov Y 1992 Ergodic systems of nballs in a billiard table Comm. Math. Phys. 146357. [16] Dolgopyat D, Keller G, and Liverani C 2007 Random Walk in Markovian Environment , to appear in Annals of Probability; Dolgopyat D and Liverani C 2007 Random Walk in Deterministically Changing Environment preprint. [17] Bricmont J and Kupiainen A 2008 in preparation. [18] Gaspard P and Gilbert T 2008 Heat conduction and Fourier’s law by consecutive local mixi ng and thermalization Phys. Rev. Lett. 101020601. [19] Nicolis G and Malek Mansour M 1984 Onset of spatial correlations in nonequilibrium systems: A master-equation description Phys. Rev. A 292845. [20] Chernov N and Markarian R 2006 Chaotic billiards Math. Surveys and Monographs 127(AMS, Providence, RI). [21] Ernst M H, Dorfman J R, Hoegy W R, and Van Leeuwen J M J 1969 Hard-sphere dynamics and binary-collision operators Physica45127; Dorfman J R and Ernst M H 1989 Hard-sphere binary-collision operators J. Stat. Phys. 57581. [22] R´ esibois P and De Leener M 1977 Classical kinetic theory of fluids (John Wiley & Sons, NewFourier’s law in many particle dispersing billiards 31 York). [23] Spohn H 1991 LargeScale Dynamics of Interacting Particles (Springer-Verlag, Berlin). [24] Gaspard P and Gilbert T 2008, On the derivation of Fourier’s law in stochastic energy exch ange systemsin preparation. [25] Helfand E 1960 Transport Coefficients from Dissipation in a Canonical Ensem blePhys. Rev. 119 1. [26] AlderBJ,GassDM,andWainwrightTE1970 Studies in molecular dynamics. VIII. The transport coefficients for a hard-sphere fluid J. Chem. Phys. 533813. [27] Garcia-Rojo R, Luding S, and Brey J J 2006 Transport coefficients for dense hard-disk systems Phys. Rev. E 74061305. [28] Tehver R, Toigo F, Koplik J, and Banavar J R 1998 Thermal walls in computer simulations Phys. Rev. E57R17. [29] Forster C, Mukamel D, and Posch H A 2004 Hard disks in narrow channels Phys. Rev. E 69 066124. [30] Gaspard P and van Beijeren H 2002 When do tracer particles dominate the Lyapunov spectrum? J. Stat. Phys. 109671.

1909.04362v3.Spin_Pumping_from_Permalloy_into_Uncompensated_Antiferromagnetic_Co_doped_Zinc_Oxide.pdf

Spin Pumping from Permalloy into Uncompensated Antiferromagnetic Co doped Zinc Oxide Martin Buchner,1,Julia Lumetzberger,1Verena Ney,1Tadd aus Schaers,1,yNi eli Da e,2and Andreas Ney1 1Institut f ur Halbleiter- und Festk orperphysik, Johannes Kepler Universit at, Altenberger Str. 69, 4040 Linz, Austria 2Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland (Dated: August 11, 2021) Heterostructures of Co-doped ZnO and Permalloy were investigated for their static and dynamic magnetic interaction. The highly Co-doped ZnO is paramagentic at room temperature and becomes an uncompensated antiferromagnet at low temperatures, showing a narrowly opened hysteresis and a vertical exchange bias shift even in the absence of any ferromagnetic layer. At low temperatures in combination with Permalloy an exchange bias is found causing a horizontal as well as vertical shift of the hysteresis of the heterostructure together with an increase in coercive eld. Furthermore, an increase in the Gilbert damping parameter at room temperature was found by multifrequency FMR evidencing spin pumping. Temperature dependent FMR shows a maximum in magnetic damping close to the magnetic phase transition. These measurements also evidence the exchange bias interaction of Permalloy and long-range ordered Co-O-Co structures in ZnO, that are barely detectable by SQUID due to the shorter probing times in FMR. I. Introduction In spintronics a variety of concepts have been devel- oped over the past years to generate and manipulate spin currents [1, 2]. Amongst them are the spin Hall eect (SHE), which originates from the spin orbit coupling [3], spin caloritronics [4] utilizing the spin seebeck eect [5] or spin transfer torque (current induced torque) due to angular momentum conservation [6] as examples. Spin pumping [7], where a precessing magnetization transfers angular momentum to an adjacent layer, proved to be a very versatile method since it has been reported for dier- ent types of magnetic orders [8{11] or electrical properties [12{14] of materials. Furthermore it could also be veri- ed in trilayer systems where the precessing ferromagnet and the spin sink, into which the angular momentum is transferred, are separated by a non-magnetic spacer [15{ 18]. This is strongly dependent on the material, while for Cu [15], Au [16], or Al [17] pumping through a few nanometers is possible an MgO barrier of 1 nm is enough to completely suppress spin pumping [18]. Spintronic devices are usually based on a ferromagnet (FM) although antiferromagnetic spintronics [19] holds the advantages of faster dynamics, less perturbation by external magnetic elds and no stray elds. The latter two are caused by the zero net magnetization of an an- tiferromagnet (AFM), which on the other hand makes them harder to manipulate. One way to control an AFM is by using an adjacent FM layer and exploiting the exchange-bias (EB) eect [20, 21]. Measuring spin- transfer torque in FM/AFM bilayer structures, is possi- Electronic address: martin.buchner@jku.at; Phone: +43-732- 2468-9651; FAX: -9696 yCurrent address: NanoSpin, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076 Aalto, Finlandble [22, 23], but challenging due to Joule heating [24{26] or possible unstable antiferromagnetic orders [27]. Anti- ferromagnets can be used either as spin source [28] or as spin sink [11, 29] in a spin pumping experiment. Thereby the spin mixing conductance, a measure for the absorp- tion of angular (spin) momentum at the interface [7], is described by intersublattice scattering at an antiferro- magnetic interface [30]. Linear response theory predicted an enhancement of spin pumping near magnetic phase transitions [31], which could recently also be veried ex- perimentally [29]. In this work we investigate the behavior of the uncom- pensated, antiferromagnetic Co xZn1-xO with x2f0.3, 0.5, 0.6g(in the following 30 %, 50 % and 60 % Co:ZnO) in contact to ferromagnetic permalloy (Py). While weakly paramagnetic at room temperature, Co:ZnO makes a phase transition to an antiferromagnetic state at a N eel temperature ( TN) dependent on the Co concentra- tion [32]. This resulting antiferromagnetism is not fully compensated which is evidenced by a narrow hysteresis and a non saturating magnetization up to 17 T [33]. Fur- thermore, Co:ZnO lms exhibit a vertical EB in complete absence of a FM layer [34]. This vertical exchange shift is dependent on the Co concentration [32], temperature and cooling eld [35] and the eld imprinted magnetization predominantly shows orbital character [36]. Note that below the coalesence limit of 20 % the vertical EB van- ishes. Co:ZnO therefore oers to study magnetic inter- actions between an uncompensated AFM and a FM Py layer. Static coupling, visible as EB, is investigated using super conducting quantum interference device (SQUID) magnetometry. The dynamic coupling across the inter- face is measured using ferromagnetic resonance (FMR) at room temperature and around the magnetic transi- tion temperatures determined from M(T) SQUID mea- surements. Element selective XMCD studies are carried out to disentangle the individual magnetic contributions. Finally heterostructures with an Al spacer were investi- gated to rule out intermixing at the interface as sourcearXiv:1909.04362v3 [cond-mat.mtrl-sci] 14 Oct 20192 for the coupling eect. II. Experimental Details Heterostructures consisting of Co:ZnO, Py and Al, as shown in Fig. 1 were fabricated on c-plane sapphire sub- strates using reactive magnetron sputtering (RMS) and pulsed laser deposition (PLD) at a process pressure of 4 10-3mbar. The dierent layers of a heterostructure are all grown in the same UHV chamber with a base pressure of 210-9mbar in order to ensure an uncontaminated interface. While Py and Co:ZnO are grown by magnetron sputtering, the Al spacer and capping layers are grown by PLD. Al and Py are fabricated at room temperature using 10 standard cubic centimeters per minute (sccm) Ar as a process gas. For the heterostructures containing a Co:ZnO layer, samples with three dierent Co concentrations of 30 %, 50 % and 60 % are grown utilizing preparation conditions that yield the best crystalline quality known for Co:ZnO single layers [32, 33, 36]. For 30 % and 50 % Co:ZnO metallic sputter targets of Co and Zn are used at an Ar:O 2ratio of 10 : 1 sccm, while for 60 % Co:ZnO no oxy- gen and a ceramic composite target of ZnO and Co 3O4 with a 3 : 2 ratio is used. The optimized growth temper- atures are 450C, 294C and 525C. Between Co:ZnO growth and the next layer a cool-down period is required, to minimize inter-diusion between Py and Co:ZnO. The static magnetic properties are investigated by SQUID magnetometry. M(H) curves are recorded at 300 K and 2 K in in-plane geometry with a maximum magnetic eld of 5 T. During cool-down either a mag- netic eld of5 T or zero magnetic eld is applied to dif- ferentiate between plus-eld-cooled (pFC), minus-eld- cooled (mFC) or zero-eld-cooled (ZFC) measurements. All measurements shown in this work have been corrected by the diamagnetic background of the sapphire substrate and care was taken to avoid well-known artifacts [37, 38]. For probing the element selective magnetic properties X-ray absorption (XAS) measurements were conducted at the XTreme beamline [39] at the Swiss Synchrotron Lightsource (SLS). From the XAS the X-ray magnetic circular dichroism (XMCD) is obtained by taking the direct dierence between XAS with left and right cir- cular polarization. The measurements were conducted with total uoresence yield under 20grazing incidence. Thereby, the maximum magnetic eld of 6.8 T was ap- plied. Both, external magnetic eld and photon helic- ity have been reversed to minimize measurement arte- facts. Again pFC, mFC and ZFC measurements were conducted applying either zero or the maximum eld in the respective direction. The dynamic magnetic properties were measured us- ing multi-frequency and temperature dependent FMR. Multi-frequency FMR is exclusively measured at room temperature from 3 GHz to 10 GHz using a short cir- cuited semi-rigid cable [40]. Temperature dependentmeasurements are conducted using an X-band resonator at 9.5 GHz. Starting at 4 K the temperature is increased to 50 K in order to be above the N eel-temperature of the Co:ZnO samples [32, 35]. At both FMR setups the mea- surements were done in in-plane direction. The measured raw data for SQUID, FMR, XAS and XMCD can be found in a following data repository [41]. III. Experimental results & Discussion FIG. 1: (a) shows the schematic setup of the samples. For the Co:ZnO layer three dierent Co concentrations of 30 %, 50 % and 60 % are used. The cross section TEM image of the 60 % Co:ZnO/Py sample as well as the electron diraction pattern of the Co:ZnO layer (b) and a magnication on the interface between Co:ZnO and Py (c) are shown. Figure 1(a) displays the four dierent types of samples:3 Co:ZnO layers, with Co concentrations of 30 %, 50 % and 60 %, are grown with a nominal thickness of 100 nm and Py with 10 nm. To prevent surface oxidation a capping layer of 5 nm Al is used. For single 60 % Co:ZnO lms the vertical-exchange bias eect was largest compared to lower Co concentrations. Therefore, for 60 % Co:ZnO samples with an additional Al layer as spacer between Co:ZnO and Py have been fabricated. The thickness of the Al spacer (1 nm, 1.5 nm and 2 nm) is in a range where the Al is reported not to suppress spin pumping eects itself [17]. TEM To get information about the interface between Py and Co:ZnO high resolution cross section transmission electron microscopy (TEM) was done. In Fig. 1(b) the cross section TEM image of 60 % Co:ZnO/Py with the electron diraction pattern of the Co:ZnO is shown. A magnication of the interface between Co:ZnO and Py is shown in Fig. 1(c). From XRD measurements [32] it is obvious that the quality of the wurtzite crystal slightly decreases for higher Co doping in ZnO. A similar be- havior is observed in TEM cross section images. While 35 % Co:ZnO shows the typical only slightly misoriented columnar grain growth [32] it is obvious from Fig. 1(b) that the crystalline nanocolumns are less well ordered for 60 % Co:ZnO. Although the electron diraction pattern conrms a well ordered wurtzite structure, the misorien- tation of lattice plains is stronger than for 35 % Co:ZnO [32], even resulting in faint Moir e fringes which stem from tilted lattice plains along the electron path. This cor- roborates previous ndings of !-rocking curves in XRD [32, 36] where the increase in the full width at half maxi- mum also evidences a higher tilting of the crystallites, i.e. an increased mosaicity. The interface to the Py layer is smooth, although it is not completely free of dislocations. Also the interface seems to be rather abrupt within one atomic layer, i.e. free of intermixing. A similar behavior is found for the interface between 50 % Co:ZnO and Py (not shown). XAS and XMCD Figure 2 shows XAS and XMCD spectra recorded at 3 K and a magnetic eld of 6.8 T at the Ni L 3/2and Co L 3/2edges of 60 % Co:ZnO/Py after pFC, mFC or ZFC. For all three cooling conditions the Ni L 3/2edges (Fig. 2(a)) show a metallic character of the Ni XAS with- out any additional ne structure characteristics for NiO and thus no sign of oxidation of the Py. Further, no dif- ferences in the XAS or the XMCD of the Ni edges of dierent cooling conditions are found. The same is ob- served for the Fe L 3/2edges, however, they are aected greatly by self-absorption processes in total uorescence yield (not shown). /s56/s52/s48 /s56/s53/s48 /s56/s54/s48 /s56/s55/s48 /s56/s56/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53 /s40/s97/s41 /s78/s105/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32 /s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84 /s32/s88/s65/s83/s32/s112/s70/s67 /s32/s88/s65/s83/s32/s90/s70/s67 /s32/s88/s65/s83/s32/s109/s70/s67 /s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s112/s70/s67 /s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s90/s70/s67 /s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s109/s70/s67/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s40/s98/s41/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s88/s77/s67/s68/s32/s40/s37/s41 /s55/s55/s48 /s55/s56/s48 /s55/s57/s48 /s56/s48/s48 /s56/s49/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53 /s67/s111/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32 /s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84/s32/s88/s65/s83/s32/s112/s70/s67 /s32/s88/s65/s83/s32/s90/s70/s67 /s32/s88/s65/s83/s32/s109/s70/s67 /s32/s88/s77/s67/s68/s32/s112/s70/s67 /s32/s88/s77/s67/s68/s32/s90/s70/s67 /s32/s88/s77/s67/s68/s32/s109/s70/s67 /s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s45/s48/s46/s48/s54/s45/s48/s46/s48/s52/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54 /s88/s77/s67/s68/s32/s40/s37/s41FIG. 2: In (a) the XMCD at the Ni L 3/2edges after pFC, mFC and ZFC for 60 % Co:ZnO/Py are shown. (b) shows the same for the Co L 3/2edges. The Co L 3/2edges in Fig. 2(b) are also greatly af- fected by the self absorption of the total uorescence yield, since it is buried below 10 nm of Py and 5 nm of Al. In contrast to Ni the XAS and XMCD at the Co L3/2edges (Fig. 2(b)) are not metallic and evidence the incorporation of Co as Co2+in the wurtzite structure of ZnO [32, 36]. The overall intensity of the Co XMCD is strongly reduced indicating a small magnetic moment per Co atom well below metallic Co. This small eective Co moment in 60 % Co:ZnO can be understood by the degree of antiferromagnetic compensation that increases with higher Co doping concentrations [32]. Furthermore, no indications of metallic Co precipitates are visible in the XAS and XMCD of the heterostructure as it would be expected for a strong intermixing at the interface to the Py. No changes between the pFC, mFC and ZFC measure- ments are visible also for the Co edges either in XAS or XMCD indicating that the spin system of the Co dopants is not altered in the exchange bias state. This corrob- orates measurements conducted at the Co K-edge [36]. After eld cooling the XMCD at the Co main absorption increased compared to the ZFC conditions. At the Co K-edge the main absorption stems from the orbital mo- ment. The spin system is only measured indirectly at the pre-edge feature which remained unaected by the cool- ing eld conditions. The data of K- and L-edges com- bined evidences that the imprinted magnetization after eld cooling is composed predominantly of orbital mo-4 ment, which is in good agreement with other EB systems [42, 43] SQUID /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s49/s48/s49 /s45/s49/s48 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s49/s48/s49 /s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s45/s49/s54/s48/s45/s49/s50/s48/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48/s51/s48/s48/s75 /s40/s98/s41/s32/s77/s47/s77/s91/s49/s48/s109/s84/s93 /s48/s72/s32/s40/s109/s84/s41/s32/s80/s121/s32 /s32/s51/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121/s40/s97/s41 /s32 /s32/s51/s48/s48/s75 /s32/s50/s75/s77/s32/s40 /s101/s109/s117/s41 /s48/s72/s32/s40/s109/s84/s41/s32/s32/s109/s70/s67 /s32/s32/s112/s70/s67 /s32/s32/s90/s70/s67/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s50/s75/s32/s97/s102/s116/s101/s114/s58 FIG. 3: At 300 K the M(H) curves of the single Py lm al- most overlaps with the M(H) curves of the heterostructures with all three Co:ZnO concentrations (a). In the inset it can be seen that there is no dierence in coercive eld for Py at 300 K and 2 K. Measuring the 60 % Co:ZnO/Py heterostruc- ture after plus, minus and zero eld cooling, horizontal and vertical exchange bias shifts are visible, as well as an increase in the coercive eld (b). The static coupling in the heterostructures was investi- gated by integral SQUID magnetometry. Measurements done at 300 K, as shown in Fig. 3(a), do not reveal a sig- nicant in uence of the Co:ZnO on the M(H) curve of Py. Just a slight increase in coercive eld from 0.1 mT to 0.4 mT is determined. Some of the M(H) curves in Fig. 3(a) are more rounded than the others. This can be attributed to slight variations in the aspect ra- tio of the SQUID pieces and thus variations in the shape anisotropy. The inset of Fig. 3(a) shows the hysteresis of the single Py lm at 300 K and 2 K, where no dierence in coercivity is visible. Please note that up to now mea- surements were conducted only in a eld range of 10 mT and directly after a magnet reset. This is done to avoid in uences of the oset eld of the SQUID [38]. At lowtemperatures, to determine the full in uence of Co:ZnO, high elds need to be applied, as it has been shown in [35]. Therefore, coercive elds obtained from low temperature measurements are corrected by the known oset eld of 1.5 mT of the SQUID [38]. Since the paramagnetic signal of Co:ZnO is close to the detection limit of the SQUID and thus, orders of mag- nitude lower than the Py signal it has no in uence on the room temperature M(H) curve. However, with an additional Co:ZnO layer a broadening of the hysteresis, a horizontal and a small vertical shift are measured at 2 K as can be seen exemplary for 60 % Co:ZnO/Py in Fig. 3(b). Similar to single Co:ZnO lms where an open- ing of theM(H) curve is already visible in ZFC mea- surements [32, 34{36] also in the heterostructure no eld cooling is needed to increase the coercive eld. /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s53/s49/s48/s49/s53/s50/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50 /s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41 /s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41 /s40/s98/s41 /s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41 /s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s54/s48/s37 /s32/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s97/s102/s116/s101/s114/s32/s90/s70/s67/s40/s97/s41 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s32/s112/s70/s67 /s32/s109/s70/s67 /s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41 /s45/s49/s50/s45/s49/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50 /s32/s112/s70/s67 /s32/s109/s70/s67/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s58 /s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s58 FIG. 4: (a) At 2 K the coercivity increases with Co concen- tration in the heterostructure. In the inset the temperature dependence of the coercivity of the 60 % Co:ZnO/Py het- erostructure is given. (b) The vertical shift (circles) and the horizontal shift (squares) depend on the Co concentration. Both shifts reverse the direction when the measurement is changed from pFC to mFC. Earlier works [32, 34] demonstrated that the hystere- sis opening and vertical shift in Co:ZnO are strongly de- pendent on the Co concentration and increase with in- creasing Co doping level. Furthermore, the EB eects are observed in the in-plane and out-of-plane direction, with a greater vertical shift in the plane. Therefore, the heterostructers with Py are measured with the mag- netic eld in in-plane direction. Figure 4(a) provides an overview of the coercive eld after ZFC for the dier-5 ent Co concentrations. The coercive eld increases from 0.1 mT for single Py to 20.6 mT for 60 % Co:ZnO/Py. Additionally, in the inset the temperature dependence of the coercive eld of the 60 % Co:ZnO/Py heterostructure is shown, since it shows the strongest increase in coercive eld. From the 20.6 mT at 2 K it rst increases slightly when warming up to 5 K. That the maximum coercivity is not at 2 K is in good agreement with measurements at single 60 % Co:ZnO lms where a maximum hysteresis opening at 7 K was determined [35]. Afterwards the co- ercive eld decreases. At the N eel temperature of 20 K a coercive eld of 11.6 mT is measured. Above T Nit de- creases even further but the coercivity is still 3.65 mT at 50 K. A coupling above T Ncould stem from long range magnetic ordered structures in Co:ZnO where rst in- dications are visible already in single Co:ZnO lms [32]. However, for single layers they are barely detectable with the SQUID. The vertical (circles) and horizontal (squares) hystere- sis shifts after pFC and mFC are shown in Fig. 4(b) for the Py samples with Co:ZnO layers. Similar to single Co:ZnO lms the vertical shift increases with rising Co concentration. The shift is given in percent of the magne- tization at 5 T to compensate for dierent sample sizes. Due to the overall higher magnetization at 5 T in combi- nation with Py this percentage for the heterostructures is lower than the vertical shift for single Co:ZnO lms. With increasing Co concentration the degree of antiferro- magnetic compensation increases [32, 35], which in turn should lead to a stronger EB coupling. This can be seen in the horizontal shift and thus EB eld which is strongest for 60 % Co:ZnO/Py and nearly gone for 30 % Co:ZnO/Py. For both kinds of shift the pFC and mFC measurements behave similar, except the change of di- rection of the shifts. Multifrequency FMR The dynamic coupling between the two layers has been investigated by multifrequency FMR measured at room temperature. The frequency dependence of the resonance position between 3 GHz and 10 GHz of the heterostruc- tures is shown in Fig. 5(a). The resonance position of Py yields no change regardless of the Co concentration in the Co:ZnO layer or its complete absence. Also in 2 nm Al/Py and 60 % Co:ZnO/2 nm Al/Py the resonance po- sition stays unchanged. The resonance position of a thin lm is given by Kittel formula [44]: f= 2p Bres(Bres+0M) (1) with the gyromagnetic ratio =gB hand magnetiza- tionM. However, any additional anisotropy adds to Bres and therefore alters eq. (1) [44]. The fact that all samples show the identical frequency dependence of the resonance position evidences that neither the gyromagnetic ratio and thus the Py g-factor are in uenced nor any addi- tional anisotropy BAniso is introduced by the Co:ZnO. By tting the frequency dependence of the resonance po- sition using the Kittel equation with the g-factor of 2.11 [45] all the samples are in the range of (700 15) kA/m, which within error bars is in good agreement with the saturation magnetization of (670 50) kA/m determined from SQUID. /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48/s50/s52/s54/s56/s49/s48 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s48/s49 /s40/s98/s41 /s32/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41 /s66 /s114/s101/s115/s32/s40/s109/s84/s41/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121 /s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s65/s108/s47/s80/s121 /s32/s80/s121 /s40/s97/s41 /s32/s32/s66 /s112/s112/s32/s40/s109/s84/s41 /s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41 /s32/s32 /s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41 /s32/s32/s110/s111/s114/s109/s46/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s66/s32/s40/s109/s84/s41/s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s32/s32/s32/s32/s102/s114/s101/s113/s46/s32/s61/s32/s54/s46/s53/s56/s71/s72/s122 /s32/s76/s111/s114/s101/s110/s116/s105/s97/s110/s32/s102/s105/s116 FIG. 5: The resonance elds determined at room temperature with multifrequency FMR are seen in (a). In the inset an ex- emplary FMR spectrum for of 50 % Co:ZnO/Py at 6.58 GHz is shown with the corresponding Lorentian t. For the linewidth (b) and the associated damping parameter (inset) an in- crease is visible for the heterostructures with higher Co con- centration in the Co:ZnO. The lines are linear ts to the data. Even though the Co:ZnO layer does not in uence the resonance position of the FMR measurement the het- erostructures exhibit an increase in linewidth. This cor- responds to a change of the damping in the system. The frequency dependence of the linewidth can be used to sep- arate the inhomogeneous from the homogeneous (Gilbert like) contributions, from which the Gilbert damping pa- rametercan be determined.
B=
Bhom+
Binhom (2)6 where
Bhom=4 f (3) No dierence in linewidth between Al/Py (open stars) and Py (full stars) is found, as can be seen in Fig. 4(b) where the peak to peak linewidth B ppis plotted over the measured frequency range for all the heterostructures. While the heterostructure with 30 % Co:ZnO/Py (green triangles) lies atop the single Py and the Al/Py lm, the linewidth increases stronger with frequency for 50 % Co:ZnO/Py (blue circles). The broadest FMR lines are measured for the 60 % Co:ZnO/Py heterostructure (red sqaures). Using the Py g-factor of 2.11 [45], can be calcu- lated from the slopes of the frequency dependence ex- tracted from the linewidths seen in Fig. 5(b): the result- ingare shown in the inset. For the single Py layer Py = (5.70.3)10-3which compares well to previously re- ported values [7]. This increases to 50= (8.00.3)10-3 for 50 % Co:ZnO/Py and even 60= (9.40.3)10-3for 60 % Co:ZnO/Py. So the damping increases by a factor of 1.64 resulting in a spin pumping contribution
= (3.70.5)10-3that stems from the angular momentum transfer at the interface of Py and Co:ZnO. By insertion of a 2 nm Al spacer layer
reduces to (0.80.5)10-3. Dependence on the Al spacer thickness To obtain information about the lengthscale of the static and dynamic coupling, heterostructures with Al spacer layers of dierent thickness (1 nm, 1.5 nm and 2 nm thick) between Py and the material beneath (sap- phire substrate or 60 % Co:ZnO) were fabricated. With- out a Co:ZnO layer the spacer underlying the Py layer does not exhibit any changes in either SQUID (not shown) or FMR (see Fig 5 (a) and (b)). The results ob- tained for the 60 % Co:ZnO/Al/Py heterostructure for the coercive eld, vertical and horizontal shift extracted fromM(H) curves are shown in Fig. 6(a), whereas the damping parameter from room temperature multifre- quency FMR measurements, analogues to Fig. 5(b), are depicted in Fig. 6(b). The horizontal shift and the increased coercive eld are caused by the coupling of FM and AFM moments in range of a few Angstrom to the interface [46{48]. There- fore, both eects show a similar decrease by the insertion of an Al spacer. While the horizontal shift and coer- cive eld are reduced signicantly already at a spacer thickness of 1 nm, the vertical shift (inset of Fig. 6(a)) is nearly independent of the Al spacer. Comparing with the XMCD spectra of Fig. 2 it can be concluded that the vertical shift in the uncompensated AFM/FM sys- tem Co:ZnO/Py stems solely from the increased orbital moment of pinned uncompensated moments in Co:ZnO and is independent of the FM moments at the interface.Furthermore, the FM moments do not exhibit any ver- tical shift and the exchange between the two layers only results in the horizontal shift. /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52 /s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100 /s32/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41 /s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52 /s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116 /s32/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41 /s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41 /s40/s98/s41 /s32/s32 /s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s65/s108/s32/s115/s112/s97/s99/s101/s114/s47/s80/s121/s40/s97/s41 /s65/s108/s47/s80/s121 FIG. 6: When an Al spacer is inserted between the Py and the Co:ZnO layer horizontal shift and coercive eld show a strong decrease already at 1 nm spacer thickness (a) while the vertical shift (inset) is not dependent on the spacer thickness. (b) shows the eect of the Al spacer on the Gilbert damping parameter, which also decreases if the spacer gets thicker than 1 nm. As shaded region the Gilbert damping parameter of a Al/Py lm is indicated within error bars. For the FMR measurments after inserting an Al spacer no eect on the resonance position is found, as was shown already in Fig. 5(a). For a 1 nm thick Al spacer the damp- ing results in = (8.80.3)10-3, which gives a
= (3.10.5)10-3. This is only a slight decrease compared to the sample without Al spacer. By increasing the spacer thicknessreduces to values just above the damping ob- tained for pure Py or Al/Py, shown as shaded region in Fig. 6(b). The 1 nm thick Al layer is thick enough to sup- press intermixing between the Co:ZnO and the Py layer as can be seen in Fig. 1(b). Together with the unchanged behavior of Al/Py without Co:ZnO damping eects due to intermixing between Al and Py can be excluded. Also, a change in two magnon scattering can be ruled out, since it would account for non-linear eects on the linewidth and contribute to
Binhom [49]. Therefore, the increase in Gilbert damping can be attributed to a dynamic cou- pling, e.g. spin pumping from Py into Co:ZnO. Further- more, the dynamic coupling mechanism is extends over a longer range than the static coupling. With 1 nm spacer the dynamic coupling is only slightly reduced whereas the static coupling is already completely suppressed.7 Temperature dependent FMR In vicinity to the magnetic phase transition temper- ature the spin pumping eciency should be at a max- imum [29, 31]. Therefore, the samples are measured inside a resonator based FMR setup, as a function of temperature. During the cooldown no magnetic eld is applied and the results shown in Fig. 7 are ZFC mea- surements. For 50 % Co:ZnO/Py the resonance posi- tions shifts of Py to lower magnetic elds as the tem- perature decreases as can be seen in Fig. 7(a). Not only the resonance position is shifting, but also the linewidth is changing with temperature as shown in Fig. 7(b). The linewidth has a maximum at a temperature of 15 K which corresponds well to T Ndetermined by M(T) SQUID measurements for a 50 % Co:ZnO layer [32]. This max- ium of the linewidth in the vicinity of T Nis also ob- served for 60 % Co:ZnO/Py and even 30 % Co:ZnO/Py, as shown in Fig. 7(c). The measured maximum of 30 % Co:ZnO/Py and 60 % Co:ZnO/Py are at 10.7 K, 19.7 K respectively and are marked with an open symbol in Fig. 7(c). For comparison the N eel temperatures de- termined from M(T) measurements [32] are plotted as dashed line. Py on the other hand shows only a slight increase in linewidth with decreasing temperature. The observed eects at low temperatures vanish for the 60 % Co:ZnO/2 nm Al/Py heterostructure. Figure 7(d) shows the temperature dependence of the resonance eld for all samples. For Py Bresonly decreases slightly whereas for 50 % and 60 % Co:ZnO a strong shift ofBrescan be observed. This shift evidences a magnetic coupling between the Py and the Co:ZnO layer. Even in the heterostructure with 30 % Co:ZnO/Py a clear de- crease in resonance position below 10 K (the previously determined T N[32]) is visible. This shift of the resonance position is only observed at low temperatures. At room temperature no shift of the resonance position at 9.5 GHz has been observed as shown in Fig. 5(a). From the low- temperature behavior of the single Py layer and eq. 1 it is obvious that the gyromagnetic ratio is not changing strongly with temperature, therefore shift of the reso- nance position in the heterostructure can be attributed to a change in anisotropy. From the SQUID measure- ments at 2 K, see Fig. 3(b) and Fig. 4(b) EB between the two layers has been determined, which acts as additional anisotropy [20] and therefore causes the shift of the reso- nance position. Both the shift of the resonance position and the maximum in FMR linewidth vanish if the Py is separated from 60 % Co:ZnO by a 2 nm Al spacer layer. So, also at low temperatures the static EB coupling and the dynamic coupling can be suppressed by an Al spacer layer. M(T) measurements indicated a more robust long- range magnetic order in 60 % Co:ZnO by a weak sepa- ration of the eld heated and ZFC curves lasting up to 200 K [32]. Additionally, the coercive eld measurements on the 60 % Co:ZnO/Py hetersotructure revealed a weak coupling above T N. However, this has not been observedfor lower Co concentrations. In the heterostructure with 30 % Co:ZnO the FMR resonance position and linewidth return quickly to the room temperature value for temper- atures above the T Nof 10 K. For both 50 % Co:ZnO/Py and 60 % Co:ZnO/Py the resonance positions are still de- creased and the linewidths are increased above their re- spective N eel temperatures and are only slowly approach- ing the room temperature value. In the 60 % Co:ZnO/Py heterostructure measurements between 100 K and 200 K revealed that a reduced EB is still present. It is known for the blocking temperatures of superparamagnetic struc- tures that in FMR a higher blocking temperature com- pared to SQUID is obtained due to much shorter probing times in FMR of the order of nanoseconds compared to seconds in SQUID [50]. Hence, large dopant congura- tions in Co:ZnO still appear to be blocked blocked on timescales of the FMR whereas they already appear un- blocked on timescales of the SQUID measurements. V. Conclusion The static and dynamic magnetic coupling of Co:ZnO, which is weakly paramagnetic at room temperature and an uncompensated AFM at low temperatures, with ferro- magnetic Py was investigated by means of SQUID mag- netometry and FMR. At room temperature no static in- teraction is observed in the M(H) curves. After cooling to 2 K an EB between the two layers is found resulting in an increase of coercive eld and a horizontal shift. Additionally, a vertical shift is present caused by the un- compensated moments in the Co:ZnO. While this vertical shift is nearly unaected by the insertion of an Al spacer layer between Co:ZnO and Py the EB vanishes already at a spacer thickness of 1 nm. The FMR measurements at room temperature re- veal an increase of the Gilbert damping parameter for 50 % Co:ZnO/Py and 60 % Co:ZnO/Py, whereas 30 % Co:ZnO/Py is in the range of an individual Py lm. At room temperature the resonance position is not aected for all the heterostructures. For the 60 % Co doped sam- ple
= 3.710-3, which is equivalent to an increase by a factor of 1.64. In contrast to the static magnetic coupling eects, an increased linewidth is still observed in the heterostructure containing a 1 nm Al spacer layer. At lower temperatures the resonance position shifts of the heterostructures to lower resonance elds, due to the additional EB anisotropy. The temperature depen- dence of the linewidth shows a maximum at tempera- tures, which by comparison with M(T) measurements correspond well to T Nof single Co:ZnO layers and thus corroborate the increase of the damping parameter and thus spin pumping eciency in vicinity to the magnetic phase transition. Furthermore, the shift of the resonance position has been observed at temperatures well above TNfor 50 % Co:ZnO/Py and 60 % Co:ZnO/Py. Up to now only indications for a long range AFM order in 60 % Co:ZnO/Py had been found by static M(T) measure-8 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41 /s40/s99/s41 /s32/s32/s110/s111/s114/s109/s46/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s66/s32/s40/s109/s84/s41/s32/s84/s32/s61/s32/s32/s32/s52/s46/s48/s75 /s32/s84/s32/s61/s32/s49/s52/s46/s57/s75 /s32/s84/s32/s61/s32/s51/s49/s46/s52/s75 /s32/s84/s32/s61/s32/s53/s48/s46/s50/s75/s40/s97/s41 /s32/s32/s66 /s112/s112/s32/s40/s109/s84/s41 /s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s84 /s78/s32/s100/s101/s116/s101/s114/s109/s105/s110/s101/s100/s32 /s102/s114/s111/s109/s32/s77/s40/s84/s41/s32/s83/s81/s85/s73/s68/s32/s91/s51/s50/s93 /s32/s32/s66 /s114/s101/s115/s32/s40/s109/s84/s41 /s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121 /s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121/s84 /s78/s32/s54/s48/s37/s84 /s78/s32/s53/s48/s37/s84 /s78/s32/s51/s48/s37 /s32/s32/s66 /s112/s112/s32/s40/s109/s84/s41 /s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121 /s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 /s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121 FIG. 7: By decreasing the temperature the resonance position of 50 % Co:ZnO/Py shifts to lower resonance elds (a) and the linewidth increases, showing a maxium at the T N(b). A similar behavior is observed for the heterostructures with 30 % and 60 % Co doping while a single Py lm does not exhibit a maximum when cooling (c). The maximum is marked as open symbol in the temperature dependence, while the T Ndetermined from M(T) [32] are shown as dashed lines. Furthermore, the resonance position of the heterostructures with Co:ZnO shifts at low temperatures (d). ments. The dynamic coupling, however, is sensitive to those interactions due to the higher time resolution in FMR resulting in a shift of the resonance position above the T Ndetermined from M(T) SQUID. Acknowledgment The authors gratefully acknowledge funding by the Austrian Science Fund (FWF) - Project No. P26164-N20 and Project No. ORD49-VO. All the mea- sured raw data can be found in the repository at http://doi.org/10.17616/R3C78N. The x-ray absorption measurements were performed on the EPFL/PSI X- Treme beamline at the Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland. 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1002.3295v1.Measurement_of_Gilbert_damping_parameters_in_nanoscale_CPP_GMR_spin_valves.pdf

Measurement of Gilbert damping paramete rs in nanoscale CPP-GMR spin-valves Neil Smith, Matthew J. Carey, and Jeffrey R. Childress. San Jose Research Center Hitachi Global Storage Technologies San Jose, CA 95120 abstract ⎯ In-situ, device level measurement of thermal mag-noise spectral linewidths in 60nm diameter CPP-GMR spin-valve stacks of IrMn/ref/Cu/free , with reference and free la yer of similar CoFe/CoFeGe alloy, are used to simultaneously determine the intrins ic Gilbert damping for both magnetic layers. It is shown that careful alignment at a "magic-angle" between free and reference layer static equilibrium magnetization can allow direct measurement of the broadband intrinsic thermal spectr a in the virtual absence of spin-torque effects which otherwi se grossly distort the spectral line shapes and require linewidth extrapolations to zer o current (which are nonetheless al so shown to agree well with the direct method). The experimental magic-angle spectra are shown to be in good qualit ative and quantitative agreement with both macros pin calculation s and micromagnetic eigenmode analysis. Despite similar composition and thickness, it is repeatedly found that the IrMn exchange pinn ed reference layer has ten times larger intrinsic Gilbert damping than that of the free-layer ) 1 . 0 ( ≈ α ) 01 . 0 ( ≈α .It is argued that the large reference layer damping results from strong, off -resonant coupling to to lossy modes of an IrMn/ref couple, rather than commonly invoked two-magnon pr ocesses. I. INTRODUCTION Spin-torque phenomena, in tunneling magnetoresistive (TMR) or giant-magnetoresistive (GMR) film stacks lithographically patterned into ~100 nm nanopillars and driven with dc electrical currents perpendicular to the plane (CPP) of the films have in recent years been the topic of numerous theoretical and experimental papers, both for their novel physics as well as potential applications for magnetic memory elements, microwave oscillators, and magnetic field sensors and/or magnetic recording heads. 1 In all cases, the electrical current density at which spin-torque instability or oscillation occurs in the constituent magnetic film layers is closely related to the magnetic damping of these ferromagnetic (FM) films This paper considers the electrical measurement of thermal mag-noise spectra to determine intrinsic damping at the device level in CPP-GMR spin-valve stacks of sub-100nm dimensions (intended for read head applications), which allows simultaneous R-H and transport characterization on the same device. Compared to traditional ferromagnetic resonance (FMR) linewidth measurements at the bulk film level, the device-level approach naturally includes finite-size and spin-pumping 2 effects characteristic of actual devices, as well as provide immunity to inhomogeneous and/or two-magnon linewidth broadening not relevant to nanoscale devices. Complimentary to spin-torque- FMR using ac excitation currents, 3 broadband thermal excitation naturally excites all modes of the system (with larger, more quantitatively modeled signal amplitudes) and allows simultaneous damping measurement in both reference and free FM layers of the spin-valve, which will be shown to lead to some new and unexpected conclusions. However, spin-torques at finite dc currents can substantially alter the absolute linewidth, and so it is necessary to account for or eliminate this effect in order to determine the intrinsic damping. 1 II. PRELIMINARIES AND MAGIC-ANGLES Fig. 1a illustrates the basic film stack structure of a prospective CPP-GMR spin-valve (SV) read sensor, which apart from the Cu spacer between free-layer (FL) and reference layer (RL), is identical in form to well-known, present day TMR sensors. In addition to the unidirectional exchange coupling between the IrMn and the pinned-layer (PL), the usual "synthetic-antiferromagnet" (SAF) structure PL/Ru/RL is meant to increase magnetostatic stability and immunity to field-induced rotation of the PL-RL couple, as well as strongly reduce its net demagnetizing field on the FL which otherwise can rotate in response to signal fields. However, for simplicity in interpreting and modeling the spectral and transport data of Sec. III , the present experiment restricts attention to devices with a single RL directly exchange-coupled to IrMn, as shown in Fig. 1b. The simplest practical model for describing the physics of the device of Fig. 1b is a macrospin model that treats the RL unit magnetization as fixed, with only the FL magnetization RLˆm ) (ˆ ) ( ˆFL t tm m ↔ as possibly dynamic in time. As was described previously,4 the linearized Gilbert equations for small deviations ) , (z ym m′ ′′ ′=′m about equilibrium x m′ ↔ˆ ˆ0 can be expressed in the primed coordinates as a 2D tensor/matrix equation5: mm mH mmm Hhmmh mm ′∂∂⋅∂∂⋅∂′∂−⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛⋅ ≡ ′Δ≡⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛− γ≡⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ γα ≡⋅∂′∂≡′=′⋅′+′⋅ + ˆ ˆ ˆ 1 00 1)ˆ () (,0 11 0,1 00 1) (ˆ) ( ) ( eff 0effFL HmV MppGpDt p t HdtdG D s tt tt tt (1) cap RL IrMnCu seedFL PL IrMn seedcap RuCu RL (a)(b)FL xz FIG. 1. (a) Cartoon of prospective CPP-GMR spin-valve sensor stack, analogous to that used for contemporary TMR read head. (b) Cartoon of simplified spin-valve stack used for present experiments, patterned into ~60nm circular pillars using e- beam lithography. In (1), is a 3D Cartesian tensor, m Hˆ/eff∂ ∂ m m′∂ ∂/ˆ is a 2 3× transformation matrix between 3D unprimed and 2D primed vectors (with its transpose) which depends only on , and is a 3D perturbation field supposed as the origin of the deviations . The magnetic moment m mˆ/∂′∂ 0ˆm ) (th ) (tm′ mΔ is an arbitrary fixed value, but is a natural choice for Sec. II. Using an explicit Slonczewski6 type expression for the spin-torque contribution, the general form for is FL) (V M ms → Δ )ˆ(effm H m J P e HHE m eΔ ≡ ⋅ ≡ θ× θ η −∂∂ Δ−= / ) 2 / ( and , ˆ ˆ cos),ˆ ˆ ( ) (cosˆ1 effeff ST FL RLFL RL ST h m mm mmH (2) for any free energy function . A positive electron current density implies electron flow from the RL to the FL. is the net spin polarization of th e current inside the Cu spacer. Oersted-field contributions to )ˆ(mE eJeffP effH will be neglected here. 2 With in (1), nontrivial solutions require s satisfy 0 ) (=thste t−′=′ m m) ( 0 | ) ( | det= + −G D s Httt . The value when defines the critical onset of spin-torque instability. Using (1), the general criticality condition is expressible as crit e eJ J≡ 0 Re=s 0 ) ( t independen=′−′+′+′ α ∝′ ′ ′ ′ ′ ′ ′ ′4434421 4434421 e e Jy z z y Jz z y y H H H H - (3a) ) cos ( ) ( 2 ) 1 (2 STθ ≡ η −η− ≅′−′′ ′ ′ ′q q qdqdqHH Hy z z y (3b) ) ( 2 / ) 1 () ( ) 2 / (2effcrit q q dq d qH H P emJz z y y eη − η −′+′ α Δ= ⇒′ ′ ′ ′ h (3c) where is the Gilbert damping. The -scaling of the terms in in (3a) follows just from the form of (2). The result in (3b) was derived earlier4 in the present approximation of rigid . αeJ RLˆm With θ the angle between and (at equilibrium), it follows from (3c) that at a "magic-angle" where the denominator vanishes, and spin-torque effects are effectively eliminated from the system at finite . To pursue this point further, explicit results for will be used from the prototypical case where th e CPP-GMR stack (Fig. 1b) is approximately symmetric about th e Cu spacer, which is roughly equivalent to the less restrictiv e situation where the RL and FL are similar materials with thicknesses that are not small compared the spin-diffusion length. For this quasi-symmetric case, both quasi-ballistic6 and fully diffusive7 transport models yield the following simple functional forms: RLˆmFLˆm magicθ ∞ →crit eJ eJ) (cosθ η ) cos 1 () (cos ) (cos) (cos] cos ) 1 ( 1 [ / ) (cos min maxminθ −Γθ η=− ≡ Δ− θ ≡ δ≡ θθ − Γ + + Γ Γ=θ η R R RR R Rr (4) which also relates η to the normalized resistance r 1 0 (≤≤r ) which is directly measurable experimentally. The transport parameter Γ is theoretically related to the Sharvin resistance6,8 or mixing conductance8 at the Cu/FL interface, but will be estimated via measurement in Sec III. Using crit eJ ) (qη from (4) in (3), magicθ and ) (magicθr vs. curves are shown in Fig. 2. Γ The "magic-angle" concept also applies to mag-noise power spectral density (PSD) at bias current , arising from thermal fluctuations in θ θ Δ = S d dr R I SV2 bias bias ] ) / ( [ biasI θabout equilibrium bias angle biasθ . Assuming in/near the film plane (FL RL 0ˆ, - m =≅′z zˆ ˆplane-normal), and requiring | , it can be shown5 from fluctuation-dissipation arguments that | | |crit bias eI I< ] ) ( [ andwhere) ( ) () (4) ()] ( [ ) ( , ) (4) ( 02 2 2 022 2 2 21 z y y z y y z zy z z y z z y yz y z z By yB H H H HH H H HH H mT kf SG D i H DmT kf S ′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ θ− ′ ′ θ ′−′+′+′ α γ = ω Δ′ ′ −′ ′ γ = ωω Δ ω + ω − ωω + ′+′ γ Δα γ≅ ⇒+ ω − ′ = ω χ χ ⋅ ⋅ χΔ γ≅tt t t t tt @ (5) Comparing (5) with (3), it is se en that the spectral linewidth ωΔis predicted to be a linear function of but with ,eJ 0 /→ ωΔedJ d when magic bias θ → θ . Since y y z zH H ′ ′ ′ ′′> > ′ (due to ~10 kOe out-of-plane demag fields) and z y y yH H ′ ′ ′ ′′> > ′ (e.g., for the measurements in Sec. III), it is only in the linewidth crit e eJ J< ωΔ that the off-diagonal terms y z z yH H ′ ′ ′ ′′ ′, can be expected to influence . Therefore, measurement of with ) (f Sθ ) (f SV magic bias θ≅ θ ideally allows direct measurement of the natural thermal-equ ilibrium mag-noise spectrum , from which can be extracted the intrinsic (i.e., -independent) Gilbert damping constant) (f Sθ eJ α. This is the subject of Sec. III. 9095100105110115120 0.20.250.30.350.40.450.5rmagic 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6θmagic (deg) ΓFL0 1112m2 m = +− Γ+ Γ+ c c FIG. 2. Graph of θmagic(blue) and rmagic= rbias(θma g i c) (red) vs. ΓFLas described by (4). The equation for cm= cos( θmagic) follows from (3) and (4). The red solid squares are measured ( ΓFL, rmagic) from Figs. 3,4 and 6. III. EXPERIMENTAL RESULTS The results to be shown below were measured on CPP-GMR- spin-valves of stack structure: seed-layers/IrMn (60A)/RL/Cu (30A)/FL/cap layers. The films were fabricated by magnetron sputtering onto AlTiC substrates at room temperature, with 2mTorr of Ar sputter gas. The bottom contact was a ~1- μm thick NiFe layer, planarized using ch emical-mechanical polishing. To increase ΔR/R, both the RL and FL were made from (CoFe) 70Ge30 magnetic alloys.9 The RL includes a thin CoFe between IrMn and CoFeGe to help maximize the exchange coupling strength, and both RL and FL include very thin CoFe at the Cu interface. The resultant product for the RL and FL were about 0.64 emu/cm 2. After deposition, SV films were annealed for 5hours at 245C in 13kOe applied field to set the exchange pinning direction. The IrMn/RL exchange pinning strength of ≈0.75 erg/cm2 was measured by vibrating sample magnetometry. After annealing, patterned devices with ≈ 60 nm diameter (measured at the FL) were fabricated using e-beam lithography and Ar ion milling. A 0.2 μm-thick Au layer was used as the top contact to devices. t Ms Fig. 3 illustrates a full measurement sequence. Devices are first pre-screened to find samp les with approximate ideal in- plane δR-H loops (Fig. 3a) for circul ar pillars: non-hysteretic, unidirectionally-square loops with parallel with the RL's exchange pinning direction ||H H= ),ˆ(x+ along with symmetric loops about when is transverse The right-shift in the 0=H⊥=H H axis).ˆ(-y||H R-δ loop indicates a large demagnetizing field of ~500 Oe from the RL on the FL. As shown previously,4 narrow-band "low"-frequency measurements (eI N- MHz) 100 ( = ≡ f PSD N , 1MHz bandwidth) can reveal spin-torque criticality as the very rapid onset of excess (1/ f-like) noise when exceeds . loops are measured with sourced from a continuous sawtooth generator (2-Hz) which also triggers 1/2 sec sweeps of an Agilent-E4440 spectrum analyzer (i n zero-span, averaging mode) for ≈50 cycles. With high sweep repeatability and virtually no -hysteresis, this averaging is sufficient so that after (quadratically) subtracting the mean | |eI | |crit eIeI N- eI eI Hz nV/ 1 ) 0 ( ≈ ≈eI N electronics noise, the resultant loops (Fig. 3b) indicate stochastic uncertainty eI N- . Hz nV/ 1 . 0 < < With 1 cos±=θ , it readily follows from (3c) and (4) that crit critcrit crit PAP ) 0 () ( I II I ee ≡ = θ≡ π = θ− = Γ (6) Hence, to estimate Γ, are measured with applied fields eI N- kOe2 . 1 , 45 . 0|| + −≈H (Fig. 3b),which more than sufficient to align antiparallel (AP), or parallel (P) to ,respectively (see Fig. 3a), thereby reducing possible sensitivity to Oersted field and/or thermal effects. (Reducing by ~200-300 Oe did not significantly change either curve.) With denoting electron flow from RL to FL, it is readily found from (3) that and for the FL. By symmetry, it must follow that and for spin-torque induced instability of the RL. This sign convention readily identifies these four critical points by inspection of the data. To account for possible small (thermal) spread in critical onset, specific values for the (excluding ) are defined by where the curves cross the FLˆmRLˆm | |||H eI N- 0>eI 0crit FL AP>-I 0crit FL P<-I 0crit RL AP<-I 0crit RL P>-I eI N- crit eIcrit RL P-I eI N- Hz nV/ 2 . 0 line, which is easily distinguished from the mA / Hz nV/ 05 . 0 ~ residual magnetic/thermal background. is estimated in Fig. 3b (and repeatedly in Figs. 4-7) to be ≈ +4.5 mA. Arbitrariness in the value of from using the crit RL P-I crit eI Hz nV/ 2 . 0 criterion is thought to only be of minor significance for , due to the rounded shape of the AP curves near this particular critical point, which may in part explain why estimated from is found to be systematically somewhat larger than crit RL AP-I eI N- RL/CuΓeI N- Cu/FLΓ . 3 However, the key results here are the 0.1-18 GHz broad-band (rms) spectra (Fig. 3c). They are measured at discrete dc bias currents with the same Miteq preamp (and in- series bias-T) used for the data, the latter being insitu gain-) ; PSD(eI f eI N-H (kOe)-1.5 -1 -0.5 0 0.5 1 1.50246810 (%)δR RRj19.2Ω (a) FIG. 3. Measurement set for 60nm device. (a) δR-H ||(black) and δR-H ⊥ (gray) loops at -5mV bias. (b) P-state N-Ieloops at H| |≈+1.2 kOe (re d), and AP-state N-Ieloops at H ||≈-0.45 kOe ( blue); FL critical currents to deter mine ΓFL(via (6)) enclosed by oval. (c) rms PSD (f, Ie) (normalized to 1 mA) with Ieas indicated by color. Thin black curves are least-squares fits via (7), fitted values for αFL, αRLlisted on top of graph. M easured rbiasand applied field Hlisted inside graph. Field strength and direction (see Fig. 9) adjusted to achieve "magic-angle". ±1.5 mA spectra shown, but not fit.02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5 nV HzPSD frequency (GHz)I = +0.4, -0.4, +0.6, +0.8, -0.8 mA -0.6, H l +750 Oeα = 0.12, 0.13, 0.11, 0.12, 0.10 R L 0.10, α = 0.011, 0.011 , 0.011 , 0.012, 0.010 F L 0.010, rbiasj 0.36normalized to 1 mA -1.5 mA+1.5 mA (c)-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81 I (mA)eH l +1.2 kOe | | nV HzPSD H l -0.45 kOe | | Γ l 4(100 MHz) RL Γ l 3.1FL (b)calibrated vs. frequency (with ≈50Ω preamp input impedance and additionally compensating the present ≈0.7 pF device capacitance) to yield quantitative ly absolute values for these (each averaged over ~100 sweeps, with subtracted post-process) . To confirm the real existence of an effective "magic-angle", the applied field H was carefully adjusted (by repeated trial and error) in both amplitude and direction to eliminate as much as possible any real-time observed dependence of the raw near the FL FMR peak (~ 6 GHz) on the polarity as well as amplitude of over a sufficient range. This procedure was somewhat tedious and delicate, and initial attempts us ing a nominally transverse field were empirically found inferior to additionally adjusting the direction of the field, here rotated somewhat toward the pinning direction for the RL. Using a mechanically-positioned permanent magnet as a field source, this field rotation was only crudely estimated at the time to be ~20-30 o (see also Sec. IV). With both H and bias-point "optimized" as such, an - series of were measured, after which the bias- resistance , and finally and were measured at a common (low) bias of −10 mV to determine (as in (4)). ) ; PSD(eI f ) 0 ; PSD(=eI f ) ; PSD(eI f eI ⊥H biasθeI ) ; PSD(eI f biasRminRmaxR biasr The key feature of the rms in Fig. 3c is that these measured spectra (excluding appear essentially independent of both the polarity and magnitude of (after 1mA-normalization), de fining a "universal" spectrum curve over the entire 18GHz bandwidth, including the unexpectedly wide, low amplitude RL-FMR peak near 14 GHz (more on this below). Because of the relatively large ) ; PSD(eI f mA) 5 . 1 + =eI eI Hz nV/ 1 ~ ) 0 ; PSD( =eI f background, these RL peaks were not well discernible during ra w spectrum measurements, and were practically revealed only after electronics background noise subtraction. As suggested in Fig. 3c, eventual breakdown of the magic-angle condition was genera lly found to first occur from spin-torque instability of the FL at larger positive . eI The spectra Fig. 4 shows the equivalent set of measurements on a physically different (tho ugh nominally identical) 60-nm device. They are found to be remarkably alike in all properties to those of Fig. 3, providing additional confirmation that the "magic-angle" method can work on real nanoscale structures to directly obtain the intrinsic in the absence of of spin-torque effects. This appears further confirmed by the close agreement of measured pairs (from data of Figs. 3,4, and 6) and the macrospin model predictions described in Fig. 2. ) 0 ; (=θ eI f S ) , (Cu/FLΓbr To obtain values for linewidth and then damping ω Δ α from the measured , regions of spectra several-GHz wide, surrounding the FL and RL FMR peaks are each nonlinear least-sqaures fitted to the functional form for) ; PSD(eI f ) 0 ; (=θ eI f S in (5). In particular, the fitting function is taken to be z z z z y yy y z z z z y yz z y y V H H HH H H HH H S f S ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ πω ′ ′ α + γ ω → ′′+′ α γ = ω Δ ′ ′ γ = ωω Δ ω + ω − ωω′ ′ + ω ω = = / ] 2 / ) ( ) / [( and) ( , with, ) ( ) (] ) / ( [ ) ( 2 fit2 peakfit 02 2 2 022 2 02 0 0 2 (7) Three fitting parameters are used: ) 0 (0 = = f S SV , fitα, and peakω , the latter being already well defined by the data itself. The substitution for y yH ′ ′′ is accurate to order , leaving 2α z zH ′ ′′ as yet unknown. With dominated by out- of-plane demagnetizing fields, depends mostly on the product y y z zH H ′ ′ ′ ′′> >′ ) (f SV z zH ′ ′′ αfit . For simplicity, fixed values and were used here, based on macrospin calculations that approximately account for device geometry and net product for FL and RL films. The fitted curves, and the values obtained for and are also included in Figs. 3c and 4c. These values are notably independent of (or show no significant trend with) . kOe 8FL=′′ ′z zH kOe, 10RL=′′ ′z zH t Ms ) ; PSD(eI f FL fitαRL fitα eI 4 Although the repeatedly found from these data is a quite typical magnitude for Gilbert damping in CoFe alloys, the extremely large, 10× greater value of is quite noteworthy, since the RL and FL are not too dissimilar in thickness and composition. Although the small amplitude of the RL-FMR peaks in Figs. 3-4 (everywhere below the raw 01 . 0FL fit≈ α 1 . 0RL fit≈ α Hz nV/ 1 electronics noise), may suggest a basic unreliability in this fitt ed value for , this concern is seemingly dismissed by the data of Fig. 5. Measured on a third (nominally identical) device, an alternative "extrapolation-method" was used, in which RL fitα-1.5 -1 -0.5 0 0.5 1 1.50246810 (%)δR RRj19.0Ω H (kOe)(a) FIG. 4. Analogous measurement set for a different (but nominally identical) 60nm device. as that shown in Fig. 3. 0 2 4 6 8 1 01 21 41 61 80.00.51.01.52.02.5 frequency (GHz)I = +0.5, -0.5, +1.0, -1.5 mA α = 0.12, 0.13, 0.10, 0.12 R L -1.0, 0.12, α = 0.012, 0.011 , 0.013 , 0.013 F L 0.012 , rbiasj 0.39nV HzPSD normalized to 1 mAH l +600 Oe (c)-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81 I (mA)eH l +1.2 kOe | | Γ l 2.7FLΓ l 4RLnV HzPSD H l -0.45 kOe | | (100 MHz) (b) 5 the applied field was purposefully reduced in magnitude (and more transversely aligned than for magic-angle measurements) to increase and thus align to be more antiparallel to . As a result, spin-torque effects at larger negative - will decrease and concomitantly enhance RL-FMR peak amplitude (and visa-versa for the FL), bringing this part of the measured spectrum above the raw el ectronics noise background. biasrFLˆm RLˆmeI ω Δ Using the same fitting function from (7), it is now necessary to extrapolate the to (Fig. 5d) in order to obtain the intrinsic damping. This method works well in the case of the RL since and the extrapolated ) (RL fit eI α 0→eI 0 | | /RL fit< αeI d d 0=eI intercept value of is necessarily larger than the measured , and hence will be (proportionately) less sensitive to uncertainty in the estim ated extrapolation slope. As can be seen from Fig. 5d, the extrapolated values for intrinsic RLα ) (RL fit eI α RLα are virtually identical to those obtain ed from the data of Figs. 3,4. The extrapolated is also quite consistent as well. The extrapolation data also confirm the expectation (noted earlier following (5)) that linewidth will vary linearly with . FLα ω ΔeI Comparing with Figs. 3c,4c, the spectra in Fig. 5c illustrate the profound effect of spin-torque on altering the linewidth and peak-height of both FL and RL FMR peaks even if the system is only moderately misaligned from the magic-angle condition. By contrast, for other frequencies (where the ωΔ term in the denominator of (5) is unimportant), the 1mA-normalized spectra are independent of . Being consistent with (5), this appears to verify that this 2nd form of fluctuation-dissipation theorem remains valid despite that the system of (1) is not in thermal equilibrium 10 at nonzero . (Alternatively stated, spin-torques lead to an asymmetric eI eI Ht , but do not alter the damping tensor Dt in (1)). The α-proportionality in the prefactor of in (5) relatedly shows that the effect of spin-torque on ) (f Sθ ωΔ is not equivalent to additional dampin g (positive or negative) as may be commonly misconstrued. It fu rther indicates that Oersted- field effects, or other -dependent terms in eI Ht not contributing to ωΔ, are insignificant in this experiment. Analogous to Figs. 4,5, the data of Figs. 6,7 are measured on CPP-GMR-SV stacks differing only by an additional 1-nm thick Dy cap layer deposited directly on top of the FL. The use of Dy in this context (presumed spin-pumping from FL to Dy, but possibly including Dy intermixing near the FL/Dy interface 11) was found in previous work12 to result in an ~3 × increase in FL- damping, then inferred from the ~3 × increase in measured . Here, a more direct measure from the FL FMR linewidth indicates a roughly similar, increase in | |crit FLI × ≈3 . 2FLα(now using somewhat thicker FL films). This ratio is closely consistent with that inferred from data measured in this experiment over a population of devices (see Table 1). Notably, the values found for | |crit FLI RLα remain virtually the same as before. Finally, Fig.8 shows results for a "synthetic-ferrimagnet" (SF) free-layer of the form FL1/Ru(8A)/FL2. The Ru spacer provides -1.5 -1 -0.5 0 0.5 1 1.50246810 (%)δR RRj19.5Ω H (kOe)(a) I (mA)e- 5 - 4 - 3 - 2 - 1 01234500.20.40.60.81 Γ l 3.3FLΓ l 3.7RLnV HzPSD (100 MHz) (b)H l -0.45 kOe | |H l +1.2 kOe | | 02468 1 0 1 2 1 4 1 6 1 80.00.51.01.5 frequency (GHz)I = -1.0, -1.5, -2.5, -3.0 mA -2.0, rbiasj 0.53 H l +500 Oe nV HzPSD normalized to 1 mA (c) FIG. 5. Measurement set for a different (but nominally identical) 60nm device as that shown in Figs. 3-4. (c) rms spectra (with least-sqaures fits) measured at larger r biasand θbi as> θmagic. (d) Ie-dependent values of αfi t(Ie) for FL (red) and RL (blue), with suggested Ie→0 extrapolation lines.0.0 0.5 1.0 1.5 2.0 2.5 3.00.000.050.100.15 α fitRL (d) |I | (mA)eFL-1.5 -1 -0.5 0 0.5 1 1.50246810 H (kOe)(%)δR RRj19.9Ω (a) -5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81 I (mA)eH l +1.2 kOe | | H l -0.45 kOe Γ l 3.2FLΓ l 4.2RLnVPSD Hz| | (100 MHz) (b) 02468 1 0 1 2 1 4 1 6 1 80.00.51.01.5 frequency (GHz)I = +0.7, -0.7, +1.0 , +1.4, -1.4 mA α = 0.11, 0.11, 0.11 , 0.11, 0.10 R Lα = 0.027 , 0.026 , 0.027 , 0.027, 0.026 F L -1.0, 0.10, 0.026, H l +700 Oe rnVPSD biasj 0.36 Hz normalized to 1 mA (c) FIG. 6. Analogous measurement set as in Figs.3-4, for (an otherwise identical) device with a 10A Dy cap layer in direct contact with the FL 6 an interfacial antiferromagnetic coupling of . Here, FL1 has a thicker CoFeGe layer than used for prior FL films, and FL2 is a relatively thin CoFe layer chosen so that ≈ 0.64 erg/cm2. Although having similar static M-H or R-H characteristics to that of the simple FL (of similar net product) used in earlier measurements, the transport of the SF-FL in regard to spin- torque effects in particular is fundamentally distinct. The basic physics of this phenomenon was described in detail previously.13 In summary, a spin-torque induced quasi-coresonance between the two natural oscillation modes of the FL1/FL2 couple in the case of negative and , can act to transfer energy out of the mode that is destabilized by spin-torque, thereby delaying the onset of criticality and substantially increasing . Indeed, the side-by-side comparison of loops provided in Fig. 8b indicate a nearly 5 × increase in , despite that remains virtually unchanged. 2erg/cm 0 . 1 ≅ FL 2 1 FL ) ( ) ( ) (FL t M t M t Ms s s ≅ − t Ms eI 0 ˆ ˆRL 1 FL> ⋅ m m | |crit FL P-IeI N- | |crit FL P-I | |crit FL AP-I For the SF-FL devices, attempts at finding the magic-angle under similar measurement conditions as used for Figs. 3c,4c, and 6c were not successful, and so the extrapolation method at similar 4 . 0bias≈ r was used instead. To improve accuracy for extrapolated-FLα , the data of Fig. 8c include measurements for mA 3 . 0 | | ≤eI (so that ) for which electronics noise overwhelms the signal from the RF FMR peaks. Showing excellent linearity of over a wide -range, the extrapolated intrinsiccrit FLI Ie< eI. vsFL fitαeI 01 . 0FL≈ α is, as expected, unchanged from before. The same is true for the extrapolated RLα as well. Table 1 summarizes the mean critical voltages (less sensitive to lithographic variations in actual device area) from a larger set of measurements. The crit FL P-I R− eI- PSD ×≈3 . 2 increase in with the use of the Dy-cap is in good agreement with that of the ratio of measured . | |crit FL P-I R FLα -1.5 -1 -0.5 0 0.5 1 1.50246810 (%)δR RRj19.5Ω H (kOe)(a) - 5 - 4 - 3 - 2 - 1 01234500.20.40.60.81 Γ l 3.3FLΓ l 4.4RL I (mA)enV HzPSD (100 MHz) (b)H l -0.45 kOe | |H l +1.2 kOe | | 0 2 4 6 8 1 01 21 41 61 80.00.51.01.52.0 frequency (GHz)I = -0.7, -1.0, -1.3, -1.8, -2.0mA -1.6 , normalized to 1 mArbiasj 0.66H l +400 OenV HzPSD (c) FIG. 7. Analogous measurement set as in Fig. 5 for a different (but nominally identical) device as that in Fig. 6 with a 10A Dy cap layer..0.0 0.5 1.0 1.5 2.00.000.050.100.15 α fitRL (d) |I | (mA)eFL-1.5 -1 -0.5 0 0.5 1 1.50246810 (%)δR RRj18.2Ω H (kOe)(a) -5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81 I (mA)eH l +1.2 kOe | | H l -0.45 kOe nVPSD Hz| | (100 MHz) (b) 0.01.02.03.04.05.06.0I = +0.15 , -0.15 , +0.3, -0.3, -1., -2., -2.5 mA 0 2 4 6 8 10 14 16 18 12-1.5., rbiasj 0.41H l +600 OenVPSD Hz normalized to 1 mA 0 2 4 6 8 1 01 21 41 61 80.00.20.40.60.81.0 frequency (GHz)I = +0.15 , -0.15 , +0.3 , -0.3, -1., -2., -2.5 mA -1.5., (c) 0.0 0.5 1.0 1.5 2.0 2.50.000.011 FL 0.026 / 0.011 / 0.011 / αFL0.050.100.15 α fit |I | (mA)eRL (d) FIG. 8. Analogous m ea surem ent set as in Figs.5, for (an otherwise identica l) devic e with a synthetic-ferrimagnet FL (SF-FL) as described in text. (b) includes for comparison N-Ieloops (in lighter color) from Fig. 3b ; arrows show SF-FL Icr itfor P-state (red) and AP-state (blue). (c) spectral data and fits are repeatedly shown (for clarity) using two different ordinate scales. 0.12 44.9 !2.0 SF-FL0.11 10.4 !0.1 Control/ αRLstack 0.11 24.5 !0.5 Dy cap 0.026 / 0.011 / 0.011 / αFL 0.11 24.5 !0.5 Dy cap 0.12 44.9 !2.0 SF-FL0.11 10.4 !0.1 Control/ αRLstack (mV)crit FLR I − Table. 1. Summary of critical voltages (measured over ≈ 8 devices each) and damping parameter values α for the present experiment. Estimated statistical uncertainty in the α-values is ~10%. IV. MICROMAGNETIC MODELLING For more quantitative comparison with experiment than afforded by the 1-macrospin model of Sec. II, a 2-macrospin model equally treating both and is now considered here as a simpler, special case of a more general micromagnetic model to be discussed below. The values , , , and will be used as simplified, combined representations (of similar thickness and ) to the actual CoFe/CoFeGe multilayer films used for the RL and FL. The magnetic films are geometrically modeled as 60 nm squares which (in the macrospin approximation) have zero shape anisotropy (like circles), but allow analytical calculation of all magnetostatic interactions. The effect of IrMn exchange pinning on the RL is simply included as a uniform field with measured . Firstly, Fig. 9b shows simulated and curves computed assuming , roughly the mean value found from the data of Sec. III. The agreement with the shape of the measured is very good (e.g., Figs. 6,7 in particular), which reflects how remarkably closely these actual devices resemble idealized (macrospin) behavior. RLˆmFLˆm emu/cc 950FL=sM nm 7FL=t emu/cc 1250RL=sM nm 5RL=t t Ms x H ˆ] ) /( [RL pin pin t M Js =2erg/cm 75 . 0pin≅ J || bias H r-⊥H r-bias 2 . 3= Γ crit FLI H R- Next, Fig. 9d shows simulated PSD curves computed (see Appendix) in the absence of spin-torque (i.e., ) (f SV ) 0ST= H , but otherwise assuming typical experimental values R=19Ω, ΔR/R=9%, and T=300K, as well as and 01 . 0FL= α 1 . 0RL=α , so to be compared with the magic-angle spectra of Figs. 3,4. Since (as stated in Sec. III) th e experimental field angle was not accurately known, the field angle was varied systematically for the simulations, and in each case the field-magnitude H was iterated until Hφ 37 . 0bias≅ r , approximately matching the mean measured value. In terms of both absolute values and the ratio of FL to RL FMR peak amplitudes, the location of (particularly for the FL), and the magnitude of H (on average 650-700 Oe from the three magic-angle data in Sec. III), the best match with experiment clearly occurs with . The agreement, both qualitatively and quantitatively, is again remarkable given the simplicit y of the 2-macrospin model. peakf o o40 30 ≤ φ ≤H Finally, results from a di scretized micromagnetic model are shown in Fig. 10. Based on Fig.9, the value was fixed, and H = 685 Oe was determined by iteration until o35= θH 37 . 0bias≅ r . The equilibrium bias-point magneti zation distribution is shown 60 nm RL FL(a) 60 nm RL FL(a) FIG. 10. Micromagnetic model results. (a) cell discretizations with arrow- heads showing magnetization orie nta tion when | H|=685 Oe and φH=35o (see Fig. 9c). (b) simulated partial rms PSD for first 7 eigenmodes (as labeled) computed individually with αFL=0.01 andαRL=0. 01, other parameter values indicated. ( c) simulated total rms PSD with αFL=0.01 and αRL=0.01 (green) or αRL=0.1(red or blue); blue curve excludes contribution from 5th(FL) eigenmode at 16 GHz. 7 02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5 frequency (GHz)nV HzPSD rbiasj 0.37 Γ =3.2 α =0.01FLH =0STφ =35HoH=685 Oe exclude #5 include # 1-7 α =0.01RL (c)α =0.1RL02468 1 0 1 2 1 4 1 6 1 8 2 00.00.51.01.52.02.5 frequency (GHz)nV HzPSDI=1mA R=19 ΩΔR/R=9% Γ =3.2 α =0.01 RLT=300K H =0ST(#1) (#2) (#3)(#4) (#5) (#6) (#7)rbiasj 0.37 φ =35HoH=685 Oe "FL"-mode "RL"-mode α =0.01 FL (b)-1.5 -1 -0.5 0 0.5 1 1.500.20.40.60.81 H (kOe)δR ΔRΓ = 3.2 (b)5 nm3nm7 nm 60 nm60 nm RLFL (c)(a) H x zymRL mFLφH 02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5 frequency (GHz)ΔR/R=9% R=19 Ω I=1mA φ =10 , H=802 Oeo Hφ =0 , H=895 Oeo H φ =20 , H=723 Oeo H φ =30 , H=657 Oeo H φ =40 , H=605 Oeo Hα =0.10RLT=300K r H =0ST j 0.37bias nVPSD α =0.01Γ =3.2Hz FL (d) FIG. 9. Two-macrospin model results. (a) cartoon of model geometry. (b) simulated δR-H loops analogous to data of Figs 3-8c. (c ) cartoon defining vector orientations (RL exchange pinned along + x direction). (d) simulated rms PSD assuming parameter values indicated, with variable | H| to maintain a fixed rbiasat each φH( as indicated by color). in Fig. 10a for this 416 cell model. Estimated values for exchange stiffness, and erg/cm 4 . 1FL μ = A erg/cm 2RL μ=A were assumed. The simulated spectra in Fig. 10b are shown one eigenmode at a time (see Appendix), for the 7 eigenmodes with predicted FMR frequencies below 20 GHz (the 8th mode is at 22.9 GHz). The 1st, 2nd, 5th, and 7th modes involve mostly FL motion, the nearly degenerate 3rd and 4th modes (and the 6th) mostly that of the RL. (The amplitudes of from the 6th or 7th mode are negligible.). For illustration purposes only, Fig. 10b assumed identical damping in each film. ) (f SV 01 . 0FL FL= α = α For Fig. 10c, the computation of is more properly computed using either 6 or all 7 eigenmodes simultaneously, which includes damping-induced coupling between the modes. Including higher order modes makes negligible change to (but rapidly increases computation time). As was observed earlier, the agreement between simulated and measured spectra in Figs. 3c,4c is good (with ) (f SV GHz) 20 ( <f SV 1 . 0RL=α ), and the simulations now include the small, secondary FL-peak near 8 GHz clearly seen in the measured data (including that of Fig. 6c), though it is somewhat more pronounced in the model results. Notably, the computed spectrum near the RL FMR peak more resembles the measurements after removing the 16-GHz 5 th mode from the calculation, as this (FL) mode does not appear to be physically present in the Figs. 3c,4c spectra. While it is perhaps expected that higher order modes in a micromagnetic simulation assuming perfectly homogeneous magnetic films would show deviations from real devices with finite grain-size, edge-roughness/damage, etc., the situation is actually more interesting. Fig. 11 shows measured spectra on yet another device (again, nominally identical to that of Figs. 3-5) in which the experiment was perhaps slightly off from the optimum magic-angle condition, as evidenced by the very small shift in the 6-GHz FL FMR peak position with polarity of . More noteworthy, however, is the clear polarity asymmetry and nonlinear-in- peak-amplitude (for in particular) of both the 8-GHz secondary FL mode and a higher order mode close to 15 GHz. (Both resemble typical spin-torque effect at angles more antiparallel than the magic-angle condition.) This similarity in behavior indicates with near certainty that this 15- GHz mode is also FL-like in origin, and is thus a demonstration of the "missing" 5 th mode predicted in Fig. 10. (In hindsight, there is now discernible a small but similar 15-GHz peak in the spectra of Fig. 4c). It is worth remembering that the "magic- angle" argument was based on a simple 1-macrospin model, and so remarkably there appears to be circumstances where this "spin-torque null" actually does apply simultaneously to both the FL and RL, as well as to higher order modes. eI eI 0>eI V. DISCUSSION In addition to the direct evidence from the measured spectral linewidth in Figs. 3-8, evidence for large Gilbert damping FL RL α> > α for the RL is also seen in the data. As ratios and are (from Figs. 3-5 data) both roughly ~7, this conclusion is semi-quantitatively consistent with the basic scaling (from (3c)) that . This, as well as the substantial, 2-3 × variation of with in Figs. 5d, and 7d, appears to rather conclusively (and expectedly) confirm that inhomogeneous broadening is not a factor in the large linewidth- inferred values of crit eI crit crit FL P RL P/- -I Icrit crit FL AP RL AP /- - I I α ∝crit eI RL fitαeI RLα found in these nanoscale spin-valves. Large increases in effective damping of "bulk" samples of ferromagnetic (FM) films in cont act with antiferromagnet (AF) exchange pinning layers has been reported previously.14-16 The excess damping was generally attributed to two-magnon scattering processes 17 arising from an inhomogeneous AF/FM interface. However, the two-magnon description applies to the case where the uniform, ( , mode is pumped by a external rf source to a high excitation ( magnon) level, which then transfers energy via two-magnon scattering into a large (quasi-continuum) number of degenerate 0=k )0ω ≡ ω ) , 0 (0ω=ω≠k k spin-wave modes, all with low (thermal) excitation levels and mutually coupled by the same two-magnon process. In this circumstance, the probability of en ergy transfer back to the uniform mode (just one among the degenerate continuum) is negligible, and the resultant one-way flow of energy out of the uniform mode resembles that of intrinsic damping to the lattice. By contrast, for the nanoscale spin-valve device, the relevant eigenmodes (Fig. 10) are discrete and generally nondegenerate. in frequency. Even for a coincide ntal case of a quasi-degenerate pair of modes (e.g., RL modes #3 and #4 in Fig. 10), both modes are equally excited to thermal equilibrium levels (as are all modes), and have similar intrinsi c damping rates to the lattice. Any additional energy transfer via a two-magnon process should flow both ways, making impossible a large (e.g., ~10× ) increase in the effective net damping of either mode. 0.00.51.01.52.0 I = +0.7, -0.7 , +1.0 , +1.4, -1.4 mA 8 Two alternative hypotheses for large RLα which are essentially independent of device size are 1) large spin-pumping effect at the IrMn/RL interface, or 2) strong interfacial exchange coupling at the IrMn/RL resulting in non-resonant coupling to high frequency modes in either the RL and/or or the IrMn film. However, these two alternatives can be distinguished since the exchange coupling strength can be greatly altered without necessarily changing the spin-pumping effect. In particular, RLα was very recently measured by conventional FMR methods -1.0, nVPSD Hz 0 2 4 6 8 1 01 21 41 61 8 frequency (GHz)r(normalized to 1 mA) j 0.35biasH l +700 Oe FIG. 11. The rms PSD measured on a physically diffe rent (but nominally identical) device as that generating the analogous "magic-angle" spectra shown in Figs. 3c and 4c. Table. 2. Summary of bulk film FMR measurements18 for reduced film stack structure: seed/IrMn( tAF)/Cu( tCu)/RL/Cu(30A)/cap. Removal of IrMn, or alternatively a lack of proper seed layer and/or use of a sufficiently thick tCu≈30A can each effectively eliminate exchange pinning strength to RL. 0.013 tAF= tCu= 0 (out-of-plane FMR)0.013 tAF=60 A , tCu= 30A0.010 tAF=6 0 A , tCu= 0 (no seed layer for IrMn)0.011 tAF= tCu= 0sample type 0.013 tAF= tCu= 0 (out-of-plane FMR)0.013 tAF=60 A , tCu= 30A0.010 tAF=6 0 A , tCu= 0 (no seed layer for IrMn)0.011 tAF= tCu= 0sample type ) / (23 RL ω Δ γ = α d H d by Mewes18 on bulk film samples (grown by us with the same RL films and IrMn annealing procedure as that of the CPP- GMR-SV devices reported herein) of the reduced stack structure: seed/IrMn( tAF)/Cu( tCu)/RL/Cu(30A)/cap. For all four cases described in Table 2, the exchange coupling was deliberately reduced to zero, and the measured was found to be nearly identical to that found here for the FL of similar CoFeGe composition. However, for the two cases with tAF = 60A, excess damping due to spin-pumping of electrons from RL into IrMn should not have been diminished (e.g., the spin diffusion length in Cu is ~100 × greater than tCu ≈ 30A). This would appear to rule out the spin-pumping hypothesis. 012 . 0RL≈ α The second hypothesis emphasizes the possibility that the energy loss takes place inside the IrMn, from oscillations excited far off resonance by locally strong interfacial exchange coupling to a fluctuating . This local interfacial exchange coupling can be much greater than , since the latter reflects a surface average over inhomogeneous spin-alignment (grain-to- grain and/or from atomic roughness) within the IrMn sub-lattice that couples to the RL. Further, though such strong but inhomogeneities coupling cannot truly be represented by a uniform acting on the RL, the similarity between measured and modeled values of ~14 GHz for the "uniform" RL eigenmode has clearly been demonstrated here. Whatever are the natural eigenmodes of the real device, the magic-angle spectrum measurements of Sec. III reflect the thermal excitation of all eigenmodes for which "one-way" intermodal energy transfer should be precluded by the condition of thermal equilibrium and the orthogonality 19 of the modes themselves. Hence, without an additional energy sink exclusive of the RL/FL spin-lattice system, the linewidth of all modes should arguably reflect the intrinsic Gilbert damping of the FL or RL films, which the data of Sec. III and Table 2 indicate are roughly equal with . Inclusion of IrMn as a combined AF/RL system, would potentially provide that extra energy loss channel for the RL modes. RLˆm exJpinJ pinH 01 . 0 ~ α 9 A rough plausibility argument for the latter may be made with a crude AF/FM model in which a 2-sublattice AF film is treated as two ferromagnetic layers (#1 and #2) occupying the same physical location. Excluding magnetostatic contributions, the free energy/area for this 3-macrospin system is taken to be x m m mx m x m m m ˆ ˆ ] [ ˆ ˆ] )ˆ ˆ ( ) ˆ ˆ [( ) ( ˆ ˆ ) ( FM FMAF AF AF pin 0 2 ex2 22 1 212 1 ⋅ − + ⋅ −⋅ + ⋅ − ⋅ J J Jt K H t Ms (8) For IrMn with Neel temperature of , the internal AF exchange field .20 With K T700N≈ Oe 10 ~ / ~7 B B AFμNT k H A, 60AF=t AF uniaxial anisotropy is estimated to be .21 A rough estimate for strong interfacial exchange is obtained by equating interface energy erg/cc 10 ~6 AFK FM) / ( 8 ~ex t A J 2 /2 exφJ to the bulk exchange energy t A/ 42φ of a hypothetical, small angle Bloch wall ) 2 0 (φ ≤ φ ≤ twisting through the FM film thickness. Taking nm 5≈t and A ~ 10-6 erg/cm yields . The value of in the last "field-like" term in (8) is more precisely chosen to maintain a constant eigenfrequency for the FM layer independent of or , thus accounting for the weaker inhomogeneous coupling averaged over an actual AF/FM interface. 2 ex erg/cm 15 ~ J 1 ex 0 ] ) ( / 1 / 1 [ ~AF−+ t K J J exJAFK As shown in Fig. 12, this crude model can explain a ~10 × increase in the FM linewidth provided á 5- and exJ2erg/cm 10 1 . 0 05 . 0 ~AF - α . It is worthily noted20 that for the 2-sublattice AF, the linewidth ) ) / ( /( 2 /AF AF AF 0 sM K H HK≡ α ≈ ω ω Δ is larger by a factor of 100 ~ / 2AF KH H compared to high order FM spin-wave modes in cases of comparable α and 0ω (with Hz 10 ~ 212 0 AF KH H γ ≈ ω for the AF). Since the lossy part of the "low" frequency susceptibility for FM or AF modes scales with ωΔ, it is suggested that the IrMn layer can effectively sink energy from the ~14 GHz RL mode despite the ~100 × disparity in their respective resonant frequencies. S ize-independent damping mechanisms for FM films exchange-coupled to AF layers such as IrMn are worthy of further, detailed study. 0 2 4 6 8 10 12 14 16 18 200.000.010.020.030.040.05 frequency (GHz)T=300K (GHz)-1/2α =0.01 FMSθFMJ =0α = 0.02, 0.05, 0.10 0.01, exAF erg cm2 J =10ex 0 2 4 6 8 10 12 14 16 18 200.000.010.020.030.040.05 frequency (GHz)T=300K α =0.01 FMJ =0exerg 1, 3, 10, 100cm2 J =ex S α =0.10 θFM AF (GHz)-1/2(a) (b) FIG. 12. Simulated rms PSD SθFM(f) for a 3-macrospin model of an AF/FM couple as described via (8) and in the text. The FM film parametrics are the same as used for macrospin RL model in Fig. 9, with αFM=0.01 and Jpin= 0.75 erg/cm2. (a) varied α AF(denoted by color) with Jex=10erg/cm2. (b) varied Jex(denoted by color) with αAF=0.1. The black curve in (a) or in (b) corresponds to Jex=0. For AF, Msis taken to be 500 emu/cc. ACKNOWLEDGMENTS The authors wish to acknowledge Jordan Katine for the e-beam lithography used to make all the measured devices, and Stefan Maat for film growth of alternative CPP spin-valve stacks useful for measurements not included here. The authors wish to thank Tim Mewes (and his student Zachary Burell) for making the bulk film FMR measurements on rather short notice. One author (NS) would like to thank Thomas Schrefl for a useful suggestion for micromagnetic modeling of an AF film. APPENDIX As was described in detail elsewhere,22 the generalization of (1) or (5) from a single macrospin to that for an N-cell micromagnetic model takes the form 1)] ( [ ) ( ,2) () ( ) ( −+ ω − ′ = ω ⋅ ⋅Δ γ≡ ω′=′⋅′+′⋅ + G D H D Sh m HmG D tt t t ttt trrtrtt imT ktdtd Bχ χ χ@ (A1) where m′r) (orh′r is an column vector built from the N 2D vectors , and 1 2×N N j... 1=′m H G Dttt and , , are matrices formed from the array of 2D tensors N N2 2× N N× ,jkDt ,jkGt and Here, and , though .jkHt jk jk D Dδ =t t jk jkG Gδ =t t .jkHt is nonlocal in cell indices j,k due to the magnetost atic interaction. The PSD for any scalar quantity is22 ) (f SQ })ˆ({j Qm j jj jN k jk jk j QQS f Sm mm d d dˆ ˆ, ) ( 2 ) ( 1 , ∂∂⋅∂′∂ ≡′ ′ ⋅ ω ⋅ ′ =∑ =t (A2) The computations for the PSD of Figs.9, 10 took ) (mrQ to be ∑ = ⋅ − Γ + + Γ⋅ − Δ=iN i i ii i iR I NQ 1bias FL RLFL RL ˆ ˆ ) 1 ( 1)ˆ ˆ 1 ( 1 m mm m averaged over the cell pairs at the RL-FL interface. 2 /N Ni= For a symmetric Ht (e.g., the set of eigenvectors ), 0ST= H m err← of the system (A1) can be defined from the following eigenvalue matrix equation n n N n ie e H Gr rtt ω = ⋅ ⋅=− 2 ... 11) ( (A3) The eigenvectors come in N complex conjugate pairs − +e err, with real eigenfrequencies . With suitably normalized ω ± ,ner matrices and are diagonal in the eigenmode basis .22. The analogue to (A1) becomes mn mn H δ =n mn mni G ω δ =/ ∑∑ ′⋅ ≡ ′ ′ ω ′ =ω χ ω χΔ γ≡ ω⋅ ⋅ ≡ ω − δ ω ω − = ω χ ∗ ∗∗ ′ ′ ′′ ′ ′∗ − n mn n n mn m Qn n n mn m m mB mnn m mn mn n mn d d S d f SDmT kSD i ,,1 , ) ( 2 ) () ( ) (2) () ( ) / 1 ( )] ( [ d ee D e rrrtr (A4) The utility of eigenmodes for computing PSD, e.g, in the computations of Fig. 10, is that only a small fraction (e.g., 7 rather than 416 eigenvector pairs) need be kept in (A4) (with all the rest simply ignored ) in order to obtain accurate results in practical frequency ranges (e.g., GHz). Despite that is (in principle) a full matrix, the reduction in matrix size for the matrix inversion to obtain at each frequency more than makes up for the cost of computing the 20<mnD ) (ω χ ) , (n nerω which need be done only once independent of frequency or α-values. REFERENCES 1 Volume 320 (2008) of the Journal of Magnetism and Magnetic Materials offers a series of review articles (with many additional references therein) covering this field. Some examples include: J. A. Katine and E. E. Fullerton, pp. 1217-1226, J. Z. Sun and D. C. Ralph, pp. 1227-1237, T. J. Silva and W. H. Rippard, pp. 1260-1271. 2Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 67, 140404(R) (2003). 3G. D. Fuchs et al. , Appl. Phys. Lett. 91, 062505 (2007). 4 N. Smith, J. A. Katine, J. R. Childress, and M. J. Carey, IEEE Trans. Magn. 41, 2935 (2005). 5 As discussed in N. Smith, Phys. Rev. B 80, 064412 (2009), the spin- pumping effect described in Ref. 2 l eads to additional contributions to the damping tensor-matrix Dt in (1) or (A1) which are anisotropic, angle-dependent, and nonlocal betw een RL and FL. For simplicity, these are here simply lumped into the Gilbert damping parameter α for either RL or FL. In this c ontext, "intrinsic " damping refers approximately to that in the linit , but also in the presence of the complete stack structure of the spin-valve device of interest. 0→eI 6 J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324 (2002). 7 N. Smith, J. Appl. Phys. 99, 08Q703 (2006). 8 Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 9 S. Maat, M. J. Carey, and J. R. Childress, Appl. Phys. Lett. 93, 143505 (2008). 10 R. A. Duine, A. S. Nunez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 (2007). 11 G. Woltersdorf, M. Kiessling, G. Meye r, J. U. Thiele, and C. H. Back, Phys. Rev. Lett. 102, 257602 (2009). 12 S. Maat, N. Smith, M. J. Carey, and J. R. Childress, Appl. Phys. Lett. 93 , 103506 (2008). 13 N. Smith, S. Maat, M. J. Carey, and J. R. Childress, Phys. Rev. Lett. 101 , 247205, (2008). 14 R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, Jr., J. Appl. Phys., 83, 7037 (1998). 15 S. M. Rezende, A. Azevedo, M. A. Lucena, and F. M. de Aguiar, Phys. Rev. B 63, 214418 (2001). 16 M. C. Weber, H. Nembach, B. Hillebrands, M. J. Carey, and J. Fassbender, J. Appl. Phys. 99, 08J308 (2006). 17 R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999). 10 18 T. Mewes, MINT Center, U. Alabama (Tuscaloosa). These bulk film FMR measurements were made in-plane, except for the last entry in Table 2 which was measured in th e out-of-plane configuration which should exclude two-magnon contributions (Ref. 17). The similarity in α -values suggests two-magnon is unim portant here, which in turn then suggests that RL-FL spin-pumping contributions in the present CPP-GMR nanopillars spin-valves (footnote (5)) that would not be present in the RL-only bulk samples are also small. Further comparisons of present measuremen ts with additional bulk film FMR results are expected to be addressed in a future publication. 19 A nonzero damping matrix Dt can technically cause a coupling of nondegenerate eigenmodes if , though this does not change the argument citing this f ootnote. This small effect can be seen in Fig. 10c as the slight change in the 6-GHz FL FMR peak height and linewidth when 0≠ ⋅ ⋅∗≠ n n me D ertr RLαis grossly varied from 0.01 to 0.1. 20 Magnetic Oscillations and Waves , A. G. Gurevich and G. A. Melkov, CRC Press (Florida, USA) 1996. Chap. 3. 21 K. O'Grady, L. E. Ferndandez- Outon, G. Vallejo-Fernandez, J. Magn. Magn. Mater. 322, 883 (2010). 22 Appendix A of N. Smith, J. Magn. Magn. Mater. 321, 531 (2009). 11

2106.11597v1.Choice_of_Damping_Coefficient_in_Langevin_Dynamics.pdf

Choice of Damping Coefficient in Langevin Dynamics Robert D. SkeelandCarsten Hartmanny Abstract. This article considers the application of Langevin dynamics to sampling and investigates how to choose the damping parameter in Langevin dynamics for the purpose of maximizing thoroughness of sampling. Also, it considers the computation of measures of sampling thoroughness. 1. Introduction. Langevindynamicsisapopulartoolformolecularsimulation. Itrequires the choice of a damping coefficient, which is the reciprocal of a diffusion coefficient. (More generally this might be a diffusion tensor.) The special case of a constant scalar diffusion coefficient is the topic of this article. The motivation for this study is a suspicion that proposed novel MCMC propagators based on Langevin dynamics (in particular, stochastic gradient methods for machine learning [4, 9]) might be obtaining their advantage at the expense of reduced sampling efficiency, as, say, measured by effective sample size. For simulations intended to model the dynamics, the appropriate choice of is based on physics. Generally, the dissipation and fluctuation terms are there to account for omitted degrees of freedom. In their common usage as thermostats, they model the effect of forces due to atoms just outside the set of explicitly represented atoms. These are essentially boundary effects, which disappear in the thermodynamic limit Natoms!1, whereNatomsis the number of explicitly represented atoms. Since the ratio of the number of boundary atoms to interior atoms is of order N1=3 atoms, it might be expected that is chosen to be proportional to N1=3 atoms. There is second possible role for the addition of fluctuation-dissipation terms in a dynamics simulation: with a small damping coefficient, these terms can also play a role in stabilizing a numerical integrator [21], which might be justified if the added terms are small enough to have an effect no greater than that of the discretization error. The bulk of molecular simulations, however, are “simply” for the purpose of drawing ran- dom samples from a prescribed distribution and this is the application under consideration here. The appropriate choice of optimizes the efficiency of sampling. A measure of this is the effective sample size N=whereNis the number of samples and is the integrated autocorrelation time. The latter is, however, defined in terms of an observable. An observable is an expectation of a specified function of the configuration, which for lack of a better term, is referred to here as a preobservable . As an added complication, the accuracy of an estimate of an integrated autocorrelation time (IAcT) depends on sampling thoroughness [13, Sec. 3], so a conservative approach is indicated. Ref. [13, Sec. 3.1] advocates the use of the maximum possible IAcT and shows how it might be a surrogate for sampling thoroughness. The max- imum possible IAcT is about the same (except for a factor of 2) as the decorrelation time of Ref. [30], defined to be “the minimum time that must elapse between configurations for them to become fully decorrelated (i.e., with respect to any quantity)”. School of Mathematics and Statistical Sciences, Arizona State University, 900 S Palm Walk, Tempe, AZ 85281, USA, E-mail: rskeel@asu.edu yInstitute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046 Cottbus, Ger- many, E-mail: hartmanc@b-tu.de 1arXiv:2106.11597v1 [stat.CO] 22 Jun 2021Therefore, for sampling, it is suggested that be chosen to achieve a high level of sampling thoroughness, as measured by the maximum possible IAcT. An initial study of this question is reported in Ref. [38, Sec. 5], and the purpose of the present article is to clarify and extend these results. To begin with, we analyse an underdamped Langevin equation with a quadratic potential energy function. (See Eq. (12) below.) The main purpose of analyzing this model problem is, of course, to obtain insight and heuristics that can be applied to general potential energy functions. Needed for choosing the optimal gamma is a substitute for the lowest frequency. For the model problem, this can be obtained from the covariance matrix for the position coordinates, which is not difficult to compute for a general potentials. And for estimating q;max, the analysis suggests using the set of all quadratic polynomials, which can be achieved using the algorithm of reference [13, Sec. 3.5]. For molecular simulation, the suggestion is that one might choose linear combinations of functions of the form j~ rj~ rij2and(~ rj~ ri)
(~ rk~ ri)where each ~ riis an atomic position or center of mass of a group of atoms. Such functions share with the potential energy function the property of being invariant under a rigid body movement. 1.1. Results and discussion. Section 5 analyzes integrated autocorrelation times for the standard model problem of a quadratic potential energy function. An expression is derived for the IAcT for any preobservable; this is applied in Sec. 5.2 to check the accuracy of a method for estimating the IAcT. In Sec. 5, we also determine the maximum IAcT, denoted by q;max, over all preobservables defined on configurations, as well as the damping coefficient that minimizesq;max. It is shown that it is polynomials of degree 2that produce the largest value ofq;max. And that choosing equal to the lowest frequency, which is half of the optimal value of for that frequency, minimizes q;max. These results extend those of Ref. [38, Sec. 5], which obtains a (less relevant) result for preobservables defined on phase space rather than configuration space. Sections 6 and 7 test the heuristics derived from the quadratic potential energy on some simple potential energy functions giving rise to multimodal distributions. Results suggest that the heuristics for choosing the maximizing preobservable and optimal gamma are effective. One of the test problems is one constructed by Ref. [23] to demonstrate the superiority of BAOAB over other Langevin integrators. Experiments for this problem in Sec. 6 are consistent with this claim of superiority. In defining “quasi-reliability” and the notion of thorough sampling, Ref. [13] makes an unmotivated leap from maximizing over preobservables that are indicator functions to maxi- mizing over arbitrary preobservables. The test problem of Sec. 7 provides a cursory look at this question, though the matter may warrant further study. Obtaining reliable estimates of the IAcT without generating huge sets of samples very much hinders this investigation. To this end, Sec. 4.1 explores an intriguing way of calculating an estimate for the phase space max, which avoids the difficult calculation of IAcTs. For the model problem, it give more accurate results for maxthan estimating IAcTs, due to the difficulty of finding a set of functions that play the same role as quadratic polynomials when maximizing IAcTs. The literature offers interesting suggestions that might help in the development of better schemes for estimating IAcTs, and it may be fruitful to recast some of 2these ideas using the formalisms employed in this article. In particular, Ref. [30] offers a novel approach based on determining whether using every th sample creates a set of independent samples. Additionally, there are several conditions on covariances [16, Theorem 3.1] that can be checked or enforced. 1.2. Related work. While the major part of the literature on Markov chain Monte Carlo (MCMC) methods with stochastic differential equations focuses on the overdamped Langevin equation (e.g. [35, 3] and the references given there), there have been significant advances, both from an algorithmic and a theoretical point of view, in understanding the underdamped Langevindynamics[34]. Forexample,inRefs. [39,7]Langevindynamicshasbeenstudiedfrom theperspectiveofthermostattingandenhancmentofspecificvibrationalmodesorcorrelations, in Refs. [8, 17, 25] Langevin dynamics has been used to tackle problems in machine learning and stochastic optimisation. From a theoretical point of view, the Langevin equation is more difficult to analyse than its overdamped counterpart, since the noise term is degenerate and the associated propagator is non-symmetric; recent work on optimising the friction coefficient for sampling is due to [11, 36, 4], theoretical analyses using both probabilistic and functional analytical methods have been conducted in [10, 5, 12]; see also [27, Secs. 2.3–2.4] and the references therein. Relevant in this regard are Refs. [20, 26, 33], in which non-reversible perturbations of the overdamped Langevin equation are proposed, with the aim of increasing the spectral gap of the propagator or reducing the asymptotic variance of the sampler. Related results on decorrelation times for the overdamped Langevin using properties of the dominant spectrum of the infinitesimal generator of the associated Markov process have been proved in [22, Sec. 4]. A key point of this article is that quantities like spectral gaps or asymptotic variances are not easily accessible numerically, therefore computing goal-oriented autocorrelation times (i.e. for specific observables that are of interest) that can be computed from simulation data is a sensible approach. With that being said, it would be a serious omission not to mention the work of Ref. [30], which proposes the use of indicator functions for subsets of configuration space in order to estimate asymptotic variance and effective sample size from autocorrelation times using trajectory data. Finally, we should also mention that many stochastic optimisation methods that are nowa- days popular in the machine learning comminity, like ADAM or RMSProp, adaptively control the damping coefficient, though in an ad-hocway, so as to improve the convergence to a local minimum. They share many features with adaptive versions of Langevin thermostats that are used in moecular dynamics [24], and therefore it comes as no surprise that the Langevin model is the basis for the stochastic modified equation approach that can be used to analyse state of the art momentum-based stochastic optimisation algorithms like ADAM [1, 28]. 2. Preliminaries. The computational task is to sample from a probability density q(q) proportional to exp(V(q)), whereV(q)is a potential energy function and is inverse temperature. In principle, these samples are used to compute an observable E[u(Q)], where Qis a random variable from the prescribed distribution and u(q)is a preobservable (possible 3an indicator function). The standard estimate is E[u(Q)]bUN=1 NN1X n=0u(Qn); where the samples Qnare from a Markov chain, for which q(q)(or a close approximation thereof) is the stationary density. Assume the chain has been equilibrated, meaning that Q0is drawn from a distribution with density q(q). An efficient and popular way to generate such a Markov chain is based on Langevin dynamics, whose equations are (1)dQt=M1Ptdt; dPt=F(Qt) dt Ptdt+q 2 MhdWt; whereF(q) =rV(q),Mis a matrix chosen to compress the range of vibrational frequencies, MhMT h=M, and Wtis a vector of independent standard Wiener processes. The invariant phase space probability density (q;p)is given by (q;p) =1 Zexp((V(q) +1 2pTM1p)); whereZ > 0is a normalisation constant that guarantees that integrates to 1. We call q(q) its marginal density for q. We suppose >0. It is common practice in molecular dynamics to use a numerical integrator, which intro- duces a modest bias, that depends on the step size
t. As an illustration, consider the BAOAB integrator [23]. Each step of the integrator consists of the following substeps: B:Pn+1=4=Pn+1 2
tF(Qn), A:Qn+1=2=Qn+1 2
tM1Pn+1=4, O:Pn+3=4= exp(
t)Pn+1=4+Rn+1=2, A:Qn+1=Qn+1=2+1 2
tM1Pn+3=4, B:Pn+1=Pn+3=4+1 2
tF(Qn+1=2), where Rn+1=2is a vector of independent Gaussian random variables with mean 0and covari- ance matrix (1exp(2
t))1M. In the following, we use the shorthand Z= (Q;P)to denote a phase space vector. It is known [16, Sec. 2] that the variance of the estimate bUNforE[u(Z)]is (2) Var[bUN] NVar[u(Z)]; which is exact relative to 1=Nin the limit N!1. Hereis theintegrated autocorrelation time (IAcT) (3) = 1 + 2+1X k=1C(k) C(0) andC(k)is the autocovariance at lag kdefined by (4) C(k) =E[(u(Z0))(u(Zk))] 4with=E[u(Z0)] =E[u(Zk). Here and in what follows the expectation E[
]is understood over all realisations of the (discretized) Langevin dynamics, with initial conditions Z0drawn from the equilibrium probability density function . 2.1. Estimating integrated autocorrelation time. Estimates of the IAcT based on es- timating covariances C(k)suffer from inaccuracy in estimates of C(k)due to a decreasing number of samples as kincreases. To get reliable estimates, it is necessary to underweight or omit estimates of C(k)for larger values of k. Many ways to do this have been proposed. Most attractive are those [16, Sec. 3.3] that take advantage of the fact that the time series is a Markov chain. Onethatisusedinthisstudyisashortcomputerprogramcalled acor[18]thatimplements a method described in Ref. [31]. It recursively reduces the series to one half its length by summing successive pairs of terms until the estimate of based on the reduced series is deemed reliable. The definition of “reliable” depends on heuristically chosen parameters. A greater number of reductions, called reducsin this paper, employs greater numbers of covariances, but at the risk of introducing more noise. 2.2. Helpful formalisms for analyzing MCMC convergence. It is helpful to introduce the linear operator Tdefined by Tu(z) =Z (z0jz)u(z0)dz0 where(z0jz)is the transition probability density for the Markov chain. Then one can express an expectation of the form E[v(Z0)u(Z1)], arising from a covariance, as E[v(Z0)u(Z1)] =hv;Tui where the inner product h
;
iis defined by (5) hv;ui=Z v(z)u(z)(z) dz: The adjoint operator Tyv(z) =1 (z)Z (zjz0)v(z0)(z0)dz0 is what Ref. [37] calls the forward transfer operator, because it propagates relative probability densities forward in time. On the other hand, Ref. [29] calls Tythe backward operator and callsTitself the forward operator. To avoid confusion, use the term transfer operator forT. The earlier work[13, 38] isin terms ofthe operator Ty. To get an expression for E[v(Z0)u(Zk)], write E[v(Z0)u(Zk)] =ZZ v(z)u(z0)k(z0jz)(z) dzdz0 wherek(z0jz)is the iterated transition probability density function defined recursively by 1(z0jz) =(zjz0)and k(z0jz) =Z (z0jz00)k1(z00jz)dz00; k = 2;3;:::: 5By induction on k Tku(z) =TTk1u(z) =Z k(z0jz)u(z0)dz0; whence, E[v(Z0)u(Zk)] =hv;Tkui: 2.2.1. Properties of the transfer operator and IAcT. It is useful to establish some prop- erties ofTand the IAcT that will be used throughout the article. In particular, we shall provide a formula for (u)in terms of the transfer operator that will be the starting point for systematic improvements and that will later on allow us to estimate by solving a generalised eigenvalue problem. Clearly,T1 = 1, and 1 is an eigenvalue of T. Here, where the context requires a function, the symbol 1 denotes the constant function that is identically 1. Where the context requires an operator, it denotes the identity operator. To remove the eigenspace corresponding to the eigenvalue= 1fromT, define the orthogonal projection operator Eu=h1;ui1 and consider instead the operator T0=TE: It is assumed that the eigenvalues ofT0satisfyjj<1, in other words, we assume that the underlying Markov chain is ergodic. Stationarity of the target density (z)w.r.t.(zjz0) implies thatTy1 = 1and thatTyT1 = 1. Therefore,TyTis a stochastic kernel. This implies that the spectral radius of TyTis 1, and, since it is a symmetric operator, one has that (6) hTu;Tui=hu;TyTuihu;ui: The IAcT, given by Eq. (3), requires autocovariances, which one can express in terms of T0as follows: (7)C(k) =h(1E)u;(1E)Tkui =h(1E)u;(1E)Tk 0ui =h(1E)u;Tk 0ui; which follows because Eand1Eare symmetric. Substituting Equation (7) into Equation (3) gives (8) (u) =h(1E)u;Dui h(1E)u;ui;whereD= 2(1T0)11: It can be readily seen that is indeed nonnegative. With v= (1T0)1u, the numerator in Eq. (8) satisfies h(1E)u;Dui=h(1E)(1T0)v;(1 +T0)vi =hv;vihTv;Tvi 0: Therefore,(u)0if(1E)u6= 0, where the latter is equivalent to u6=E[u]being not a constant. 63. Sampling Thoroughness and Efficiency. Less than “thorough” sampling can degrade estimates of an IAcT. Ref. [13, Sec. 1] proposes a notion of “quasi-reliability” to mean the absence of evidence in existing samples that would suggest a lack of sampling thoroughness. A notion of sampling thoroughness begins by considering subsets Aof configuration space. The probability that Q2Acan be expressed as the expectation E[1A]where 1Ais the indicator function for A. A criterion for thoroughness might be that (9) jc1APr(Q2A)jtolwherec1A=1 NNX n=11A(Qn): This is not overly stringent, since it does not require that there are any samples in sets Aof probabilitytol. The next step in the development of this notion is to replace the requirement jc1APr(Q2 A)jtolby something more forgiving of the random error in c1A. For example, we could require instead that (Var[c1A])1=20:5tol; which would satisfy Eq. (9) with 95% confidence, supposing an approximate normal distribu- tion for the estimate. (If we are not willing to accept the Gaussian assumption, Chebychev’s inequality tells us that we reach 95% confidence level if we replace the right hand side by 0:05tol.) Now letAbe the integrated autocorrelation time for 1A. Because Var[c1A]A1 NVar[1A(Z)] =A1 NPr(Z2A)(1Pr(Z2A)) 1 4NA; it is enough to have (1=4N)A(1=4)tol2for all sets of configurations Ato ensure thorough sampling (assuming again Gaussianity). The definition of good coverage might then be ex- pressed in terms of the maximum (1A)over allA. Note that the sample variance may not be a good criterion if all the candidate sets Ahave small probability Pr(Z2A), in which case it is rather advisable to consider the relativeerror [6]. Ref. [13, Sec 3.1] then makes a leap, for the sake of simplicity, from considering just indi- cator functions to arbitrary functions. This leads to defining q;max= supVar[u(Q)]>0(u). The condition Var[u(Q)]>0is equivalent to (1E)u6= 0. A few remarks on the efficient choice of preobservables are in order. Remark 1. Generally, if there are symmetries present in both the distribution and the pre- observables of interest, this may reduce the amount of sampling needed. Such symmetries can be expressed as bijections qfor whichu( q(q)) =u(q)andq( q(q)) =q(q). Examples in- clude translational and rotational invariance, as well as interchangeability of atoms and groups of atoms. Let qdenote the set of all such symmetries. The definition of good coverage then 7need only include sets A, which are invariant under all symmetries q2 q. The extension from indicator sets 1Ato general functions leads to considering Wq=fu(q)ju( q(q)) =u(q) for all q2 qgand defining q;max= sup u2W0q(u) whereW0 q=fu2WqjVar[u(Q)]>0g. Remark 2. Another consideration that might dramatically reduce the set of relevant preob- servables is the attractiveness of using collective variables =(q)to characterize structure and dynamics of molecular systems. This suggests considering only functions defined on collective variable space, hence, functions of the form u((q)). 4. Computing the Maximum IAcT. The difficulty of getting reliable estimates for (u)in order to compute the maximum IAcT makes it interesting to consider alternative formulation. 4.1. A transfer operator based formulation. Although, there is little interest in sampling functionsofauxiliaryvariableslikemomenta, itmaybeusefultoconsiderphasespacesampling efficiency. Specifically, a maximum over phase space is an upper bound and it might be easier to estimate. Putting aside exploitation of symmetries, the suggestion is to using max= supVar[u(Z)]>0(u). One has, with a change of variables, that ((1T0)v) =2(v) where 2(v) =h(1T)v;(1 +T)vi h(1T)v;(1T)vi: This follows from h(1E)(1T0)v;(1T0)vi=h(1T)v;(1T)vEvi=h(1T)v;(1T)vi. Therefore, max= sup Var[(1T0)v(Z)]>0((1T0)v) = sup Var[(1T0)v(Z)]>02(v) = sup Var[v(Z)]>02(v): The last step follows because (1T0)is nonsingular. Needed for an estimate of 2(v)ishTv;Tvi. To evaluatehTv;Tvi, proceed as follows: Let Z0 n+1be an independent realization of Zn+1fromZn. In particular, repeat the step, but with an independent stochastic process having the same distribution. Then (10)E[v(Z1)v(Z0 1)] =Z Z v(z)v(z0)Z (zjz00)(z0jz00)(z00)dz00dzdz0 =hTv;Tvi: For certain simple preobservables and propagators having the simple form of BAOAB, the samplesv(Zn)v(Z0 n)might be obtained at almost no extra cost, and their accuracy improved and their cost reduced by computing conditional expectations analytically. 8This approach has been tested on the model problem of Sec. 5, a Gaussian process, and found to be significantly better than the use of acor. Unfortunately, this observation is not generalisable: For example, for a double well potential, it is difficult to find preobservables v(z), giving a computable estimate of maxwhich comes close to an estimate from using acor withu(z) =z1. Another drawback is that the estimates, though computationally inexpensive, require ac- cessing intermediate values in the calculation of a time step, which are not normally an output option of an MD program. Therefore we will discuss alternatives in the next two paragraphs. 4.2. A generalised eigenvalue problem. Letu(z)be a row vector of arbitary basis func- tionsui(z),i= 1;2;:::; imaxthat span a closed subspace of the Hilbert space associated with the inner product h
;
idefined by (5) and consider the linear combination u(z) =u(z)Tx. One has (u) =h(1E)u;Dui h(1E)u;ui=xTDx xTC0x where D=h(1E)u;DuTiand C0=h(1E)u;uTi: If the span of the basis is sufficiently extensive to include preobservables having the greatest IAcTs (e.g. polynomials, radial basis functions, spherical harmonics, etc.), the calculation of maxreduces to that of maximizing xTDx=(xTC0x)over all x, which is equivalent to solving the symmetric generalized eigenvalue problem (11)1 2(D+DT)x=C0x: It should be noted that the maximum over all linear combinations of the elements of u(z)can be arbitrarily greater than use of any of the basis functions individually. Moreover, in practice, the coefficients in (11) will be random in that they have to be estimated from simulation data, which warrants special numerical techniques. These techniques, including classical variance reduction methods, Markov State Models or specialised basis functions, are not the main focus of this article and we therefore refer to the articles [19, 32], and the references given there. Remark 3. B records different notions of reversibility of the transfer operator that entail spe- cific restrictions on the admissible basis functions that guarantee that the covariance matrices, and thus C0, remain symmetric. 4.3. The use of acor.It is not obvious how to use an IAcT estimator to construct matrix off-diagonal elements Dij=h(1E)ui;DuT ji,j6=i, from the time series fu(Zm)g. Nevertheless, it makes sense to use arcoras a preprocessing or predictor step to generate an initial guess for an IAcT. The acorestimate for a scalar preobservable u(z)has the form b=bD=bC0 where bC0=bC0(fu(Zn)^Ug;fu(Zn)^Ug) 9and bD=bD(fu(Zn)^Ug;fu(Zn)^Ug) arebilinearfunctionsoftheirargumentsthatdependonthenumberofreductions reducswhere ^Udenotes the empirical mean of fu(Zm)g. The tests reported in Secs. 5–7 then use the following algorithm. (In what follows we assume thatfu(Zm)ghas been centred by subtracting the empirical mean.) Algorithm 1 Computing the IAcT For each basis function, compute b, and record the number of reductions, set reducsto the maximum of these. Then compute D= (Dij)ijfrombD(fui(zm)g;fuj(zn)g)with a number of reductions equal toreducs. ifD+DThas a non-positive eigenvalue then redo the calculation using reducs1reductions. end if Ref. [13, Sec. 3.5] uses a slightly different algorithm that proceeds as follows: Algorithm 2 Computing the IAcT as in [13, Sec. 3.5] Setreducsto the value of reducsfor the basis function having the largest estimated IAcT. Then run acorwith a number of reductions equal to reducsto determine a revised Dand a maximizing x. ForuTx, determine the number of reductions reducs0. ifreducs0<reducs then, redo the calculation with reducs =reducs0and repeat until the value of reducsno longer decreases. end if In the experiments reported here, the original algorithm sometimes does one reduction fewer than the new algorithm. Remark 4. Theoretically, the matrix D+DTis positive definite. If it is not, that suggests that the value of reducsis not sufficiently conservative, in which case reducsneeds to be reduced. A negative eigenvalue might also arise if the Markov chain does not converge due to a stepsize
tthat is too large. This can be confirmed by seeing whether the negative eigenvalue persists for a larger number of samples. 5. Analytical Result for the Model Problem. The question of optimal choice for the damping coefficient is addressed in Ref. [38, Sec. 5.] for the standard model problem F(q) = Kq, whereKis symmetric positive definite, for which the Langevin equation is (12)dQt=M1Ptdt; dPt=KQtdt Ptdt+q 2 MhdWt: 10Changing variables Q0=MT hQandP0=M1 hPand dropping the primes gives dQt=Ptdt, dPt=M1 hKMT hQtdt Ptdt+p 2 =dWt: With an orthogonal change of variables, this decouples into scalar equations, each of which has the form dQt=Ptdt;dPt=!2Qtdt Ptdt+p 2 =dWt where!2is an eigenvalue of M1 hKMT h, or, equivalently, an eigenvalue of M1K. Changing to dimensionless variables t0=!t, 0= =!,Q0= (m)1=2!Q,P0= (=m)1=2P, and dropping the primes gives (13) dQt=Ptdt;dPt=Qtdt Ptdt+p 2 dWt: ForanMCMCpropagator, assumeexactintegrationwithstepsize
t. FromRef.[38, Sec.5.1], one hasT= (e
tL)y= exp(
tLy)where Lyf=p@ @qfq@ @pf p@ @pf+ @2 @p2f: The Hilbert space defined by the inner product from Eq. (5) has, in this case, a decomposition into linear subspaces Pk= spanfHem(q)Hen(p)jm+n=kg(denoted by P0 kin Ref. [38, Sec. 5.3]). Let uT k= [Hek(q)He0(p);Hek1(q)He1(p); :::; He0(q)Hek(p)]; and, in particular, uT 1= [q; p]; uT 2= [q21; qp; p21]; uT 3= [q33q;(q21)p; q(p21); p33p]; uT 4= [q46q2+ 3;(q33q)p;(q21)(p21); q(p33p); p46p+ 3]: With a change of notation from Ref. [38, Sec. 5.3], LuT k=uT kAk, with Akgiven by (14) Ak=2 666640 1 k ... ......k 1k 3 77775: One can show, using arguments similar to those in [38, Sec. 5.3], that Pkclosed under ap- plication ofLy. Therefore,LyuT k=uT kBkfor somek+ 1byk+ 1matrix Bk. Forming the inner product of ukwith each side of this equation gives Bk=C1 k;0huk;LyuT kiwhere Ck;0=huk;uT ki. It follows that Bk=C1 k;0huk;LyuT ki=C1 k;0hLuk;uT ki 11and LyuT k=uT kC1 k;0AT kCk;0: The Hermite polynomials ukare orthogonal and Ck;0= diag(k!0!;(k1)!1!; :::; 0!k!): Also,EuT k=0T. Accordingly, T0uT k=TuT k=uT kC1 k;0exp(
tAT k)Ck;0 and (15) DuT k=uT kC1 k;0Dk where Dk=Ck;0 2(IC1 k;0exp(
tAT k)Ck;0)1I =coth(
t 2AT k)Ck;0: A formula for (u)is possible if u(q)can be expanded in Hermite polynomials as u=P1 k=1ckHek. Then, from Eq. (15), DHek2Pk, not to mention Hek2Pk. Using these facts and the mutual orthogonality of the subspaces Pk, it can be shown that (16) (u) =P1 k=1k!c2 k(Hek)P1 k=1k!c2 k: From this it follows that maxu(u) = maxk(Hek). Since Hek=uT kxwithx= [1;0;:::; 0]T, one has (17) (Hek) = (Dk)11=(Ck;0)11= (coth(
t 2Ak))11: Asymptotically (Hek) =(2=
t)(A1 k)11, in the limit as
t!0. In particular, (18) A1 1= 1 1 0 and (19) A1 2=1 2 2 4 2+ 12 1 0 0 1 0 13 5: Writing(Hek)as an expansion in powers of
t, (Hek) =Tk( )=
t+O(
t); 12Figure 1. From top to bottom on the right Tk( )vs. ,k= 1;2;3;4 one hasT1( ) = 2 andT2( ) = + 1= . Fig. 1 plots Tk( ),k= 1;2;3;4,1=2 4. Empirically, maxkTk=Tmaxdef= maxfT1;T2g. Restoring the original variables, one has q;max=Tmax( =!)=(!
t) +O(!
t): The leading term increases as !decreases, so q;maxdepends on the lowest frequency !1. And q;maxisminimizedat =!1, whichishalfofthecriticalvalue = 2!1. Contrastthiswiththe result [38, Sec. 5.] for the phase space maximum IAcT, which is minimized for = (p 6=2)!1. Remark 5. The result is consistent with related results from [4, 12] that consider optimal damping coefficients that maximise the speed of convergence measured in relative entropy. Specifically, calling t=N(t;t)the law of the solution to (13), with initial conditions (Qt;Pt) = (q;p); see A for details. Then, using [2, Thm. 4.9], we have KL(t;)Mexp(2t); whereM2(1;1)anddenotes the spectral abcissa of the matrix Ain A, i.e. the negative real part of the eigenvalue that is closest to the imaginary axis. Here KL(f;g) =Z logf(z) g(z)f(z)dz denotes the relative entropy (or: Kullback-Leibler divergence) between two phase space proba- bility densities fandg, assuming that Z fg(z)=0gf(z)dz= 0: (Otherwise we set KL(f;g) =1.) It is a straightforward calculation to show that the maximum value for(that gives the fastest decay of KL(t;)) is attained at = 2, which is in agreement 13with the IAcT analysis. For analogous statements on the multidimensional case, we refer to [4]. We should mention that that there may be cases, in which the optimal damping coefficient may lead to a stiff Langevin equation, depending on the eigenvalue spectrum of the Hessian of the potential energy function. As a consequence, optimizing the damping coefficient may reduce the maximum stable step size
tthat can be used in numerical simulations. 5.1. Application to more general distributions. Note that for the model problem, the matrixKcan be extracted from the covariance matrix Cov[Q] = (1=)K1: Therefore, as a surrogate for the lowest frequency !1, and as a recommended value for , consider using = (min(M1K))1=2= (max(Cov[Q]M))1=2: 5.2. Sanity check. As a test of the accuracy of acorand the analytical expression (16), the IAcT is calculated by acorfor a time series generated by the exact analytical propagator (given in A) for the reduced model problem given by Eq. (12). For the preobservable, we choose u(q) =He3(q)=p 3!He2(q)=p 2! where He2(q) =q21andHe3(q) =q33qare Hermite polynomials of degree 2 and 3; as damping coefficient, we choose = 2, which is the critical value; the time increment is
t= 0:5, which is about 1/12th of a period. In this and the other results reported here, equilibrated initial values are obtained by running for 50000 burn-in steps. As the dependence of the estimate on Nis of interest here, we runM= 103independent realisations for each value of N, from which we can estimate the relative error N((u)) =p Var[(u)] E[(u)]; which we expect to decay as N1=2. Fig. 2 shows the relative error in the estimated IAcT (u) forN= 213,214, ..., 222. The least-squares fit of the log relative error as a function of logN has slopem= 0:4908. Thus we observe a nearly perfect N1=2decay of the relative error, in accordance with the theoretical prediction. 6. A simple example. The procedure to determine the optimal damping coefficient in the previous section is based on linear Langevin systems. Even though the considerations of Section 5 do not readily generalize to nonlinear systems, it is plausible to use the harmonic approximation as a proxy for more general systems, since large IAcT values are often due to noise-induced metastability, in which case local harmonic approximations inside metastable regions are suitable. For estimating the maximum IAcT, the model problem therefore suggests the use of linear, quadratic and cubic functions of the coordinates, where the latter is suitable to capture the possible non-harmonicity of the potential energy wells in the metastable regime. The first test problem, which is from Ref. [23], possesses an asymmetric multimodal dis- tribution. It uses U(q) =1 4q4+ sin(1 + 5q)and= 1, and it generates samples using BAOAB 14104105106 N10-210-1relative error Nm=-0.4908 N (M=103)Figure 2. Relative error in estimated IAcT as a function of sample size N. The relative error N=p Var[]=E[]has been computed by averaging over M= 103independent realisations of each simulation. with a step size
t= 0:2, which is representative of step sizes used in Ref. [23]. Fig. 3 plots with dotted lines the unnormalized probability density function. 6.1. Choice of basis. A first step is to find a preobservable that produces a large IAcT. It would be typical of actual practice to try to select a good value for . To this end, choose = = 1:276, To obtain this value, do a run of sample size N= 2
106using = 1, as in one of the tests in Ref. [23]. With a sample size N= 107, the maximum IAcT is calculated for polynomials of increasing degree using the approach described in Secs. 4.2–4.3. Odd degrees produces somewhat greater maxima than even degrees. For cubic, quintic, and septic polynomials, maxhas values 59.9, 63.9, 65.8, respectively As a check that the sample size is adequate, the calculations are redone with half the sample size. Fig. 3 shows how the maximizing polynomial evolves as its degree increases from 3 to 5 to 7. 6.2. Optimal choice of damping coefficient. The preceding results indicate that septic polynomials are a reasonable set of functions for estimating q;max. For 25 values of , ranging from 0.2 to 5, the value of q;maxwas thus estimated, each run consisting of N= 107samples. The optimal value is = 1:8 = 1:4 , which is close the heuristic choice for a damping coefficient. Fig. 4 plots q;maxvs. the ratio = . With respect to this example, Ref. [23, Sec. 5] states, “We were concerned that the im- proved accuracy seen in the high regime might come at the price of a slower convergence to equilibrium”. The foregoing results indicate that the value = 1used in one of the tests is near the apparent optimal value = 1:8. Hence, the superior accuracy of BAOAB over other methods observed in the low regime does not come at the price of slower convergence. 15Figure 3. In dotted lines is the unnormalized probability density function. From top to bottom on the right are the cubic, quintic, and septic polynomials that maximize the IAcT over all polynomials of equal degree. Figure 4.q;maxvs. = using septic polynomials as preobservables 7. Sum of three Gaussians. The next, perhaps more challenging, test problem uses the sum of three (equidistant) Gaussians for the distribution, namely. exp(V(x;y)) = exp(((xd)2+y2)=2) + exp(((x+d=2)2+ (yp 3d=2)2)=2) + exp(((x+d=2)2+ (y+p 3d=2)2)=2)) wheredis a parameter that measures the distance of the three local minima from the origin. Integrating the Langevin system using BAOAB with a step size
t= 0:5as for the model problem, which is what V(x;y)becomes if d= 0. Shown in Fig. 5 are the first 8
104points of a trajectory where d= 4:8. 7.1. Choice of basis. To compare maxfor different sets of preobservables, choose = = 0:261, and with so chosen, run the simulation with d= 4:8forN= 107steps. To 16Figure 5. A typical time series for a sum of three Gaussians compute , run the simulation for N= 2
106steps with = 1(which is optimal for d= 0). Here are the different sets of preobservables and the resulting values of max: 1. linear polynomials of xandy, for which max= 18774, 2. quadratic polynomials of xandy, for which max= 19408, 3. linear combinations of indicator functions f1A;1B;1Cgfor the three conformations A=f(x;y) :jyjp 3xg B=f(x;y) :y0andyp 3xg C=f(x;y) :y0andyp 3xg; for whichmax= 18492, 4.1Aalone, for which = 12087, 5.1Balone, for which = 5056, 6.1Calone, for which = 4521. As consequence of these results, the following section uses quadratic polynomials to estimate q;max. 7.2. Optimal choice of damping coefficient. Shown in Fig. 6 is a plot of q;maxvs. the ratio = . To limit the computing time, we set the parameter to d= 4:4rather than 4.8 as in Sec. 7.1; for d= 4:4, we have ?= 0:285, obtained using the same protocol as does Sec. 7.1. We consider 0:05 2:2in increments of 0.01 from 0.05 to 0.2, and in increments of 0.1 from 0.2 to 2.2. Each data point is based on a run of N= 2
107time steps. Even though the variance of the estimator is not negligible for our choice of simulation parameters, it is clearly visible that the minimum of q;maxis attained at  . 8. Conclusions. We have discussed the question of how to choose the damping coefficient in (underdamped) Langevin dynamics that leads to efficient sampling of the stationary proba- bilitydistributionorexpectationsofcertainobservableswithrespecttothisdistribution. Here, efficient sampling is understood as minimizing the maximum possible (worst-case) integrated 17Figure 6.q;maxvs. the ratio =  autocorrelation time (IAcT). We propose a numerical method that is based on the concept of phase space preobservables that span a function space over which the worst-case IAcT is computed using trajectory data; the optimal damping coefficient can then chosen on the basis of this information. Based on heuristics derived from a linear Langevin equation, we derive rules of thumb for choosing good preobservables for more complicated dynamics. The results for the linear model problem are in agreement with recent theoretical results on Ornstein-Uhlenbeck processes with degenerate noise, and they are shown to be a good starting point for a systematic analysis of nonlinear Langevin samplers. Appendix A. Analytical propagator for reduced model problem. This section derives the analytical propagator for Eq. (13). In vector form, the equation is dZt=AZdt+bdWtwhereA=0 1 1  andb= [0;p2 ]T. The variation of parameters solution is Zt= etAZ0+Rtwhere Rt=Zt 0e(ts)Abdt: The stochastic process Rtis Gaussian with mean zero and covariance matrix  =E[RtRT t] =Zt 0e(ts)AbbTe(ts)ATdWt: To evaluate this expressions, use A=XX1where X=1 1 + ; X1=1  + 1 1 ;  = diag( ; +), =1 2( );and=p 24!2: 18Noting that exp( t) = exp( t=2)(cosh(t=2)sinh(t=2)), one has etA= e t=2cosht 21 0 0 1 + e t=2t 2sinhct 2 2 2  ; where sinhcs= (sinhs)=s. Then  =XZt 0e(ts)X1bbTXTe(ts)dtXT =2 2XZt 0e(ts)11 1 1 e(ts)dtXT =2 2X2 6641e2 t 2 1e t 1e t 1e2 +t 2 +3 775XT: Noting that exp(2 t) = exp( t)(1 + 2 sinh2(t=2))2 sinh(t=2) cosh(t=2)), one has  = (1e t)1 0 0 1 t2 2e t(sinhct 2)2 2 2  + te tsinhct 2cosht 21 0 0 1 : Appendix B. Different notions of reversibility. We briefly mention earlier work and discuss different reversiblity concepts for transfer operators. B.1. Quasi-reversibility. Ref. [13, Sec. 3.4] introduces a notion of quasi-reversibility. A transfer operator Tis quasi-reversible if Ty=RyTR whereRis an operator such that R2= 1. This somewhat generalizes the (suitably modified) definitions in Refs. [13, 38]. The principal example of such an operator is Ru=uRwhere Ris a bijection such that RR= idanduR=uforu2W, e.g, momenta flipping. The value of the notion of quasi-reversibility is that it enables the construction of basis functions that lead to a matrix of covariances that possesses a type of symmetric structure [38, Sec. 3.1]. This property is possessed by “adjusted” schemes that employ an acceptance test, and by the limiting case
t!0of unadjusted methods like BAOAB. B.2. Modified detailed balance. A quite different generalization of reversibility, termed “modified detailed balance”, is proposed in Ref. [14] as a tool for making it a bit easier to prove stationarity. Modified detailed balance is introduced in Ref. [14] as a concept to make it easier to prove stationarity. In terms of the transfer operator, showing stationarity means showing that F1 = 1, where 1is the constant function 1. 19Ref. [14, Eq. (15)] defines modified detailed balance in terms of transition probabilities. The definition is equivalent to F=R1FyR1under the assumption that Rpreserves the stationary distribution. This readily generalizes to (20) F=R2FyR1 whereR1andR2are arbitrary except for the assumption that each of them preserve the stationary distribution. Stationarity follows from Eq. (20) because Fy1 = 1for any adjoint transfer operator and R11 =R21 = 1by assumption. Reference [14] has errors, which are corrected in Ref. [15]. 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1611.03886v1.The_destabilizing_effect_of_external_damping__Singular_flutter_boundary_for_the_Pfluger_column_with_vanishing_external_dissipation.pdf

The destabilizing eect of external damping: Singular utter boundary for the P uger column with vanishing external dissipation Mirko Tommasinia, Oleg N. Kirillova,b, Diego Misseronia, Davide Bigonia aUniversit a di Trento, DICAM, via Mesiano 77, I-38123 Trento, Italy bRussian Academy of Sciences, Steklov Mathematical Institute, Gubkina st. 8, 119991 Moscow, Russia Abstract Elastic structures loaded by nonconservative positional forces are prone to instabilities in- duced by dissipation: it is well-known in fact that internal viscous damping destabilizes the marginally stable Ziegler's pendulum and P uger column (of which the Beck's column is a special case), two structures loaded by a tangential follower force. The result is the so-called `destabilization paradox', where the critical force for utter instability decreases by an order of magnitude when the coecient of internal damping becomes innitesimally small. Until now external damping, such as that related to air drag, is believed to provide only a stabi- lizing eect, as one would intuitively expect. Contrary to this belief, it will be shown that the eect of external damping is qualitatively the same as the eect of internal damping, yielding a pronounced destabilization paradox. Previous results relative to destabilization by external damping of the Ziegler's and P uger's elastic structures are corrected in a denitive way leading to a new understanding of the destabilizating role played by viscous terms. Keywords: P uger column, Beck column, Ziegler destabilization paradox, external damping, follower force, mass distribution 1. Introduction 1.1. A premise: the Ziegler destabilization paradox In his pioneering work Ziegler (1952) considered asymptotic stability of a two-linked pendulum loaded by a tangential follower force P, as a function of the internal damping in the viscoelastic joints connecting the two rigid and weightless bars (both of length l, Fig. 1(c)). The pendulum carries two point masses: the mass m1at the central joint and the Email addresses: mirko.tommasini@unitn.it (Mirko Tommasini), kirillov@mi.ras.ru (Oleg N. Kirillov), diego.misseroni@unitn.it (Diego Misseroni), davide.bigoni@unitn.it (Davide Bigoni) Corresponding author: Davide Bigoni, davide.bigoni@unitn.it; +39 0461 282507 Preprint submitted to Elsevier October 19, 2021arXiv:1611.03886v1 [physics.class-ph] 1 Oct 2016Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 massm2mounted at the loaded end of the pendulum. The follower force Pis always aligned with the second bar of the pendulum, so that its work is non-zero along a closed path, which provides a canonical example of a nonconservative positional force. For two non-equal masses ( m1= 2m2) and null damping, Ziegler found that the pendulum is marginally stable and all the eigenvalues of the 2 2 matrix governing the dynamics are purely imaginary and simple, if the load falls within the interval 0 P <P u, where P u=7 2p 2k l2:086k l; (1) andkis the stiness coecient, equal for both joints. When the load Preaches the value P u, two imaginary eigenvalues merge into a double one and the matrix governing dynamics becomes a Jordan block. With the further increase of Pthis double eigenvalue splits into two complex conjugate. The eigenvalue with the positive real part corresponds to a mode with an oscillating and exponentially growing amplitude, which is called utter, or oscilla- tory, instability. Therefore, P=P umarks the onset of utter in the undamped Ziegler's pendulum. When the internal linear viscous damping in the joints is taken into account, Ziegler found another expression for the onset of utter: P=Pi, where Pi=41 28k l+1 2c2 i m2l3; (2) andciis the damping coecient, assumed to be equal for both joints. The peculiarity of Eq. (2) is that in the limit of vanishing damping, ci! 0, the utter load Pitends to the value 41 =28k=l1:464k=l, considerably lower than that calculated when damping is absent from the beginning, namely, the P ugiven by Eq. (1). This is the so-called `Ziegler's destabilization paradox' (Ziegler, 1952; Bolotin, 1963). The reason for the paradox is the existence of the Whitney umbrella singularity on the boundary of the asymptotic stability domain of the dissipative system (Bottema, 1956; Krechetnikov and Marsden, 2007; Kirillov and Verhulst, 2010)3. In structural mechanics, two types of viscous dampings are considered: (i.) one, called `internal', is related to the viscosity of the structural material, and (ii.) another one, called `external', is connected to the presence of external actions, such as air drag resistance during 3In the vicinity of this singularity, the boundary of the asymptotic stability domain is a ruled surface with a self-intersection, which corresponds to a set of marginally stable undamped systems. For a xed damping distribution, the convergence to the vanishing damping case occurs along a ruler that meets the set of marginally stable undamped systems at a point located far from the undamped instability threshold, yielding the singular utter onset limit for almost all damping distributions. Nevertheless, there exist particular damping distributions that, if xed, allow for a smooth convergence to the utter threshold of the undamped system in case of vanishing dissipation (Bottema, 1956; Bolotin, 1963; Banichuk et al., 1989; Kirillov and Verhulst, 2010; Kirillov, 2013). 2Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 oscillations. These two terms enter the equations of motion of an elastic rod as proportional respectively to the fourth spatial derivative of the velocity and to the velocity of the points of the elastic line. Of the two dissipative terms only the internal viscous damping is believed to yield the Ziegler destabilization paradox (Bolotin, 1963; Bolotin and Zhinzher, 1969; Andreichikov and Yudovich, 1974). 1.2. A new, destabilizing role for external damping Dierently from internal damping, the role of external damping is commonly believed to be a stabilizing factor, in an analogy with the role of stationary damping in rotor dynamics (Bolotin, 1963; Crandall, 1995). A full account of this statement together with a review of the existing results is provided in Appendix A. Since internal and external damping are inevitably present in any experimental realization of the follower force (Saw and Wood, 1975; Sugiyama et al., 1995; Bigoni and Noselli, 2011), it becomes imperative to know how these factors aect the utter boundary of both the P uger column and of the Ziegler pendulum with arbitrary mass distribution. These structures are fully analyzed in the present article, with the purpose of showing: (i.) that external damping is a destabilizing factor, which leads to the destabilization paradox for all mass distributions; (ii.) that surprisingly, for a nite number of particular mass distributions, the utter loads of the externally damped structures converge to the utter load of the undamped case (so that only in these exceptional cases the destabilizing eect is not present); and (iii.) that the destabilization paradox is more pronounced in the case when the mass of the column or pendulum is smaller then the end mass. Taking into account also the destabilizing role of internal damping, the results presented in this article demonstrate a completely new role of external damping as a destabilizing eect and suggest that the Ziegler destabilization paradox has a much better chance of being observed in the experiments with both discrete and continuous nonconservative systems than was previously believed. 2. Ziegler's paradox due to vanishing external damping The linearized equations of motion for the Ziegler pendulum (Fig. 1(c)), made up of two rigid bars of length l, loaded by a follower force P, when both internal and external damping are present, have the form (Plaut and Infante, 1970; Plaut, 1971) Mx+ciDi_x+ceDe_x+Kx= 0; (3) where a superscript dot denotes time derivative and ciandceare the coecients of internal and external damping, respectively, in front of the corresponding matrices DiandDe Di=21 1 1 ;De=l3 68 3 3 2 ; (4) 3Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 andMandKare respectively the mass and the stiness matrices, dened as M=m1l2+m2l2m2l2 m2l2m2l2 ;K=Pl+ 2k Plk k k ; (5) in whichkis the elastic stiness of both viscoelastic springs acting at the hinges. 16 16 16 163/c112 /c112/c112 /c112 3/c112 5/c112 7/c112 /c970/c112 88 420.00 -0.05-0.10 -0.15 -0.20 -0.25 -0.30/c68F DA BC P m2l m1l k, c , ciek, c , ciea) c)b) 16 16 16 163/c112 /c112/c112 /c112 3/c112 5/c112 7/c112 /c970/c112 88 421.52.02.53.0 FA BCD m =2m12 internalexternalidealFLUTTER 1.2502.086 1.464 Figure 1: (a) The (dimensionless) tangential force F, shown as a function of the (transformed via cot = m1=m2) mass ratio , represents the utter domain of (dashed/red line) the undamped, or `ideal', Ziegler pendulum and the utter boundary of the dissipative system in the limit of vanishing (dot-dashed/green line) internal and (continuous/blue line) external damping. (b) Discrepancy
Fbetween the critical utter load for the ideal Ziegler pendulum and for the same structure calculated in the limit of vanishing external damping. The discrepancy quanties the Ziegler's paradox. Assuming a time-harmonic solution to the Eq. (3) in the form x=uetand introducing the non-dimensional parameters =l kp km2; E =cel2 pkm2; B =ci lpkm2; F =Pl k;  =m2 m1; (6) an eigenvalue problem is obtained, which eigenvalues are the roots of the characteristic polynomial p() = 364+ 12(15B+ 2E+ 3B+E)3+ (36B2+ 108BE + 7E272F+ 180+ 36)2+ 6(5EF+ 12B+ 18E)+ 36: (7) 4Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 In the undamped case, when B= 0 andE= 0, the pendulum is stable, if 0 F <F u, unstable by utter, if F uFF+ u, and unstable by divergence, if F >F+ u, where F u() =5 2+1 21p: (8) In order to plot the stability map for all mass distributions 0  <1, a parameter 2[0;=2] is introduced, so that cot =1and hence F u() =5 2+1 2cotp cot: (9) The curves (9) form the boundary of the utter domain of the undamped, or `ideal', Ziegler's pendulum shown in Fig. 1(a) (red/dashed line) in the load versus mass distribution plane (Oran, 1972; Kirillov, 2011). The smallest utter load F u= 2 corresponds to m1=m2, i.e. to==4. Whenequals=2, the mass at the central joint vanishes ( m1= 0) and F u=F+ u= 5=2. Whenequals arctan (0 :5)0:464, the two masses are related as m1= 2m2andF u= 7=2p 2. In the case when only internal damping is present ( E= 0) the Routh-Hurwitz criterion yields the utter threshold as (Kirillov, 2011) Fi(;B) =252+ 6+ 1 4(5+ 1)+1 2B2: (10) For= 0:5 Eq. (10) reduces to Ziegler's formula (2). The limit for vanishing internal damping is lim B!0Fi(;B) =F0 i() =252+ 6+ 1 4(5+ 1): (11) The limitF0 i() of the utter boundary at vanishing internal damping is shown in green in Fig. 1(a). Note that F0 i(0:5) = 41=28 andF0 i(1) = 5=4. For 0<1the limiting curve F0 i() has no common points with the utter threshold F u() of the ideal system, which indicates that the internal damping causes the Ziegler destabilization paradox for every mass distribution. In a route similar to the above, by employing the Routh-Hurwitz criterion, the critical utter load of the Ziegler pendulum with the external damping Fe(;E) can be found Fe(;E) =122219+ 5 5(81)+7(2+ 1) 36(81)E2 (2+ 1)p 35E2(35E2792+ 360) + 1296(281 2130+ 25) 180(81) and its limit calculated when E!0, which provides the result F0 e() =122219+ 5(2+ 1)p 2812130+ 25 5(81): (12) 5Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 The limiting curve (12) is shown in blue in Fig. 1(a). It has a minimum min F0 e() = 28 + 8p 141:933 at= (31 + 7p 14)=750:763. Remarkably, for almost all mass ratios, except two (marked as A and C in Fig. 1(a)), the limit of the utter load F0 e() isbelow the critical utter load F u() of the undamped system. It is therefore concluded that external damping causes the discontinuous decrease in the critical utter load exactly as it happens when internal damping vanishes. Qualitatively , the eect of vanishing internal and external damping is the same . The only dierence is the magnitude of the discrepancy: the vanishing internal damping limit is larger than the vanishing external damping limit, see Fig. 1(b), where
F() =Fe()F u() is plotted. 0.00B 0.010.020.030.040.050.060.070.080.090.10 0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 Eb) a) E0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.02.002.102.202.302.40 2.052.152.252.35 FFLUTTER FLUTTER 2.086 Figure 2: Analysis of the Ziegler pendulum with xed mass ratio, =m2=m1= 1=2: (a) contours of the utter boundary in the internal/external damping plane, ( B;E), and (b) critical utter load as a function of the external damping E(continuous/blue curve) along the null internal damping line, B= 0, and (dot- dashed/orange curve) along the line B= 8=123 + 5p 2=164
E. For example,
F0:091 at the local minimum for the discrepancy, occurring at the point B with 0:523. The largest nite drop in the utter load due to external damping occurs at==2, marked as point D in Fig. 1(a,b):
F=11 201 20p 2810:288: (13) For comparison, at the same value of , the utter load drops due to internal damping of exactly 50%, namely, from 2 :5 to 1:25, see Fig. 1(a,b). As a particular case, for the mass ratio = 1=2, considered by Plaut and Infante (1970) and Plaut (1971), the following limit utter load is found F0 e(1=2) = 2; (14) 6Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c112 /c970/c112 88 42 16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c112 /c970/c112 88 42b) a) -1.0/c98 -0.8-0.6-0.4-0.20.00.2 1.52.02.53.0 FA C0.111 0.524-2/15FLUTTERideal Figure 3: Analysis of the Ziegler pendulum. (a) Stabilizing damping ratios () according to Eq. (19) with the points A and C corresponding to the tangent points A and C in Fig. 1(a) and to the points A and C of vanishing discrepancy
F= 0 in Fig. 1(b). (b) The limits of the utter boundary for dierent damping ratios have: two or one or none common points with the utter boundary (dashed/red line) of the undamped Ziegler pendulum, respectively when  < 0:111 (continuous/blue curves), 0:111 (continuous/black curve), and  >0:111 (dot-dashed/green curves). only slightly inferior to the value for the undamped system, F u(1=2) = 7=2p 22:086. This discrepancy passed unnoticed in (Plaut and Infante, 1970; Plaut, 1971) but gives evi- dence to the destabilizing eect of external damping. To appreciate this eect, the contours of the utter boundary in the ( B;E) - plane are plotted in Fig. 2(a) for three dierent values ofF. The contours are typical of a surface with a Whitney umbrella singularity at the origin (Kirillov and Verhulst, 2010). At F= 7=2p 2 the stability domain assumes the form of a cusp with a unique tangent line, B=E, at the origin, where =8 123+5 164p 20:108: (15) For higher values of Fthe utter boundary is displaced from the origin, Fig. 2(a), which indicates the possibility of a continuous increase in the utter load with damping. Indeed, along the direction in the ( B;E) - plane with the slope (15) the utter load increases as F(E) =7 2p 2 +47887 242064+1925 40344p 2 E2+o(E2); (16) see Fig. 2(b), and monotonously tends to the undamped value as E!0. On the other hand, along the direction in the ( B;E) - plane specied by the equation B= 0, the following 7Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 condition is obtained F(E) = 2 +14 99E2+o(E2); (17) see Fig. 2(b), with the convergence to a lower value F= 2 asE!0. In general, the limit of the utter load along the line B=EwhenE!0 is F() =5042+ 1467+ 104(4 + 21)p 5762+ 1728+ 121 30(1 + 14)7 2p 2; (18) an equation showing that for almost all directions the limit is lower than the ideal utter load. The limits only coincide in the sole direction specied by Eq. (15), which is dierent from theE-axis, characterized by = 0. As a conclusion, pure external damping yields the destabilization paradox even at = 1=2, which was unnoticed in (Plaut and Infante, 1970; Plaut, 1971). In the limit of vanishing external ( E) and internal ( B) damping, a ratio of the two =B=E exists for which the critical load of the undamped system is attained, so that the Ziegler's paradox does not occur. This ratio can therefore be called `stabilizing', it exists for every mass ratio =m2=m1, and is given by the expression () =1 3(101)(1) 252+ 6+ 1+1 12(135)(3+ 1) 252+ 6+ 11=2: (19) Eq. (19) reduces for = 1=2 to Eq. (15) and gives =2=15 in the limit !1 . With the damping ratio specied by Eq. (19) the critical utter load has the following Taylor expansion near E= 0: F(E;) =F u() +()(5+ 1)(41+ 7) 6(252+ 6+ 1)E2 +6363+ 3852118+ 25 288(252+ 6+ 1)E2+o(E2); (20) yielding Eq. (16) when = 1=2. Eq. (20) shows that the utter load reduces to the undamped case when E= 0 (called `ideal' in the gure). When the stabilizing damping ratio is null, = 0, convergence to the critical utter load of the undamped system occurs by approaching the origin in the ( B;E) - plane along the E - axis. The corresponding mass ratio can be obtained nding the roots of the function () dened by Eq. (19). This function has only two roots for 0 <1, one at0:273 (or 0:267, marked as point A in Fig. 3(a)) and another at 2:559 (or1:198, marked as point C in Fig. 3(a)). Therefore, if = 0 is kept in the limit when the damping tends to zero, the limit of the utter boundary in the load versus mass ratio plane will be obtained as a curve showing two common points with the utter boundary of the undamped system, exactly at the mass 8Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 E Eb) a) 0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.02.052.072.09 2.062.08F 0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.0B 0.000 -0.025-0.0500.0250.0500.0750.100 2.102.112.122.132.142.15 2.07F=2.10F=2.07 F=2.04FLUTTER FLUTTER B=0 Figure 4: Analysis of the Ziegler pendulum with xed mass ratio, 2:559: (a) contours of the utter boundary in the internal/external damping plane, ( B;E), and (b) critical utter load as a function of external dampingE(continuous/blue curve) along the null internal damping line, B= 0. ratios corresponding to the points denoted as A and C in Fig. 1(a), respectively characterized byF2:417 andF2:070. If for instance the mass ratio at the point C is considered and the contour plots are analyzed of the utter boundary in the ( B;E) - plane, it can be noted that at the critical utter load of the undamped system, F2:07, the boundary evidences a cusp with only one tangent coinciding with the Eaxis, Fig. 4(a). It can be therefore concluded that at the mass ratio2:559 the external damping alone has a stabilizing eect and the system does not demonstrate the Ziegler paradox due to small external damping, see Fig. 4(b), where the the utter load F(E) is shown. Looking back at the damping matrices (4) one may ask, what is the property of the damping operator which determines its stabilizing or destabilizing character. The answer to this question (provided by (Kirillov and Seyranian, 2005b; Kirillov, 2013) via perturbation of multiple eigenvalues) involves all the three matrices M(mass), D(damping), and K (stiness). In fact, the distributions of mass, stiness, and damping should be related in a specic manner in order that the three matrices ( M,D,K) have a stabilizing eect (see Appendix B for details). 3. Ziegler's paradox for the P uger column with external damping The Ziegler's pendulum is usually considered as the two-dimensional analog of the Beck column, which is a cantilevered (visco)elastic rod loaded by a tangential follower force (Beck, 9Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 1952). Strictly speaking, this analogy is not correct because the Beck column has a dierent mass distribution (the usual mass distribution of the Ziegler pendulum is m1= 2m2) and this mass distribution yields dierent limiting behavior of the stability threshold (Section 2). For this reason, in order to judge the stabilizing or destabilizing in uence of external damping in the continuous case and to compare it with the case of the Ziegler pendulum, it is correct to consider the Beck column with the point mass at the loaded end, in other words the so-called `P uger column' (P uger, 1955). A viscoelastic column of length l, made up of a Kelvin-Voigt material with Young modulus Eand viscosity modulus E, and mass per unit length mis considered, clamped at one end and loaded by a tangential follower force Pat the other end (Fig. 5(c)), where a point mass Mis mounted. The moment of inertia of a cross-section of the column is denoted by Iand a distributed external damping is assumed, characterized by the coecient K. Small lateral vibrations of the viscoelastic P uger column near the undeformed equilib- rium state is described by the linear partial dierential equation (Detinko, 2003) EI@4y @x4+EI@5y @t@x4+P@2y @x2+K@y @t+m@2y @t2= 0; (21) wherey(x;t) is the amplitude of the vibrations and x2[0;l] is a coordinate along the column. At the clamped end ( x= 0) Eq. (21) is equipped with the boundary conditions y=@y @x= 0; (22) while at the loaded end ( x=l), the boundary conditions are EI@2y @x2+EI@3y @t@x2= 0; EI@3y @x3+EI@4y @t@x3=M@2y @t2: (23) Introducing the dimensionless quantities =x l;  =t l2q EI m; p =Pl2 EI;  =M ml; =E El2q EI m; k =Kl2p mEI(24) and separating the time variable through y(;) =lf() exp(), the dimensionless bound- ary eigenvalue problem is obtained (1 + )@4 f+p@2 f+ (k+2)f= 0; (1 + )@2 f(1) = 0; (1 + )@3 f(1) =2f(1); f(0) =@f(0) = 0; (25) 10Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 dened on the interval 2[0;1]. A solution to the boundary eigenvalue problem (25) was found by Pedersen (1977) and Detinko (2003) to be f() =A(cosh(g2)cos(g1)) +B(g1sinh(g2)g2sin(g1)) (26) with g2 1;2=p p24(+k)(1 + )p 2(1 + ): (27) Imposing the boundary conditions (25) on the solution (26) yields the characteristic equation
() = 0 needed for the determination of the eigenvalues , where
() = (1 + )2A1(1 + )A22(28) and A1=g1g2 g4 1+g4 2+ 2g2 1g2 2coshg2cosg1+g1g2(g2 1g2 2) sinhg2sing1
; A2= (g2 1+g2 2) (g1sinhg2cosg1g2coshg2sing1): (29) 16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c1120/c112 88 42pa) external intenalr8101214161820 Mml Pb) 20.0 19.518.017.517.0 16/c112/c112 80external /c97FLUTTER c)ideal idealA B Figure 5: Analysis of the P uger column [scheme reported in (c)]. (a) Stability map for the P uger's column in the load-mass ratio plane. The dashed/red curve corresponds to the stability boundary in the undamped case, the dot-dashed/green curve to the case of vanishing internal dissipation ( = 1010andk= 0 ) and the continuous/blue curve to the case of vanishing external damping ( k= 1010and = 0). (b) detail of the curve reported in (a) showing the destabilization eect of external damping: small, but not null. 11Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 Transforming the mass ratio parameter in Eq. (28) as = tanwith2[0;=2] allows the exploration of all possible ratios between the end mass and the mass of the column covering the mass ratios from zero (= 0) to innity ( ==2). The former case, without end mass, corresponds to the Beck column, whereas the latter corresponds to a weightless rod with an end mass, which is known as the `Dzhanelidze column' (Bolotin, 1963). It is well-known that the undamped Beck column loses its stability via utter at p 20:05 (Beck, 1952). In contrast, the undamped Dzhanelidze's column loses its stability via divergence at p20:19, which is the root of the equation tanpp=pp(Bolotin, 1963). These values, corresponding to two extreme situations, are connected by a marginal stability curve in the ( p;)-plane that was numerically evaluated in (P uger, 1955; Bolotin, 1963; Oran, 1972; Sugiyama et al., 1976; Pedersen, 1977; Ryu and Sugiyama, 2003). The instability threshold of the undamped P uger column is shown in Fig. 5 as a dashed/red curve. 16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c1120/c112 88 428.0 7.010.0 9.011.012.013.014.015.016.017.018.019.020.021.022.023.024.025.0 STABILITY STABILITY /c103=10 , k=0/UNI207b¹⁰/c103=0.050, k=0 /c103=0.100, k=0k=5, =0 /c103k=10 , =0 /UNI207b¹⁰/c103 k=10, =0 /c103 k1 0/c61/c103=/UNI207b¹⁰ /c97p k 0.010/c61/c49/c44 /c103 = Figure 6: Evolution of the marginal stability curve for the P uger column in the ( ;p) - plane in the case ofk= 0 and tending to zero (green curves in the lower part of the graph) and in the case of = 0 andk tending to zero (blue curves in the upper part of the graph). The cases of k= = 1010and ofk= 1 and = 0:01 are reported with continuous/red lines. 12Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 For every xed value 2[0;=2), the undamped column loses stability via utter when an increase in pcauses the imaginary eigenvalues of two dierent modes to approach each other and merge into a double eigenvalue with one eigenfunction. When plies above the dashed/red curve, the double eigenvalue splits into two complex eigenvalues, one with the positive real part, which determines a utter unstable mode. At==2 the stability boundary of the undamped P uger column has a vertical tangent and the type of instability becomes divergence (Bolotin, 1963; Oran, 1972; Sugiyama et al., 1976). Settingk= 0 in Eq. (28) the location in the ( ;p)-plane of the marginal stability curves can be numerically found for the viscoelastic P uger column without external damping, but for dierent values of the coecient of internal damping , Fig. 6(a). The thresholds tend to a limit which does not share common points with the stability boundary of the ideal column, as shown in Fig. 5(a), where this limit is set by the dot-dashed/green curve. The limiting curve calculated for = 1010agrees well with that obtained for = 103 in (Sugiyama et al., 1995; Ryu and Sugiyama, 2003). At the point = 0, the limit value of the critical utter load when the internal damping is approaching zero equals the well-known value for the Beck's column, p10:94. At==4 the limiting value becomes p7:91, while for the case of the Dzhanelidze column ( ==2) it becomes p7:49. An interesting question is what is the limit of the stability diagram for the P uger column in the (;p)-plane when the coecient of internal damping is kept null ( = 0), while the coecient of external damping ktends to zero. The answer to this question was previously known only for the Beck column ( = 0), for which it was established, both numerically (Bolotin and Zhinzher, 1969; Plaut and Infante, 1970) and analytically (Kirillov and Seyranian, 2005a), that the utter threshold of the externally damped Beck's column is higher than that obtained for the undamped Beck's column (tending to the ideal value p20:05, when the external damping tends to zero). This very particular example was at the basis of the common and incorrect opinion (maintained for decades until now) that the external damping is only a stabilizing factor, even for non- conservative loadings. Perhaps for this reason the eect of the external damping in the P uger column has, so far, simply been ignored. The evolution of the utter boundary for = 0 andktending to zero is illustrated by the blue curves in Fig. 6. It can be noted that the marginal stability boundary tends to a limiting curve which has two common tangent points with the stability boundary of the undamped P uger column, Fig. 5(b). One of the common points, at = 0 andp20:05, marked as point A, corresponds to the case of the Beck column. The other corresponds to 0:516 and p16:05, marked as point B. Only for these two `exceptional' mass ratios the critical utter load of the externally damped P uger column coincides with the ideal value when k!0. Remarkably, for all other mass ratios the limit of the critical utter load for the vanishing external damping is located below the ideal value, which means that the P uger column fully demonstrates the Ziegler destabilization paradox due to vanishing external damping , exactly as it does in the case of the vanishing internal damping, see Fig. 5(a), where the two limiting 13Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 curves are compared. Note that the discrepancy in case of vanishing external damping is smaller than in case of vanishing internal damping, in accordance with the analogous result that was established in Section 2 for the Ziegler pendulum with arbitrary mass distribution. As for the discrete case, also for the P uger column the utter instability threshold calculated in the limit when the external damping tends to zero has only two common points with the ideal marginal stability curve. The discrepancy is the most pronounced for the case of Dzhanelidze column at==2, where the critical load drops from p20:19 in the ideal case to p16:55 in the case of vanishing external damping. 4. Conclusions Since the nding of the Ziegler's paradox for structures loaded by nonconservative follower forces, internal damping (due to material viscosity) was considered a destabilizing factor, while external damping (due for instance to air drag resistance) was believed to merely provide a stabilization. This belief originates from results obtained only for the case of Beck's column, which does not carry an end mass. This mass is present in the case of the P uger's column, which was never analyzed before from the point of view of the Ziegler paradox. A revisitation of the Ziegler's pendulum and the analysis of the P uger column has revealed that the Ziegler destabilization paradox occurs as related to the vanishing of the external damping, no matter what is the ratio between the end mass and the mass of the structure. Results presented in this article clearly show that the destabilizing role of external damping was until now misunderstood, and that experimental proof of the destabilization paradox in a mechanical laboratory is now more plausible than previously thought. Moreover, the fact that external damping plays a destabilizing role may have important consequences in structural design and this opens new perspectives for energy harvesting devices. Acknowledgements The authors gratefully acknowledge nancial support from the ERC Advanced Grant In- stabilities and nonlocal multiscale modelling of materials FP7-PEOPLE-IDEAS-ERC-2013- AdG (2014-2019). References References Andreichikov, I. P., Yudovich, V. I., 1974. Stability of viscoelastic bars. Izv. AN SSSR. Mekh. Tv. Tela 9(2), 78{87. Banichuk, N. V., Bratus, A. S., Myshkis, A. D. 1989. Stabilizing and destabilizing eects in nonconservative systems. PMM USSR 53(2), 158{164. 14Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 Beck, M. 1952. Die Knicklast des einseitig eingespannten, tangential gedr uckten Stabes. Z. angew. Math. Phys. 3, 225. Bigoni, D., Noselli, G., 2011. Experimental evidence of utter and divergence instabilities induced by dry friction. J. Mech. Phys. Sol. 59, 2208{2226. Bolotin, V. V., 1963. Nonconservative Problems of the Theory of Elastic Stability. Pergamon Press, Oxford. Bolotin, V. V., Zhinzher, N. I., 1969. Eects of damping on stability of elastic systems subjected to nonconservative forces. Int. J. Solids Struct. 5, 965{989. Bottema, O., 1956. The Routh-Hurwitz condition for the biquadratic equation. Indag. Math. 18, 403{406. Chen, L. W., Ku, D. M., 1992. Eigenvalue sensitivity in the stability analysis of Beck's column with a concentrated mass at the free end. J. Sound Vibr. 153(3), 403{411. Crandall, S. H., 1995. The eect of damping on the stability of gyroscopic pendulums. Z. Angew. Math. Phys. 46, S761{S780. Detinko, F. M., 2003. Lumped damping and stability of Beck column with a tip mass. Int. J. Solids Struct. 40, 4479{4486. Done, G. T. S., 1973. Damping congurations that have a stabilizing in uence on non- conservative systems. Int. J. Solids Struct. 9, 203{215. Kirillov, O. N., 2011. Singularities in structural optimization of the Ziegler pendulum. Acta Polytechn. 51(4), 32{43. Kirillov, O. N., 2013. Nonconservative Stability Problems of Modern Physics. De Gruyter Studies in Mathematical Physics 14. De Gruyter, Berlin. Kirillov, O. N., Seyranian, A. P., 2005. The eect of small internal and external damping on the stability of distributed nonconservative systems. J. Appl. Math. Mech. 69(4), 529{552. Kirillov, O. N., Seyranian, A. P., 2005. Stabilization and destabilization of a circulatory system by small velocity-dependent forces, J. Sound Vibr. 283(35), 781{800. Kirillov, O. N., Seyranian, A. P., 2005. Dissipation induced instabilities in continuous non- conservative systems, Proc. Appl. Math. Mech. 5, 97{98. Kirillov, O. N., Verhulst, F., 2010. Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella? Z. Angew. Math. Mech. 90(6), 462{488. 15Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 Krechetnikov, R., Marsden, J. E., 2007. Dissipation-induced instabilities in nite dimensions. Rev. Mod. Phys. 79, 519{553. Oran, C., 1972. On the signicance of a type of divergence. J. Appl. Mech. 39, 263{265. Panovko, Ya. G., Sorokin, S. V., 1987. On quasi-stability of viscoelastic systems with the follower forces, Izv. Acad. Nauk SSSR. Mekh. Tverd. Tela. 5, 135{139. Pedersen, P. 1977. In uence of boundary conditions on the stability of a column under non-conservative load. Int. J. Solids Struct. 13, 445{455. P uger, A., 1955. Zur Stabilit at des tangential gedr uckten Stabes. Z. Angew. Math. Mech. 35(5), 191. Plaut, R. H., 1971. A new destabilization phenomenon in nonconservative systems. Z. Angew. Math. Mech. 51(4), 319{321. Plaut, R. H., Infante, E. F., 1970. The eect of external damping on the stability of Beck's column. Int. J. Solids Struct. 6(5), 491{496. Ryu, S., Sugiyama, Y., 2003. Computational dynamics approach to the eect of damping on stability of a cantilevered column subjected to a follower force. Comp. Struct. 81, 265{271. Saw, S. S., Wood, W. G., 1975. The stability of a damped elastic system with a follower force. J. Mech. Eng. Sci. 17(3), 163{176. Sugiyama, Y., Kashima, K., Kawagoe, H., 1976. On an unduly simplied model in the non-conservative problems of elastic stability. J. Sound Vibr. 45(2), 237{247. Sugiyama, Y., Katayama, K., Kinoi, S. 1995. Flutter of cantilevered column under rocket thrust. J. Aerospace Eng. 8(1), 9{15. Walker, J. A. 1973. A note on stabilizing damping congurations for linear non-conservative systems. Int. J. Solids Struct. 9, 1543{1545. Wang, G., Lin, Y. 1993. A new extension of Leverrier's algorithm. Lin. Alg. Appl. 180, 227{238. Zhinzher, N. I. 1994. Eect of dissipative forces with incomplete dissipation on the stability of elastic systems. Izv. Ross. Akad. Nauk. MTT 1, 149{155. Ziegler, H. 1952. Die Stabilit atskriterien der Elastomechanik. Archive Appl. Mech. 20, 49{56. 16Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 Appendix A. - The stabilizing role of external damping and the destabilizing role of internal damping A critical review of the relevant literature is given in this Appendix, with the purpose of explaining the historical origin of the misconception that the external damping introduces a mere stabilizing eect for structures subject to utter instability. Plaut and Infante (1970) considered the Ziegler pendulum with m1= 2m2, without internal damping (in the joints), but subjected to an external damping proportional to the velocity along the rigid rods of the double pendulum4. In this system the critical utter load increases with an increase in the external damping, so that they presented a plot showing that the utter load converges to a value which is very close to P u. However, they did not calculate the critical value in the limit of vanishing external damping, which would have revealed a value slightly smaller than the value corresponding to the undamped system5. In a subsequent work, Plaut (1971) conrmed his previous result and demonstrated that internal damping with equal damping coecients destabilizes the Ziegler pendulum, whereas external damping has a stabilizing eect, so that it does not lead to the destabilization paradox. Plaut (1971) reports a stability diagram (in the external versus internal damping plane) that implicitly indicates the existence of the Whitney umbrella singularity on the boundary of the asymptotic stability domain. These conclusions agreed with other studies on the viscoelastic cantilevered Beck's column (Beck, 1952), loaded by a follower force which displays the paradox only for internal Kelvin-Voigt damping (Bolotin and Zhinzher, 1969; Plaut and Infante, 1970; Andreichikov and Yudovich, 1974; Kirillov and Seyranian, 2005a) and were supported by studies on the abstract settings (Done, 1973; Walker, 1973; Kirillov and Seyranian, 2005b), which have proven the stabilizing character of external damping, assumed to be proportional to the mass (Bolotin, 1963; Zhinzher, 1994). The P uger column [a generalization of the Beck problem in which a concentrated mass is added to the loaded end, P uger (1955), see also Sugiyama et al. (1976), Pedersen (1977), and Chen and Ku (1992)] was analyzed by Sugiyama et al. (1995) and Ryu and Sugiyama (2003), who numerically found that the internal damping leads to the destabilization paradox for all ratios of the end mass to the mass of the column. The role of external damping was investigated only by Detinko (2003) who concludes that large external damping provides a stabilizing eect. The stabilizing role of external damping was questioned only in the work by Panovko and Sorokin (1987), in which the Ziegler pendulum and the Beck column were considered with a dash-pot damper attached to the loaded end (a setting in which the external damper can be seen as something dierent than an air drag, but as merely an additional structural 4Note that dierent mass distributions were never analyzed in view of external damping eect. In the absence of damping, stability investigations were carried out by Oran (1972) and Kirillov (2011). 5In fact, the utter load of the externally damped Ziegler pendulum with m1= 2m2, considered by Plaut and Infante (1970) and Plaut (1971) tends to the value P= 2 which is smaller than P u2:086, therefore revealing the paradox. 17Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 element, as suggested by Zhinzher (1994)). In fact the dash-pot was shown to always yield the destabilization paradox, even in the presence of internal damping, no matter what the ratio is between the coecients of internal and external damping (Kirillov and Seyranian, 2005c; Kirillov, 2013). In summary, there is a well-established opinion that external damping stabilizes struc- tures loaded by nonconservative positional forces. Appendix B. - A necessary condition for stabilization of a general 2 d.o.f. system Kirillov and Seyranian (2005b) considered the stability of the system Mx+"D_x+Kx= 0; (A.1) where">0 is a small parameter and M=MT,D=DT, and K6=KTare real matrices of ordern. In the case n= 2, the characteristic polynomial of the system (A.1), q(;") = det( M2+"D+K); can be written by means of the Leverrier algorithm (adopted for matrix polynomials by Wang and Lin (1993)) in a compact form: q(;") = det M4+"tr(DM)3+ (tr( KM) +"2detD)2+"tr(KD)+ det K;(A.2) where D=D1detDandK=K1detKare adjugate matrices and tr denotes the trace operator. Let us assume that at "= 0 the undamped system (A.1) with n= 2 degrees of freedom be on the utter boundary, so that its eigenvalues are imaginary and form a double complex- conjugate pair =i!0of a Jordan block. In these conditions, the real critical frequency !0at the onset of utter follows from q(;0) in the closed form (Kirillov, 2013) !2 0=r detK detM: (A.3) A dissipative perturbation "Dcauses splitting of the double eigenvalue i!0, which is described by the Newton-Puiseux series (") =i!0ip h"+o("), where the coecient his determined in terms of the derivatives of the polynomial q(;") as h:=dq d"1 2@2q @21 "=0;=i!0=tr(KD)!2 0tr(DM) 4i!0detM: (A.4) Since the coecient his imaginary, the double eigenvalue i!0splits generically into two com- plex eigenvalues, one of them with the positive real part yielding utter instability (Kirillov 18Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215 doi: http://dx.doi.org/10.1016/j.jmps.2016.03.011 and Seyranian, 2005b). Consequently, h= 0 represents a necessary condition for"Dto be astabilizing perturbation (Kirillov and Seyranian, 2005b). In the case of the system (3), with matrices (5), it is readily obtained !2 0=k l2pm1m2: (A.5) Assuming D=Di, eq. (A.4) and the representations (5) and (A.5) yield h=hi:=i m1l252p+ 1 4; (A.6) so that the equation hi= 0 has as solution the complex-conjugate pair = (34i)=25. Therefore, for every real mass distribution 0 the dissipative perturbation with the matrix D=Diof internal damping results to be destabilizing. Similarly, eq. (A.4) with D=Deand representations (A.5), (5), and F=F u() yield h=he:=il 48m18211p 36+ 5p 2; (A.7) so that the constraint he= 0 is satised only by the two following real values of  A0:273; C2:559: (A.8) The mass distributions (A.8) correspond exactly to the points A and C in Fig. 1, which are common for the utter boundary of the undamped system and for that of the dissipative system in the limit of vanishing external damping. Consequently, the dissipative perturbation with the matrix D=Deof external damping can have a stabilizing eect for only two particular mass distributions (A.8). Indeed, as it is shown in the present article, the external damping is destabilizing for every 0, except for =Aand=C. Consequently, the stabilizing or destabilizing eect of damping with the given matrix D is determined not only by its spectral properties, but also by how it `interacts' with the mass and stiness distributions. The condition which selects possibly stabilizing triples ( M,D, K) in the general case of n= 2 degrees of freedom is therefore the following tr(KD) =!2 0tr(DM): (A.9) 19

2303.03852v1.Electrically_tunable_Gilbert_damping_in_van_der_Waals_heterostructures_of_two_dimensional_ferromagnetic_metals_and_ferroelectrics.pdf

Page 1 of 15 Electrically tunable Gilbert damping in van der Waals heterostructures of two- dimensional ferromagnetic meta ls and ferroelectrics Liang Qiu,1 Zequan Wang,1 Xiao-Sheng Ni,1 Dao-Xin Yao1,2 and Yusheng Hou 1,* AFFILIATIONS 1 Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State Key Laboratory of Optoelectronic Materials and Technologies, Center for Neutron Science and Technology, School of Physics, Sun Yat-Sen University, Guangzhou, 510275, China 2 International Quantum Academy, Shenzhen 518048, China ABSTRACT Tuning the Gilbert damping of ferromagnetic (FM) metals via a nonvolatile way is of importance to exploit and design next-generation novel spintronic devices. Through systematical first-principles calculations, we study the magnetic properties of the van der Waals heterostructure of two-dimensional FM metal CrTe 2 and ferroelectric (FE) In2Te3 monolayers. The ferromagnetism of CrTe 2 is maintained in CrTe 2/In2Te3 and its magnetic easy axis can be switched from in-plane to out- of-plane by reversing the FE polarization of In 2Te3. Excitingly, we find that the Gilbert damping of CrTe 2 is tunable when the FE polarization of In 2Te3 is reversed from upward to downward. By analyzing the k-dependent contributions to the Gilbert damping, we unravel that such tunability results from the changed intersections between the bands of CrTe 2 and Fermi level on the reversal of the FE polarizations of In 2Te3 in CrTe 2/In2Te3. Our work provides a n appealing way to electrically tailor Gilbert dampings of two-dimensional FM metals by contacting them with ferroelectrics. *Authors to whom correspondence should be addressed: [Yusheng Hou, houysh@mail.sysu.edu.cn] This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 2 of 15 Since the atomically thin long-range ferromagnetic ( FM) orders at finite temperatures are discovered in CrI 31 monolayer (ML) and Cr 2Ge2Te62 bilayer, two- dimensional (2D) van der Waals (vdW) FM materials have attracted intensive attention.3-5 Up to now, many novel vdW ferromagnets such as Fe 3GeTe 2,6 Fe5GeTe 2,7 VSe 28,9 and MnSe 210 have been synthesized in experiments. Due to the intrinsic ferromagnetism in these vdW FM materials, it is highly fertile to engineer emergent phenomena through magnetic proximity effect in their heterostructures.11 For instance , an unprecedented control of the spin and valley pseudospins in WSe 2 ML is reported in CrI 3/WSe 2.12 By contacting the thin films of three-dimensional topological insulators and graphene with CrI 3, high-temperature quantum anomalous Hall effect and vdW spin valves are proposed in CrI 3/Bi2Se3/CrI 313 and CrI 3/graphene/CrI 3,14 respectively. On the other hand, the magnetic properties of these vdW FM materials can also be controlled by means of external perturbations such as gating and moiré patterns.3 In CrI 3 bilayer, Huang et al. observed a voltage-controlled switching between antiferromagnetic (AFM) and FM states.15 Via an ionic gate, Deng et al. even increased the Curie temperature (TC) of the thin flake of vdW FM metal Fe 3GeTe 2 to room temperature, which is much higher than its bulk TC.6 Very recently, Xu et al. demonstrated a coexisting FM and AFM state in a twisted bilayer CrI 3.16 These indicate that vdW FM materials are promising platforms to design and implement spintronic devices in the 2D limit.4,11 Recently, of great interest is the emergent vdW magnetic material CrTe 2 which is a new platform for realizing room-temperature intrinsic ferromagnetism.17,18 Especially, CrTe 2 exhibits greatly tunable magneti sm. In the beginning, its ground state is believed to be the nonmagnetic 2 H phase,19 while several later researches suggest that either the FM or AFM 1 T phases should be the ground state of CrTe 2.17,18,20- 23 Currently, the consensus is that the structural ground state of CrTe 2 is the 1 T phase. With respect to its magnetic ground state, a first-principles study shows that the FM and AFM ground states in CrTe 2 ML depend on its in-plane lattice constants.24 It is worth noting that the TC of FM CrTe 2 down to the few-layer limit can be higher than 300 K,18 making it have wide practical application prospects in spintronics. This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 3 of 15 Building heterostructures of FM and ferroelectric (FE) materials offers an effective way to control nonvolatile magnetism via an electric field. Experimentally, Eerenstein et al. presented an electric-field-controlled nonvolatile converse magnetoelectric effect in a multiferroic heterostructure La0.67Sr0.33MnO 3/BaTiO 3.25 Later, Zhang et al. reported an electric-field-driven control of nonvolatile magnetization in a heterostructure of FM amorphous alloy Co40Fe40B20 and FE Pb(Mg 1/3Nb2/3)0.7Ti0.3O3.26 Theoretically, Chen et al. demonstrated based on first-principles calculations that the interlayer magnetism of CrI 3 bilayer in CrI 3/In2Se3 is switchable between FM and AFM couplings by the nonvolatile control of the FE polarization direction of In 2Se3.27 In spite of these interesting findings, using FE substrates to electrically tune the Gilbert damping of ferromagnets, an important factor determining the operation speed of spintronic devices, is rarely investigated in 2D FM/FE vdW heterostructures. Therefore, it is of great importance to explore the possibility of tuning the Gilbert damping in such kind of heterostructures. In this work, we first demonstrate that the magnetic ground state of 1 T-phase CrTe 2 ML will change from the zigzag AFM (denoted as z-AFM) to FM orders with increasing its in-plane lattice constants. By building a vdW heterostructure of CrTe 2 and FE In2Te3 MLs, we show that the magnetic easy axis of CrTe 2 can be tuned from in-plane to out- of-plane by reversing the FE polarization of In 2Te3, although its ferromagnetism is kept . Importantly, we find that the Gilbert damping of CrTe 2 is tunable with a wide range on reversing the FE polarization of In 2Te3 from upward to downward. Through looking into the k-dependent contributions to the Gilbert damping, we reveal that such tunability originates from the changed intersections between the bands of CrTe 2 and Fermi level when the FE polarizations of In 2Te3 is reversed in CrTe 2/In2Te3. Our work demonstrates that putting 2D vdW FM metals on FE substrates is an attractive method to electrically tune their Gilbert dampings. CrTe 2, a member of the 2D transition metal dichalcogenide family, can potentially crystalize into several different layered structures such as 1 T, 1Td, 1H and 2 H phases.28 It is believed that the 1 T phase is the most stable among all of the se possible phases in This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 4 of 15 both bulk and ML. This phase has a hexagonal lattice and belongs to the P3_m1 space group, with each Cr atom surrounded by the octahedrons of Te atoms (Fig. 1a). In view of the hot debates on the magnetic ground state in CrTe 2 ML, we establish a 2× 2√3 supercell and calculate the total energies of several different magnetic structures (Fig. S1 in Supplementary Materials) when its lattice constant varies from 3.65 to 4.00 Å. As shown in Fig. 1b, our calculations show that z-AFM order is the magnetic ground state when the lattice constant is from 3.65 to 3.80 Å. By contrast, the FM order is the magnetic ground state when the lattice constant is in the range from 3.80 to 4.00 Å. Note that our results are consistent with the experimentally observed z- AFM23 and FM29 orders in CrTe 2 with a lattice constant of 3.70 and 3.95 Å, respectively. Since we are interested in the Gilbert damping of ferromagnets and the experimentally grow n CrTe 2 on ZrTe 2 has a lattice constant of 3.95 Å,29 we will focus on CrTe 2 ML with this lattice constant hereinafter. FIG. 1. (a) Side (the top panel) and top (the bottom panel) views of CrTe 2 ML. The NN and second- NN exchange paths are shown by red arrows in the top view. (b) The phase diagram of the magnetic ground state of CrTe 2 ML with different lattice constants. Insets show the schematic illustrations of the z-AFM and FM orders. The up and down spins are indicated by the blue and red balls, respectively. The stars highlight the experimental lattice constants of CrTe 2 in Ref.23 and Ref.29. This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 5 of 15 To obtain an deeper understanding on the ferromagnetism of CrTe 2 ML, we adopt a spin Hamiltonian consisting of Heisenberg exchange couplings and single-ion magnetic anisotropy (SIA) as follows:30 𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2 𝑖 (1) In Eq. (1), J1 and J2 are the nearest neighbor (NN) and second- NN Heisenberg exchange couplings. Note that a negative (positive) J means a FM (AFM) Heisenberg exchange couples. Besides, A parameterizes the SIA term. First of all, our DFT calculations show that the magnetic moment of CrTe 2 ML is 3.35 μB/Cr, consistent with previous DFT calculations.31 As shown in Table I, the calculated J1 and J2 are both FM and J1 is much stronger than J2. Both FM J1 and J2 undoubtedly indicate that CrTe 2 ML has a FM magnetic ground state. Finally, the SIA parameter A is obtained by calculating the energy difference between two FM states with out-of-plane and in-plane magnetizations. Our calculations obtain A=1.81 meV/Cr, indicating that CrTe 2 ML has an out-of-plane magnetic easy axis. Hence, our calculations show that CrTe 2 ML exhibits an out-of- plane FM order, consistent with experimental observations.29 TABLE I. Listed are the in-plane lattice constant s a, Heisenberg exchange couplings J (in unit of meV) and SIA (in unit of meV/Cr) of CrTe 2 ML and CrTe 2/In2Te3. System a (Å) J1 J2 A CrTe 2 3.95 -24.56 -0.88 1.81 CrTe 2/In2Te3(↑) 7.90 -20.90 -1.80 -1.44 CrTe 2/In2Te3(↓) 7.90 -19.33 -0.88 0.16 To achieve an electrically tunable Gilbert damping in CrTe 2 ML, we establish its vdW heterostructure with F E In2Te3 ML. In building this heterostructure, w e stack a 2×2 supercell of CrTe 2 and a √3 ×√3 supercell of In2Te3 along the (001) direction. Because the magnetic properties of CrTe 2 ML are the primary topic and the electronic properties of In2Te3 ML are basically not affected by a strain (Fig. S2), we stretch the lattice constant of the latter to match that of the former. Fig. 2a shows the most stable This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 6 of 15 stacking configuration in CrTe 2/In2Te3 with an upward FE polarization [denoted as CrTe 2/In2Te3(↑)]. At the interface in this configuration, one of four Cr atoms and one of four Te atoms at the bottom of CrTe 2 sits on the top of the top-layer Te atom s of In2Te3. In CrTe 2/In2Te3 with a downward FE polarization [denoted as CrTe 2/In2Te3(↓ )], the stacking configuration at its interface is same as that in CrTe 2/In2Te3(↑ ). The only difference between CrTe 2/In2Te3(↑ ) and CrTe 2/In2Te3(↓ ) is that the middle-layer Te atoms of In 2Te3 in the former is farther to CrTe 2 than that in the latter (Fig. 2a and 2c). It is noteworthy that the bottom-layer Te atoms of CrTe 2 do not stay at a plane anymore in the relaxed CrTe 2/In2Te3 (see more details in Fig. S3), suggesting non-negligible interactions between CrTe 2 and In 2Te3. FIG. 2. (a) The schematic stacking configuration and (b) charge density difference 𝛥ρ of CrTe 2/In2Te3(↑). (c) and (d) same as (a) and (b) but for CrTe 2/In2Te3(↓). In (b) and (d), color bar indicates the weight of negative (blue) and positive (red) charge density differences. (e) The total DOS of CrTe 2/In2Te3. (f) and (g) show the PDOS of CrTe 2 and In2Te3 in CrTe 2/In2Te3, respectively. In (e)-(g), upward and downward polarizations are indicated by black and red lines, respectively. To shed light on the effect of the FE polarization of In 2Te3 on the electronic property of CrTe 2/In2Te3, we first investigate the spatial distribution of charge density This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 7 of 15 difference 2 2 3 2 2 3 CrTe In Te CrTe In Te = − − with different FE polarization directions. As shown in Fig. 2b and 2d, we see that there is an obvious charge transfer at the interfaces of both CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓), which is further confirmed by the planar averaged 𝛥ρ (Fig. S4). Additionally, the charge transfer in CrTe 2/In2Te3(↑) is distinctly less than th at in CrTe 2/In2Te3(↓). Fig. 2e shows that the total density of states (DOS) near Fermi level are highly different in CrTe 2/In2Te3(↑ ) and CrTe 2/In2Te3(↓). By projecting the DOS onto CrTe 2 and In 2Te3, Fig. 2f shows that the projected DOS (PDOS) of CrTe 2 in CrTe 2/In2Te3(↑) is larger than that in CrTe 2/In2Te3(↓) at Fermi level. Interestingly, the PDOS of In 2Te3 in CrTe 2/In2Te3(↑) is larger than that in CrTe 2/In2Te3(↓) below Fermi level while the situation is inversed above Fermi level (Fig. 2g). By looking into the five Cr- d orbital projected DOS in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) (Fig. S5), we see that there are obviously different occupations for xyd, 22xyd− and 223zrd− orbitals near Fermi level. All of these imply that the reversal of the FE polarization of In 2Te3 may have an unignorable influence on the magnetic properites of CrTe 2/In2Te3. Due to the presence of the FE In 2Te3, the inversion symmetry is inevitably broken and nonzero Dzyaloshinskii-Moriya interactions (DMIs) may exist in CrTe 2/In2Te3. In this case, we add a DMI term into Eq. (1) to investigate the magnetism of CrTe 2/In2Te3 and the corresponding spin Hamiltonian is in the form of32 𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ +∑ 𝑫𝑖𝑗⋅ (𝑆 𝑖× 𝑆 𝑗) ⟨𝑖𝑗⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2 𝑖 (2). In Eq. (2), Dij is the DMI vector of the NN Cr-Cr pairs. As the C6-rotational symmetry with respect to Cr atoms in CrTe 2 is reduced to the C3-rotational symmetry, the NN DMIs are split into four different DMIs (Fig. S6). For simplicity, the J1 and J2 are still regard ed to be six-fold. From Table I, we see that the NN J1 of both CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) are still FM but slightly smaller than that of free-standing CrTe 2 ML. Moreover, the second- NN FM J2 is obviously enhanced in CrTe 2/In2Te3(↑) compared with CrTe 2/In2Te3(↓) and free-standing CrTe 2 ML. To calculated the NN DMIs, we build a √3×√3 supercell of CrTe 2/In2Te3 and the four-state method33 is employed here. As This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 8 of 15 listed in Table S1, the FE polarization direction of In 2Te3 basically has no qualitative effect on the DMIs in CrTe 2/In2Te3 although it affects their magnitudes. More explicitly, the magnitudes of the calculated DMIs range from 1.22 to 2.81 meV, which are about one order smaller than the NN J1. Finally, we find that the SIA of CrTe 2/In2Te3 is strongly dependent on the FE polarization of In 2Te3. When In 2Te3 has an upward FE polarization, the SIA of CrTe 2/In2Te3(↑) is negative, indicating an in-plane magnetic easy axis. However, when the FE polarization of In 2Te3 is downward, CrTe 2/In2Te3(↓) has a positive SIA, indicating an out-of-plane magnetic easy axis. It is worth noting that CrTe 2/In2Te3(↓) has a much weak SIA than the free-standing CrTe 2 ML, although they both have positive SIAs. The different Heisenberg exchange couplings, DMIs and SIAs in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) clearly unveil that the magnetic properties of CrTe 2 are tuned by the FE polarization of In2Te3. To obtain the magnetic ground state of CrTe 2/In2Te3, MC simulations are carried out. As shown in Fig. S7, CrTe 2/In2Te3(↑) has an in-plane FM magnetic ground state whereas CrTe 2/In2Te3(↓ ) has an out-of-plane one. Such magnetic ground states are understandable. Firstly, the ratios between DMIs and the NN Heisenberg exchange couplings are small and most of them are out of the typical range of 0.1–0.2 for the appearance of magnetic skyrmions.34 Secondly, the SIAs of the CrTe 2/In2Te3(↑) and the CrTe 2/In2Te3(↓ ) prefer in-plane and out-of-plane magnetic easy axes, respectively . Taking them together, we obtain that the FM Heisenberg exchange couplings dominate over the DMIs and thus give rise to a FM magnetic ground state with its magnetization determined by the SIA,35 consistent with our MC simulated results. Figure 3a shows the Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward and downward FE polarizations of In2Te3. Similar to previous studies,36,37 the Gilbert dampings of CrTe 2/In2Te3 decrease first and then increase as the scattering rate Γ increases. Astonishingly, the Gilbert dampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) are distinctly different at the same scattering rate Γ ranging from 0.001 to 1.0 eV . To have a more intuitive sense on the effect of the FE polarizations of In 2Te3 on the Gilbert dampings in CrTe 2/In2Te3, we calculate the ratio = at any given Γ, where This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 9 of 15 ( ) is the Gilbert damping of CrTe 2/In2Te3(↑) [CrTe 2/In2Te3(↓)]. As shown in Fig. 3b, the ratio 𝜂 ranges from 6 to around 1.3 with increasing Γ. As the FE polarization of In 2Te3 can be switched from upward to downward by an external electric field, the Gilbert damping of CrTe 2/In2Te3 is electrically tunable in practice. FIG. 3. (a) The Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward (black line) and downward (red line) FE polarizations of In 2Te3. (b) The Gilbert damping ratio 𝜂 as a function of the scattering rate Γ. To gain a deep insight into how the FE polarization of In 2Te3 tunes the Gilbert damping in CrTe 2/In2Te3, we investigate the k-dependent contributions to the Gilbert dampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). As shown in Fig. 4a and 4b, the bands around Fermi level have qiute different intermixing between CrTe 2 and In 2Te3 states when the FE polarizaiton of In 2Te3 is reversed. Explicitly, there are obvious intermixing below Fermi leve in CrTe 2/In2Te3(↑) while the intermixing mainly takes place above Fermi level in CrTe 2/In2Te3(↓). Especially, the bands intersected by Fermi level are at different k points in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). Through looking into the k- dependent contributions to the ir Gilbert dampings (Fig. 4c and 4d), we see that large contributions are from the k points (highlighted by arrows in Fig. 4) at which the bands of CrTe 2 cross Fermi level. In addition, these large contributions are different. Such k- dipendent contribution to Gilbert dampings is understandable. Based on the scattering This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 10 of 15 theory of Gilbert damping 38, Gilbert damping parameter is calculated using the following Eq. (3) 36 ( ) ( ) , , , , , , (3)kk k i k j k j k i F k i F k j k ij SE E E EM u u = − − − HH, where EF is Fermi level and Ek,i is the enery of band i at a given k point. Due to the delta ( ) ( ) ,, F k i F k jE E E E −− , only the valence and conduction bands near Fermi level make dominant contribution to the Gilbert damping. Additionally, their contributions also depend on factor , , , ,kk k i k j k j k iuu HH. Overall , through changing the intersections between the bands of CrTe 2 and Fermi level, the reversal of the FE polarization of In 2Te3 can modulate the contributions to Gilbert damping. Consequently, the total Gilbert dampings are different in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). FIG. 4. (a) Band structure calculated with spin-orbit coupling and (c) the k-dependent contributions to the Gilbert damping in CrTe 2/In2Te3(↑). (b) and (d) same as (a) and (c) but for CrTe 2/In2Te3(↓). In (a) and (b), Fermi levels are indicated by horizontal dash lines and the states from CrTe 2 and In 2Te3 are shown by red and blue, respectively. From experimental perspectives, the fabrication of CrTe 2/In2Te3 vdW heterostructure should be feasible. On the one hand, CrTe 2 with the lattice constant of This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 11 of 15 3.95 Å has been successfully grown on ZrTe 2 substrate by the molecular beam epitaxy.29 On the other hand, In 2Te3 is also synthesized.39 Taking these and the vdW nature of CrTe 2 and In 2Te3 together, a practical scheme of growing CrTe 2/In2Te3 is sketched in Fig. S8 : first grow CrTe 2 ML on ZrTe 2 substrate29 and then put In2Te3 ML on CrTe 2 to form the desired CrTe 2/In2Te3 vdW heterostructure. In summary, by constructing a vdW heterostructure of 2D FM metal CrTe 2 and FE In2Te3 MLs, we find that the magnetic properties of CrTe 2 are engineered by the reversal of the FE polariton of In 2Te3. Although the ferromagnetism of CrTe 2 is maintained in the presence of the FE In2Te3, its magnetic easy axis can be tuned from in-plane to out- of-plane by reversing the FE polarization of In 2Te3. More importantly, the Gilbert damping of CrTe 2 is tunable with a wide range when reversing the FE polarization of In2Te3 from upward to downward. Such tunability of the Gilbert damping in CrTe 2/In2Te3 results from the changed intersections between the bands of CrTe 2 and Fermi level on reversing the FE polarizations of In 2Te3. Our work introduces a remarkably useful method to electrically tune the Gilbert dampings of 2D vdW FM metals by contacting them with ferroelectrics, and should stimulate more experimental investigations in this realm. See the supplementary material for the details of computational methods31,36,40- 50 and other results mentioned in the main text. This project is supported by National Nature Science Foundation of China (No. 12104518, 92165204, 11974432), NKRDPC-2018YFA0306001, NKRDPC- 2022YFA1402802, GBABRF-2022A1515012643 and GZABRF-202201011118 . Density functional theory calculations are performed at Tianhe- II. AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 12 of 15 Author Contributions Liang Qiu : Investigation (equal); Methodology (equal); Writing –original draft (equal). Zequan Wang : Methodology (equal). Xiao -sheng Ni : Investigation (equal); Methodology (equal). Dao-Xin Yao : Supervision (equal); Funding acquisition (equal); Investigation (equal); Writing – review &editing (equal). Yusheng Hou : Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration(equal); Resources (equal); Supervision (equal); Writing – review &editing (equal). DATA A V AILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1 B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, and D. H. Cobden, Nature 546, 270 (2017). 2 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). 3 K. S. Burch, D. Mandrus, and J.-G. Park, Nature 563, 47 (2018). 4 C. Gong and X. Zhang, Science 363, eaav4450 (2019). 5 Q. H. Wang, A. Bedoya-Pinto, M. Blei, A. H. Dismukes, A. Hamo, S. Jenkins, M. Koperski, Y . Liu, Q.-C. Sun, and E. J. Telford, ACS nano 16, 6960 (2022). 6 Y . Deng, Y . Yu, Y . Song, J. Zhang, N. 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However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401

1506.00723v1.Current_Driven_Motion_of_Magnetic_Domain_Wall_with_Many_Bloch_Lines.pdf

Journal of the Physical Society of Japan LETTERS Current-Driven Motion of Magnetic Domain Wall with Many Bloch Lines Junichi Iwasaki1and Naoto Nagaosa1;2y 1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan 2RIKEN Center for Emergent Matter Science (CEMS),Wako, Saitama 351-0198, Japan The current-driven motion of a domain wall (DW) in a ferromagnet with many Bloch lines (BLs) via the spin transfer torque is studied theoretically. It is found that the motion of BLs changes the current-velocity (j-v) characteristic dramatically. Especially, the critical current density to overcome the pinning force is reduced by the factor of the Gilbert damping coecient even compared with that of a skyrmion. This is in sharp contrast to the case of magnetic eld driven motion, where the existence of BLs reduces the mobility of the DW. Domain walls (DWs) and bubbles1,2)are the spin tex- tures in ferromagnets which have been studied inten- sively over decades from the viewpoints of both funda- mental physics and applications. The memory functions of these objects are one of the main focus during 70's, but their manipulation in terms of the magnetic eld faced the diculty associated with the pinning which hinders their motion. The new aspect introduced recently is the current-driven motion of the spin textures.3,4)The ow of the conduction electron spins, which follow the direc- tion of the background localized spin moments, moves the spin texture due to the conservation of the angu- lar momentum. This eect, so called the spin transfer torque, is shown to be eective to manipulate the DWs and bubbles compared with the magnetic eld. Magnetic skyrmion5,6)is especially an interesting object, which is a swirling spin texture acting as an emergent particle protected by the topological invariant, i.e., the skyrmion numberNsk, dened by Nsk=1 4Z d2rn(r)
@n(r) @x@n(r) @y (1) with n(r) being the unit vector representing the direc- tion of the spin as a function of the two-dimensional spa- tial coordinates r. This is the integral of the solid angle subtended by n, and counts how many times the unit sphere is wrapped. The solid angle and skyrmion number Nskalso play essential role when one derives the equation of motion for the center of mass of the spin texture, i.e., the gyro-motion is induced by Nskin the Thiele equation, where the rigid body motion is assumed.7,8) Beyond the Thiele equation,7)one can derive the equa- tion of motion of a DW in terms of two variables, i.e., the wall-normal displacement q(t;; ) and the wall- magnetization orientation angle (t;; ) (see Fig. 1) iwasaki@appi.t.u-tokyo.ac.jp ynagaosa@ap.t.u-tokyo.ac.jp ψqFig. 1. Schematic magnetization distribution of DW with many Bloch lines. whereandare general coordinates specifying the point on the DW:9)   = 2M 1h _q
_ vs ?
vs k(@k )i ;(2)  q=2M 1h _ +
1_q+vs k(@k )
1vs ?i ; (3) Here, _ means the time-derivative. kand?indicate the components parallel and perpendicular to the DW respectively. Mis the magnetization, is the gyro- magnetic ratio, and ,
are the energy per area and thickness of the DW. vsis the velocity of the conduction electrons, which produces the spin transfer torque. is the Gilbert damping constant, and represents the non- adiabatic eect. These equations indicate that qand are canonical conjugate to each other. This is understood by the fact that the generator of the spin rotation nor- mal to the DW, which is proportional to sin in Fig. 1, drives the shift of q. (Note that is measured from the xed direction in the laboratory coordinates.) In order to reduce the magnetostatic energy, the spins in the DW tend to align parallel to the DW, i.e., Bloch wall. When the DW is straight, this structure is coplanar and has no solid angle. From the viewpoint of eqs. (2) 1arXiv:1506.00723v1 [cond-mat.mes-hall] 2 Jun 2015J. Phys. Soc. Jpn. LETTERS and (3), the angle is xed around the minimum, and slightly canted when the motion of qoccurs, i.e., _ = 0. However, it often happens that the Bloch lines (BLs) are introduced into the DW as shown schematically in Fig. 1. The angle rotates along the DW and the N eel wall is locally introduced. It is noted here that the solid angle becomes nite in the presence of the BLs. Also with many BLs in the DW, the translation of BLs activates the motion of the angle , i.e., _ 6= 0, which leads to the dramatic change in the dynamics. In the following, we focus on the straight DW which extends along x-direction and is uniform in z-direction. Thus, the general coordinates here are ( ;) = (x;z). q(t;x;z ) is independent of the coordinates q(t;x;z ) = q(t), and the functional derivative =q in eq. (3) be- comes the partial derivative @=@q . In the absence of BLs, we set (t;x;z ) = (t), and= in eq. (2) also becomes@=@ . Then the equation of motion in the ab- sence of BL is @ @ = 2M 1h _q
_ vs ?i ; (4) @ @q=2M 1h _ +
1_q
1vs ?i ; (5) With many BLs, the sliding motion of Bloch lines along DW, which activates _ , does not change the wall energy, i.e.,= in eq. (2) vanishes.2)Here, for simplicity, we consider the periodic BL array with the uniform twist (t;x;z ) = (xp(t))=~
where ~
is the distance between BLs, which leads to 0 = 2M 1h _q+
~
1_pvs ?
~
1vs ki ;(6) @ @q=2M 1h ~
1_p+
1_q+~
1vs k
1vs ?i ; (7) First, let us discuss the magnetic eld driven motion without current. The eect of the external magnetic eld Hextis described by the force @=@q =2MHextin eqs. (5) and (7). vs kandvs ?are set to be zero. In the absence of BL, as mentioned above, the phase is static _ = 0 with the slight tilt of the spin from the easy-plane, and one obtains from eq. (5) _q=
Hext : (8) This is a natural result, i.e., the mobility is inversely proportional to the Gilbert damping . is determined by eq. (4) with this value of the velocity _ q. In the presence of many BLs, eqs. (6) and (7) give the velocities of DW and BL sliding driven by the magnetic eld as _q= 1 +2
Hext; (9)_p=1 1 +2~
Hext: (10) Comparing eqs. (8) and (9), the mobility of the DW is re- duced by the factor of 2sinceis usually much smaller than unity. We also note that the velocity of the BL slid- ing _pis larger than that of the wall _ qby the factor of . Physically, this means that the eect of the external magnetic eld Hextmostly contributes to the rapid mo- tion of the BLs along the DW rather than the motion of the DW itself. These results have been already reported in refs.2,9,10) Now let us turn to the motion induced by the current vs. In the absence of BL, again we put _ = 0 in eqs. (4) and (5). Assuming that there is no pinning force or ex- ternal magnetic eld, i.e., @=@q = 0, one obtains from eq. (5) _q= vs ?; (11) and eq. (4) determines the equilibrium value of . When the pinning force @=@q =Fpinis nite, there appears a threshold current density ( vs ?)cwhich is determined by putting _q= 0 in eq. (5) as (vs ?)c=
2MFpin; (12) which is inversely proportional to .11)Since eq. (11) is independent of vs k, the threshold current density vs k c is vs k c=1. In the presence of the many BLs, on the other hand, eqs. (6) and (7) give @ @q=2M 11 +2 
1_q 1 + 
1vs ? ~
1vs k ; (13) which is the main result of this paper. From eq. (13), the current-velocity characteristic in the absence of both the pinning and the external eld ( @=@q =0) is _q=1 + 1 +2vs ? 1 +2
~
1vs k 'vs ?+ ()
~
1vs k; (14) where the fact ;1 is used in the last step. If we neglect the term coming from vs k, the current-velocity relation becomes almost independent of andin sharp contrast to eq. (11). This is similar to the univer- sal current-velocity relation in the case of skyrmion,12) where the solid angle is nite and also the transverse motion to the current occurs. Note that vs kslightly con- tributes to the motion when 6=, while it does not in the absence of BL. Even more dramatic is the critical current density in the presence of the pinning 2J. Phys. Soc. Jpn. LETTERS 30 20 101525 520 1015 5 3.0 2.0 1.01.52.5 0.50.6 0.4 0.20.30.5 0.10.4 0.2 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6 4000 2000 10000 8000 6000 4000 2000 10000 8000 6000qq qq t tt tw/o BL w/ BLs0.429 0.707Pinning q(a) (c) (d)(b) Fig. 2. The wall displacement qas a fucntion of tfor the DWs without BL and with BLs. (a) vs ?= 22 :0. The inset shows the pinning force Fpin. (b) vs ?= 21 :0. (c) vs ?= 0:0043. (d) vs ?= 0:0042. (@=@q =Fpin). When we apply only the current per- pendicular to the DW, i.e., vs k= 0, putting _ q= 0 in eq. (13) determines the threshold current density as (vs ?)c=
2M 1 +Fpin; (15) which is much reduced compared with eq. (12) by the factor of 1+1. Note that ( vs ?)cin eq. (15) is even smaller than the case of skyrmion12)by the factor of . Similarly, the critical current density of the motion driven byvs kis given by  vs k c= ~
2M jjFpin; (16) which can also be smaller than eq. (12). Next we look at the numerical solutions of q(t) driven by the current vs ?perpendicular to the wall under the pinning force. We assume the following pinning force: (
=2M)Fpin(q) =v(q=
) exp (q=
)2 (see the in- set of Fig. 2(a)). We employ the unit of
= v= 1 and the parameters ( ;) are xed at ( ;) = (0:01;0:02). Here, we compare two DWs without BL and with BLs. The maximum value of the pinning force (
=2M)Fpin max= 0:429 determines the threshold current density (vs ?)cas (vs ?)c= 21:4 and (vs ?)c= 0:00429 in the absence of BL and in the presence of many BLs, respec- tively. In Fig. 2(a), both DWs overcome the pinning at the current density vs ?= 22:0, although the velocity of the DW without BL is suppressed in the pinning poten- tial. At the current density vs ?= 21:0 below the threshold value in the absence of BL, the DW without BL is pinned, while that with BLs still moves easily (Fig. 2(b)). The velocity suppression in the presence of BLs is observed at much smaller current density vs ?= 0:0043 (Fig. 2(c)), and nally it stops at vs ?= 0:0042 (Fig. 2(d)). All the discussion above relies on the assumption thatthe wall is straight and rotates uniformly. When the bending of the DW and non-uniform distribution of BLs are taken into account, the average velocity and the threshold current density take the values between two cases without BL and with many BLs. The situation changes when the DW forms closed loop, i.e., the do- main forms a bubble. The bubble with many BLs and largejNskjis called hard bubble because the repulsive interaction between the BLs makes it hard to collapse the bubble.2)At the beginning of the motion, the BLs move along the DW, which results in the tiny critical cur- rent. In the steady state, however, the BLs accumulate in one side of the bubble.13,14)Then, the conguration of the BLs is static and the Thiele equation is justied as long as the force is slowly varying within the size of the bubble. The critical current density ( vs)cis given by (vs)c/Fpin=Nsk(Nsk(1): the skyrmion number of the hard bubble), and is reduced by the factor of Nsk compared with the skyrmion with Nsk=1. In conclusion, we have studied the current-induced dynamics of the DW with many BLs. The nite _ in the steady motion activated by BLs sliding drastically changes the dynamics, which has already been reported in the eld-driven case. In contrast to the eld-driven case, where the mobility is suppressed by introducing BLs, that in the current-driven motion is not necessarily suppressed. Instead, the current-velocity relation shows universal behavior independent of the damping strength and non-adiabaticity . Furthermore, the threshold current density in the presence of impurities is tiny even compared with that of skyrmion motion by the factor of . These ndings will stimulate the development of the racetrack memory based on the DW with many BLs. Acknowledgments We thank W. Koshibae for useful discus- sion. This work is supported by Grant-in-Aids for Scientic Re- search (S) (No. 24224009) from the Ministry of Education, Cul- ture, Sports, Science and Technology of Japan. J. I. was supported by Grant-in-Aids for JSPS Fellows (No. 2610547). 1) A. Hubert and R. Sch afer, Magnetic Domains: The Analysis of Magnetic Microstructures (Springer-Verlag, Berlin, 1998). 2) A. P. Malozemo and J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979). 3) J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1{L7 (1996). 4) L. Berger, Phys. Rev. B 54, 9353{9358 (1996). 5) S. M uhlbauer et al., Science 323, 915 (2009). 6) X. Z. Yu et al., Nature 465, 901 (2010). 7) A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 8) K. Everschor et al., Phys. Rev. B 86, 054432 (2012). 9) J. C. Slonczewski, J. Appl. Phys. 45, 2705 (1974). 10) A. P. Malozemo and J. C. Slonczewski, Phys. Rev. Lett. 29, 952 (1972). 11) G. Tatara et al., J. Phys. Soc. Japan 75, 64708 (2006). 12) J. Iwasaki, M. Mochizuki and N. Nagaosa, Nat. Commun. 4, 1463 (2013). 13) G. P. Vella-Coleiro, A. Rosencwaig and W. J. Tabor, Phys. Rev. Lett. 29, 949 (1972) 14) A. A. Thiele, F. B. Hagedorn and G. P. Vella-Coleiro, Phys. 3J. Phys. Soc. Jpn. LETTERS Rev. B 8, 241 (1973). 4

2308.05955v2.Dynamical_Majorana_Ising_spin_response_in_a_topological_superconductor_magnet_hybrid_by_microwave_irradiation.pdf

Dynamical Majorana Ising spin response in a topological superconductor-magnet hybrid by microwave irradiation Yuya Ominato,1, 2Ai Yamakage,3and Mamoru Matsuo1, 4, 5, 6 1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China. 2Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050, Japan. 3Department of Physics, Nagoya University, Nagoya 464-8602, Japan 4CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan 6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: March 20, 2024) We study a dynamical spin response of surface Majorana modes in a topological superconductor- magnet hybrid under microwave irradiation. We find a method to toggle between dissipative and non-dissipative Majorana Ising spin dynamics by adjusting the external magnetic field angle and the microwave frequency. This reflects the topological nature of the Majorana modes, enhancing the Gilbert damping of the magnet, thereby, providing a detection method for the Majorana Ising spins. Our findings illuminate a magnetic probe for Majorana modes, paving the path to innovative spin devices. Introduction.— The quest for Majoranas within matter stands as one of the principal challenges in the study of condensed matter physics, more so in the field of quan- tum many-body systems [1]. The self-conjugate nature of Majoranas leads to peculiar electrical characteristics that have been the subject of intensive research, both theoretical and experimental [2]. In contrast, the focus of this paper lies on the magnetic properties of Majoranas, specifically the Majorana Ising spin [3–8]. A distinctive characteristic of Majorana modes, appearing as a surface state in topological superconductors (TSC), is its exceed- ingly strong anisotropy, which makes it behave as an Ising spin. In particular, this paper proposes a method to ex- plore the dynamical response of the Majorana Ising spin through the exchange interaction at the magnetic inter- face, achieved by coupling the TSC to a ferromagnet with ferromagnetic resonance (FMR) (as shown in Fig.1 (a)). FMR modulation in a magnetic hybrid system has at- tracted much attention as a method to analyze spin ex- citations in thin-film materials attached to magnetic ma- terials [9, 10]. Irradiating a magnetic material with mi- crowaves induces dynamics of localized spin in magnetic materials, which can excite spins in adjacent thin-film materials via the magnetic proximity effect. This setup is called spin pumping, and has been studied intensively in the field of spintronics as a method of injecting spins through interfaces [11, 12]. Recent studies have theoret- ically proposed that spin excitation can be characterized by FMR in hybrid systems of superconducting thin films and magnetic materials [13–18]. Therefore, it is expected to be possible to analyze the dynamics of surface Majo- rana Ising spins using FMR in hybrid systems. In this work, we consider a TSC-ferromagnetic insula- tor (FI) hybrid system as shown in Fig. 1 (a). The FMR is induced by microwave irradiation on the FI. At the interface between the TSC and the FI, the surface Ma- (b) (c)(a) FI~~ ~~Microwave ϑS Y, yX xZhdcHex TSC (d) hdchdc+δhα+δα Hz FIG. 1. (a) The TSC-FI hybrid schematic reveals how, under resonance frequency microwave irradiation, localized spins commence precessional motion, consequently initiating the dynamical Majorana Ising spin response at the TSC inter- face. (b) In the TSC context, the liaison between a spin-up electron and a spin-down hole with the surrounding sea of spin-triplet Cooper pairs drastically modulate their proper- ties; notably, a spin-down hole can engage with a spin-triplet Cooper pair, thereby inheriting a negative charge. (c) No- tably, spin-triplet Cooper pairs amass around holes and scat- ter around electrons, thereby eroding the rigid distinction be- tween the two. (d) The interplay between the Majorana mode and the localized spin manipulates the FMR spectrum, trig- gering a frequency shift and linewidth broadening. jorana modes interact with the localized spins in the FI. As a result, the localized spin dynamics leads to the dy- namical Majorana Ising spin response (DMISR), which means the Majorana Ising spin density is dynamically in- duced, and it is possible to toggle between dissipative and non-dissipative Majorana Ising spin dynamics by adjust- ing the external magnetic field angle and the microwave frequency. Furthermore, the modulation of the localizedarXiv:2308.05955v2 [cond-mat.mes-hall] 19 Mar 20242 spin dynamics due to the interface interaction leads to a frequency shift and a linewidth broadening, which reflect the properties of the Majorana Ising spin dynamics. This work proposes a setup for detecting Majorana modes and paves the way for the development of quantum comput- ing and spin devices using Majoranas. Model.— We introduce a model Hamiltonian Hconsist- ing of three terms H=HM+HFI+Hex. (1) The first, second, and third terms respectively describe the surface Majorana modes on the TSC surface, the bulk FI, and the proximity-induced exchange coupling. Our focus is on energy regions significantly smaller than the bulk superconducting gap. This focus allows the spin ex- citation in the TSC to be well described using the surface Majorana modes. The subsequent paragraphs provide detailed explanations of each of these three terms. The first terms HMdescribes the surface Majorana modes, HM=1 2Z drψT(r) ℏvˆkyσx−ℏvˆkxσy ψ(r),(2) where r= (x, y),ˆk= (−i∂x,−i∂y),vis a constant velocity, and σ= (σx, σy, σz) are the Pauli matrices. The two component Majorana field operator is given by ψ(r) = ( ψ→(r), ψ←(r))T, with the spin quantization axis along the xaxis. The Majorana field operators sat- isfy the Majorana condition ψσ(r) =ψ† σ(r) and the an- ticommutation relation {ψσ(r), ψσ′(r)}=δσσ′δ(r−r′) where σ, σ′=→,←. We can derive HMby using surface- localized solutions of the BdG equation based on the bulk TSC Hamiltonian. The details of the derivation of HM are provided in the Supplemental Material [19]. A notable feature of the surface Majorana modes is that the spin density is Ising like, which we call the Majo- rana Ising spin [3–8]. The feature follows naturally from the Majorana condition and the anticommutation rela- tion. The Majorana Ising spin density operator is given bys(r) := ψT(r)(σ/2)ψ(r) = (0 ,0,−iψ→(r)ψ←(r)) (See the Supplemental Material for details [19]). The anisotropy of the Majorana Ising spin is the hallmark of the surface Majorana modes on the TSC surface. The second term HFIdescries the bulk FI and is given by the ferromagnetic Heisenberg model, HFI=− JX ⟨n,m⟩Sn·Sm−ℏγhdcX nSZ n, (3) where J>0 is the exchange coupling constant, Snis the localized spin at site n,⟨n, m⟩means summation for near- est neighbors, γis the electron gyromagnetic ratio, and hdcis the static external magnetic field. We consider the spin dynamics of the localized spin under microwave irra- diation, applying the spin-wave approximation. This al- lows the spin excitation to be described by a free bosonic operator, known as a magnon [20].The third term Hexrepresents the proximity exchange coupling at the interface between the TSC and the FI, Hex=−Z drX nJ(r,rn)s(r)·Sn=HZ+HT,(4) HZ=−cosϑZ drX nJ(r,rn)sz(r)SZ n, (5) HT=−sinϑZ drX nJ(r,rn)sz(r)SX n, (6) where the angle ϑis shown in Fig. 1 (a). HZis the coupling along the precession axis and HTis the coupling perpendicular to the precession axis. In our setup, HZ leads to gap opening of the energy spectrum of the surface Majorana modes and HTgives the DMISR under the microwave irradiation. Dynamical Majorana Ising spin response.— We con- sider the microwave irradiation on the FI. The coupling between the localized spins and the microwave is given by V(t) =−ℏγhacX n SX ncosωt−SY nsinωt
,(7) where hacis the microwave amplitude, and ωis the mi- crowave frequency. The microwave irradiation leads to the precessional motion of the localized spin. When the frequency of the precessional motion and the microwave coincide, the FMR occurs. The FMR leads to the DMISR via the exchange interaction. The DMISR is character- ized by the dynamic spin susceptibility of the Majorana modes, ˜ χzz(q, ω), defined as ˜χzz(q, ω) :=Z dre−iq·rZ dtei(ω+i0)tχzz(r, t),(8) where χzz(r, t) := −(L2/iℏ)θ(t)⟨[sz(r, t), sz(0,0)]⟩ with the interface area L2and the spin den- sity operator in the interaction picture, sz(r, t) = ei(HM+HZ)t/ℏsz(r)e−i(HM+HZ)t/ℏ. For the exchange cou- pling, we consider configuration average and assume ⟨P nJ(r,rn)⟩ave=J1, which means that HZis treated as a uniform Zeeman like interaction and the interface is specular [21]. Using eigenstates of Eq. (2) and after a straightforward calculation, the uniform spin susceptibil- ity is given by ˜χzz(0, ω) =−X k,λ|⟨k, λ|σz|k,−λ⟩|2f(Ek,λ)−f(Ek,−λ) 2Ek,λ+ℏω+i0, → −Z dED (E)E2−M2 2E2f(E)−f(−E) 2E+ℏω+i0, (9) where |k, λ⟩is an eigenstate of HMwith eigenenergy Ek,λ=λp (ℏvk)2+M2, (λ=±).M=J1Scosϑis the Majorana gap, f(E) = 1 /(eE/kBT+ 1) is the Fermi3 distribution function, and D(E) is the density of states given by D(E) =L2 2π(ℏv)2|E|θ(|E| − |M|), (10) with the Heaviside step function θ(x). It is important to note that the behavior of the uniform spin susceptibil- ity is determined by the interband contribution, which is proportional to the Fermi distribution function, i.e., the contribution of the occupied states. This mechanism is similar to the Van Vleck paramagnetism [22]. The con- tribution of the occupied states often plays a crucial role in topological responses [23]. Replacing the localized spin operators with their statis- tical average values, we find the induced Majorana Ising spin density, to the first order of J1S, is given by Z dr⟨sz(r, t)⟩= ˜χzz 0(0,0)J1Scosϑ + Re[˜ χzz 0(0, ω)]hac αhdcJ1Ssinϑsinωt, (11) where ˜ χzz 0(0,0) is the spin susceptibility for M= 0. The first term originates from HZand gives a static spin den- sity, while the second term originates from HTand gives a dynamic spin density. Figure 2 shows the induced Ising spin density as a function of time at several angles. As shown in Eq. (11), the Ising spin density consists of the static and dynamic components. The dynamic compo- nent is induced by the precessional motion of the local- ized spin, which means one can induce the DMISR using the dynamics of the localized spin. The inset in Fig. 2 shows Im˜ χzz(0, ω) as a function of ϑat a fixed frequency. When the frequency ℏωis smaller than the Majorana gap, Im˜ χzz(0, ω) is zero. Once the frequency overcomes the Majorana gap, Im˜ χzz(0, ω) be- comes finite. The implications of these behaviors are that if the magnon energy is smaller than the Majorana gap, there is no energy dissipation due to the DMISR. How- ever, once the magnon energy exceeds the Majorana gap, finite energy dissipation associated with the DMISR oc- curs at the surface of the TSC. Therefore, one can toggle between dissipative and non-dissipative Majorana Ising spin dynamics by adjusting the precession axis angle and the microwave frequency. FMR modulation.— The retarded component of the magnon Green’s function is given by GR(rn, t) = −(i/ℏ)θ(t)⟨[S+ n(t), S− 0(0)]⟩with the interaction picture S± n(t) =eiHFIt/ℏS± ne−iHFIt/ℏ. The FMR signal is char- acterized by the spectral function defined as A(q, ω) :=−1 πIm"X ne−iq·rnZ dtei(ω+i0)tGR(rn, t)# . (12) SSImχzz(0, ω) ˜⟨s z⟩ 2 1ωtϑ FInon-dissipativenon-dissipativedissipativedissipativeTSC FITSC000.00.51.0 π/4 π/2 0 π/4 π/20 ϑ2π πFIG. 2. The induced Ising spin density, with a unit ˜χzz 0(0,0)J1S, is presented as a function of ωtandϑ. The frequency and temperature are set to ℏω/J1S= 1.5 and kBT/J 1S= 0.1, respectively. The coefficient, hac/αhdc, is set to 0 .3. The static Majorana Ising spin density arises from HZ. When the precession axis deviates from the di- rection perpendicular to the interface, the precessional mo- tion of the localized spins results in the dynamical Majorana Ising spin response (DMISR). Energy dissipation due to the DMISR is zero for small angles ϑas the Majorana gap ex- ceeds the magnon energy. However, once the magnon energy overcomes the Majorana gap, the energy dissipation becomes finite. Therefore, one can toggle between dissipative and non- dissipative DMISR by adjusting ϑ. For uniform external force, the spectral function is given by A(0, ω) =2S ℏ1 π(α+δα)ω [ω−γ(hdc+δh)]2+ [(α+δα)ω]2. (13) The peak position and width of the FMR signal is given byhdc+δhandα+δα, respectively. hdcandαcorre- spond to the peak position and the linewidth of the FMR signal of the FI alone. δhandδαare the FMR modu- lations due to the exchange interaction HT. We treat HM+HFI+HZas an unperturbed Hamiltonian and HT as a perturbation. In this work, we assume the specular interface, where the coupling J(r,rn) is approximated asDP n,n′J(r,rn)J(r′,rn′)E ave=J2 1. The dynamics of the localized spins in the FI is modulated due to the interaction between the localized spins and the Majo- rana Ising spins. In our setup, the peak position and the linewidth of the FMR signal are modulated and the FMR4 modulation is given by δh= sin2ϑSJ2 1 2NγℏRe˜χzz(0, ω), (14) δα= sin2ϑSJ2 1 2NℏωIm˜χzz(0, ω), (15) where Nis the total number of sites in the FI. These for- mulas were derived in the study of the FMR in magnetic multilayer systems including superconductors. One can extract the spin property of the Majorana mode from the data on δhandδα. Because of the Ising spin anisotropy, the FMR modulation exhibits strong anisotropy, where the FMR modulation is proportional to sin2ϑ. Figure 3 shows the FMR modulations (a) δαand (b) δh. The FMR modulation at a fixed frequency increases with angle ϑand reaches a maximum at π/2, as can be read from Eqs. (14) and (15). When the angle ϑis fixed and the frequency ωis increased, δαbecomes finite above a certain frequency at which the energy of the magnon coincides with the Majorana gap. When ϑ < π/ 2 and ℏω≈2M,δαlinearly increases as a function of ωjust above the Majorana gap. The localized spin damping is enhanced when the magnon energy exceeds the Majorana gap. At ϑ=π/2 and ω≈0, the Majorana gap vanishes andδαis proportional to ω/T. In the high frequency region ℏω/J 1S≫1,δαconverges to its upper threshold. The frequency shift δhis almost independent of ωand has a finite value even in the Majorana gap. This behav- ior is analogous to the interband contribution to the spin susceptibility in strongly spin-orbit coupled band insula- tors, and is due to the fact that the effective Hamiltonian of the Majorana modes includes spin operators. It is im- portant to emphasize that although the Majorana modes have spin degrees of freedom, only the zcomponent of the spin density operator is well defined. This is a hallmark of Majorana modes, which differs significantly from elec- trons in ordinary solids. Note that δhis proportional to the energy cutoff, which is introduced to converge energy integral for Re˜ χzz(0, ω). The energy cutoff corresponds to the bulk superconducting gap, which is estimated as ∆∼0.1[meV] ( ∼1[K]). Therefore, our results are ap- plicable in the frequency region below ℏω∼0.1[meV] (∼30[GHz]). In addition, we assume that Majorana gap is estimated to be J1S∼0.01[meV] ( ∼0.1[K]). Discussion.— Comparing the present results with spin pumping (SP) in a conventional metal-ferromagnet hy- brid, the qualitative behaviors are quite different. In con- ventional metals, spin accumulation occurs due to FMR. In contrast, in the present system, no corresponding spin accumulation occurs due to the Ising anisotropy. Also, in the present calculations, the proximity-induced exchange coupling is assumed to be an isotropic Heisenberg-like coupling. However, in general, the interface interaction can also be anisotropic. Even in such a case, it is no qual- itative change in the case of ordinary metals, although a 0.00.5 (a) (b) ϑℏω/J1S 0 π/4 π/2024 ϑℏω/J1S 0 π/4 π/2024δ α δ h10 0FIG. 3. The temperature is set to kBT/J 1S= 0.1. (a) The damping modulation δαonly becomes finite when the magnon energy exceeds the Majorana gap; otherwise, it van- ishes. This behavior corresponds to the energy dissipation of the Majorana Ising spin. (b) The peak shift is finite, except forϑ= 0, and is almost independent of ω. This behavior resembles the spin response observed in strongly spin-orbit coupled band insulators, where the interband contribution to spin susceptibility results in a finite spin response, even within the energy gap. correction term due to anisotropy is added [24]. There- fore, the Ising anisotropy discussed in the present work is a property unique to the Majorana modes and can characterize the Majorana excitations. Let us comment on the universal nature of the toggling between non-dissipative and dissipative dynamical spin responses observed in our study. Indeed, such toggling becomes universally feasible when the microwave fre- quency and the energy gap are comparable, and when the Hamiltonian and spin operators are non-commutative, indicating that spin is not a conserved quantity. The non-commutativity can be attributed to the presence of spin-orbit couplings [25–27], and spin-triplet pair corre- lations [28]. Microwave irradiation leads to heating within the FI, so that thermally excited magnons due to the heating could influence the DMISR. Phenomena resulting from the heating, which can affect interface spin dynamics, in- clude the spin Seebeck effect (SSE) [29], where a spin current is generated at the interface due to a tempera- ture difference. In hybrid systems of normal metal and FI, methods to separate the inverse spin Hall voltage due to SP from other signals caused by heating have been well studied [30]. Especially, it has been theoretically proposed that SP and SSE signals can be separated us- ing a spin current noise measurement [24]. Moreover, SP coherently excites specific modes, which qualitatively dif- fers from SSE induced by thermally excited magnons [14]. Therefore, even if heating occurs in the FI in our setup, the properties of Majorana Ising spins are expected to be captured. Details of the heating effect on the DMISR will be examined in the near future. We also mention the experimental feasibility of our the- oretical proposals. As we have already explained, the FMR modulation is a very sensitive spin probe. Indeed, the FMR modulation by surface states of 3D topological5 insulators [31] and graphene [32–36] has been reported experimentally. Therefore, we expect that the enhanced Gilbert damping due to Majorana Ising spin can be ob- servable in our setup when the thickness of the ferromag- netic insulator is sufficiently thin. Finally, it is pertinent to mention the potential candi- date materials where surface Majorana Ising spins could be detectable. Notably, UTe 2[37], Cu xBi2Se3[38, 39], SrxBi2Se3and Nb xBi2Se3[40] are reported to be in a p- wave superconducting state and theoretically can host surface Majorana Ising spins. Recent NMR measure- ments indicate that UTe 2could be a bulk p-wave su- perconductor in the Balian-Werthamer state [41], which hosts the surface Majorana Ising spins with the per- pendicular Ising anisotropy, as considered in this work. AxBi2Se3(A= Cu, Sr, Nb) is considered to possess in- plane Ising anisotropy [8], differing from the perpendic- ular Ising anisotropy explored in this work. Therefore, we expect that it exhibits anisotropy different from that demonstrated in this work. Conclusion.— We present herein a study of the spin dynamics in a topological superconductor (TSC)-magnet hybrid. Ferromagnetic resonance under microwave irra- diation leads to the dynamically induced Majorana Ising spin density on the TSC surface. One can toggle between dissipative and non-dissipative Majorana Ising spin dy- namics by adjusting the external magnetic field angle and the microwave frequency. Therefore, our setup provides a platform to detect and control Majorana excitations. We expect that our results provide insights toward the development of future quantum computing and spintron- ics devices using Majorana excitations. Acknowledgments.— The authors are grateful to R. Shindou for valuable discussions. This work is partially supported by the Priority Program of Chinese Academy of Sciences, Grant No. XDB28000000. We acknowl- edge JSPS KAKENHI for Grants (Nos. JP20K03835, JP21H01800, JP21H04565, and JP23H01839). SUPPLEMENTAL MATERIAL Surface Majorana modes In this section, we describe the procedure for deriv- ing the effective Hamiltonian of the surface Majorana modes. We start with the bulk Hamiltonian of a three- dimensional topological superconductor. Based on the bulk Hamiltonian, we solve the BdG equation to demon- strate the existence of a surface-localized solution. Us- ing this solution, we expand the field operator and show that it satisfies the Majorana condition when the bulk excitations are neglected. As a result, on energy scales much smaller than the bulk superconducting gap, the low-energy excitations are described by surface-localized Majorana modes. The above procedure is explained inmore detail in the following. Note that we use rfor three- dimensional coordinates and r∥for two-dimensional ones in the Supplemental Material. We start with the mean-field Hamiltonian given by HSC=1 2Z drΨ† BdG(r)HBdGΨBdG(r), (16) withr= (x, y, z ). We consider the Balian-Werthamer (BW) state, in which the pair potential is given by ∆ˆk=∆ kF ˆk·σ iσywith the bulk superconducting gap ∆. Here, we do not discuss the microscopic origin of the pair correlation leading to the BW state. As a result, the BdG Hamiltonian HBdGis given by HBdG= εˆk−EF 0 −∆ kFˆk−∆ kFˆkx 0 εˆk−EF∆ kFˆkx∆ kFˆk+ −∆ kFˆk+∆ kFˆkx−εˆk+EF 0 ∆ kFˆkx∆ kFˆk− 0 −εˆk+EF , (17) with ˆk±=ˆky±iˆkz,ˆk=−i∇, and εˆk=ℏ2ˆk2 2m. The four component Nambu spinor ΨBdG(r) is given by ΨBdG(r) := Ψ→(r) Ψ←(r) Ψ† →(r) Ψ† ←(r) , (18) with the spin quantization axis along the xaxis. The matrices of the spin operators are represented as σx=1 0 0−1 , (19) σy= 0 1 1 0 , (20) σz=0−i i0 . (21) The fermion field operators satisfy the anticommutation relations {Ψσ(r),Ψσ′(r′)}= 0, (22) {Ψσ(r),Ψ† σ′(r′)}=δσσ′δ(r−r′), (23) with the spin indices σ, σ′=→,←. To diagonalize the BdG Hamiltonian, we solve the BdG equation given by HBdGΦ(r) =EΦ(r). (24) We assume that a solution is written as Φ(r) =eik∥·r∥f(z) u→ u← v→ v← , (25)6 withk∥= (kx, ky) and r∥= (x, y). If we set the four components vector to satisfy the following equation (Ma- jorana condition) 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 u→ u← v→ v← =± u→ u← v→ v← , (26) we can obtain a surface-localized solution. If we take a positive (negative) sign, we obtain a solution localized on the top surface (bottom surface). As we will consider solutions localized on the bottom surface below, we take a negative sign. Finally, we obtain the normalized eigen- vectors of the BdG equation given by Φλ,k∥(r) =eik∥·r∥ √ L2fk∥(z)uλ,k∥, (27) with fk∥(z) =Nk∥sin(k⊥z)e−κz, (28) Nk∥=s 4κ(k2 ⊥+κ2) k2 ⊥, (29) κ=m∆ ℏ2kF, (30) k⊥=q k2 F−k2 ∥−κ2, (31) and u+,k∥= u+,→k∥ u+,←k∥ v+,→k∥ v+,←k∥ =1√ 2 sinϕk∥+π/2 2 −cosϕk∥+π/2 2 −sinϕk∥+π/2 2 cosϕk∥+π/2 2 ,(32) u−,k∥= u−,→k∥ u−,←k∥ v−,→k∥ v−,←k∥ =1√ 2 −cosϕk∥+π/2 2 −sinϕk∥+π/2 2 cosϕk∥+π/2 2 sinϕk∥+π/2 2 .(33) The eigenenergy is given by Eλ,k∥=λ∆k∥/kF. We can show that the eigenvectors satisfy u−,−k∥=u+,k∥. (34) Consequently, the field operator is expanded as ΨBdG(r) =X k∥ γk∥eik∥·r∥ √ L2+γ† k∥e−ik∥·r∥ √ L2 ×fk∥(z)u+,k∥+ (bulk modes) ,(35) where γk∥(γ† k∥) is the quasiparticle creation (annihila- tion) operator with the eigenenergy E+,k∥. Substitutingthe above expression into Eq. (16) with omission of bulk modes and performing the integration in the z-direction, we obtain the effective Hamiltonian for the surface states HM=1 2Z dr∥ψT(r∥) ℏvˆkyσx−ℏvˆkxσy ψ(r∥),(36) where v= ∆/ℏkFand we introduced the two component Majorana field operator ψ(r∥) =ψ→(r∥) ψ←(r∥) , (37) satisfying the Majorana condition ψσ(r∥) =ψ† σ(r∥), (38) and the anticommutation relation n ψσ(r∥), ψσ′(r′ ∥)o =δσσ′δ(r∥−r′ ∥). (39) The spin density operator of the Majorana mode is given by s(r∥) =ψ†(r∥)σ 2ψ(r∥). (40) Thexcomponent is given by sx(r∥) = ψ† →(r∥), ψ† ←(r∥)
1/2 0 0−1/2ψ→(r∥) ψ←(r∥) =1 2 ψ† →(r∥)ψ→(r∥)−ψ† ←(r∥)ψ←(r∥) =1 2 ψ2 →(r∥)−ψ2 ←(r∥) = 0. (41) In a similar manner, the yandzcomponents are given by sy(r∥) = ψ† →(r∥), ψ† ←(r∥)
0 1/2 1/2 0ψ→(r∥) ψ←(r∥) =1 2 ψ† →(r∥)ψ←(r∥) +ψ† ←(r∥)ψ→(r∥) =1 2 ψ→(r∥), ψ←(r∥) = 0, (42) and sz(r∥) = ψ† →(r∥), ψ† ←(r∥)
0−i/2 i/2 0ψ→(r∥) ψ←(r∥) =−i 2 ψ† →(r∥)ψ←(r∥)−ψ† ←(r∥)ψ→(r∥)
=−iψ→(r∥)ψ←(r∥), (43) respectively. As a result, the spin density operator is given by s(r∥) = 0,0,−iψ→(r∥)ψ←(r∥)
. (44) One can see that the spin density of the Majorana mode is Ising like.7 Majorana Ising spin dynamics In this section, we calculate the Ising spin density in- duced on the TSC surface by the proximity coupling Hex. Hexconsists of two terms, HZandHT.HZleads to the static spin density and HTleads to the dynamic spin density. First, we calculate the static spin density. Next, we calculate the dynamic spin density. The total spin density operator is given by sz tot=Z dr∥sz(r∥). (45) The statistical average of the static spin density is calcu- lated as ⟨sz tot⟩=−X k∥M 2Ek∥ f(Ek∥)−f(−Ek∥) → −L 2πℏv2Z∆ MEdEZ2π 0dϕM 2E[f(E)−f(−E)] =−Z∆ 0dED (E)f(E)−f(−E) 2EM. (46) At the zero temperature limit T→0, the static spin density is given by ⟨sz tot⟩=1 2L2 2π(ℏv)2(∆−M)M≈˜χzz 0(0,0)M, (47) where ˜ χzz 0(0,0) = D(∆)/2 and we used ∆ ≫M. The dynamic spin density is given by the perturbative force HT(t) =Z dr∥sz(r∥)F(r∥, t), (48) where F(r∥, t) is given by F(r∥, t) =−sinϑX nJ(r∥,rn) SX n(t) ≈ −sinϑJ1Sγhacp (ω−γhdc)2+α2ω2cosωt =:Fcosωt. (49) The time dependent statistical average of the Ising spin density, to the first order of J1S, is given by Z dr∥ sz(r∥, t) =Z dr∥Z dr′ ∥Z dt′χzz(r∥−r′ ∥, t′)F(r′ ∥, t−t′) = Re ˜χzz(0, ω)Fe−iωt ≈Re[˜χzz 0(0, ω)]Fcosωt, (50) where we used Re˜ χzz 0(0, ω)≫Im˜χzz 0(0, ω). The real part of ˜χzz(0, ω) is given by Re˜χzz(0, ω) =−PZ dED (E)E2−M2 2E2f(E)−f(−E) 2E+ℏω, (51)where Pmeans the principal value. When the integrand is expanded with respect to ω, the lowest order correc- tion term becomes quadratic in ω. In the frequency range considered in this work, this correction term is signifi- cantly smaller compared to the static spin susceptibility Re˜χzz(0,0). Therefore, the spin susceptibility exhibits almost no frequency dependence and remains constant as a function of ω. The imaginary part of ˜ χzz(0, ω) is given by Im˜χzz(0, ω) =πD(ℏω/2)(ℏω/2)2−M2 2(ℏω/2)2[f(−ℏω/2)−f(ℏω/2)]. (52) FMR modulation due to the proximity exchange coupling In this section, we provide a brief explanation for the derivation of the FMR modulations δhandδα. The FMR modulations can be determined from the retarded com- ponent of the magnon Green’s function, which is given by ˜GR(k, ω) =2S/ℏ ω−ωk+iαω−(2S/ℏ)ΣR(k, ω),(53) where we introduce the Gilbert damping constant αphe- nomenologically. In the second-order perturbation calcu- lation with respect to HT, the self-energy is given by ΣR(k, ω) =−sinϑ 22X q∥|˜J(q∥,k)|2˜χzz(q∥, ω),(54) where ˜J(q∥,0) is given by ˜J(q∥,k) =1 L2√ NZ dr∥X nJ(r∥,rn)ei(q∥·r∥+k·rn) (55) The pole of ˜GR(k, ω) signifies the FMR modulations, including both the frequency shift and the enhanced Gilbert damping. These are given by δh=2S γℏReΣR(0, ω), δα =−2S ℏωImΣR(0, ω).(56) From the above equations and Eq. 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1604.07552v1.First_principles_studies_of_the_Gilbert_damping_and_exchange_interactions_for_half_metallic_Heuslers_alloys.pdf

arXiv:1604.07552v1 [cond-mat.mtrl-sci] 26 Apr 2016First principles studies of the Gilbert damping and exchang e interactions for half-metallic Heuslers alloys Jonathan Chico,1,∗Samara Keshavarz,1Yaroslav Kvashnin,1Manuel Pereiro,1Igor Di Marco,1Corina Etz,2Olle Eriksson,1Anders Bergman,1and Lars Bergqvist3,4 1Department of Physics and Astronomy, Materials Theory Divi sion, Uppsala University, Box 516, SE-75120 Uppsala, Sweden 2Department of Engineering Sciences and Mathematics, Materials Science Division, Lule˚ a University of Technolo gy, Lule˚ a, Sweden 3Department of Materials and Nano Physics, School of Informa tion and Communication Technology, KTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden 4SeRC (Swedish e-Science Research Center), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden (Dated: September 28, 2018) Heusler alloys havebeen intensivelystudied dueto thewide varietyof properties thatthey exhibit. One of these properties is of particular interest for techno logical applications, i.e. the fact that some Heusler alloys are half-metallic. In the following, a syste matic study of the magnetic properties of three different Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with Z = (Al, Si, Ga, Ge) is per- formed. A key aspect is the determination of the Gilbert damp ing from first principles calculations, with special focus on the role played by different approximat ions, the effect that substitutional disorder and temperature effects. Heisenberg exchange inte ractions and critical temperature for the alloys are also calculated as well as magnon dispersion r elations for representative systems, the ferromagnetic Co 2FeSi and the ferrimagnetic Mn 2VAl. Correlations effects beyond standard density-functional theory are treated using both the local spin density approximation including the Hubbard Uand the local spin density approximation plus dynamical mea n field theory approx- imation, which allows to determine if dynamical self-energ y corrections can remedy some of the inconsistencies which were previously reported for these a lloys. I. INTRODUCTION The limitations presented by traditional electronic de- vices, such as Joule heating, which leads to higher en- ergyconsumption, leakagecurrentsandpoorscalingwith size amongothers1, havesparkedprofoundinterest in the fields of spintronics and magnonics. Spintronics applica- tions rely in the transmission of information in both spin and charge degrees of freedom of the electron, whilst in magnonics information is transmitted via magnetic exci- tations, spin waves or magnons. Half-metallic materials with a large Curie temperature are of great interest for these applications. Due to the fact that they are con- ductors in only one of the spin channels makes them ideal candidates for possible devices2. Half-metals also have certain advantages for magnonic applications, due to the fact that they are insulators in a spin channel and thus can have a smaller total density of states at the Fermi energy than metals. This can result into a small Gilbert damping, which is an instrumental prerequisite for magnonic applications3. The name “full Heusler alloys”refer to a set of com- pounds with formula X 2YZ with X and Y typically being transition metals4. The interest in them stems from the factthattheirpropertiescanbecompletelydifferentfrom those of their constituents. Heusler compounds can be superconducting5(Pd2YSn), semiconductors6(TiCoSb), half-metallic7(Co2MnSi), and can show a wide array of magnetic configurations: ferromagnetic7(Co2FeSi), fer- rimagnetic8(Mn2VAl) or antiferromagnetic9(CrMnSb). Due to such a wide variety of behaviours, full Heusleralloys have been studied in great detail since their dis- covery in 1903, leading to the discovery of new Heusler families such as the half-Heuslers, with formula XYZ, and the inverse Heuslers, with formula X 2YZ. The lat- ter tend to exhibit a different crystal structure and have been predicted to show quite remarkable properties10. Many Heusler alloys have also been predicted to be half-metallic, in particular Co 2MnSi has been the focus ofmany theoreticaland experimental works7,11,12, due to its large Curie temperature of 985 K13, half-metallicity and low damping parameter, which makes it an ideal candidate for possible spintronic applications. Despite the large amount of research devoted to the half-metallic Heusleralloys,suchasCo 2MnSi, onlyrecentlytheoretical predictions of the Gilbert damping parameter have been made for some Heusler alloys14,15. In the present work first principle calculations of the full Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with Z = (Al, Si, Ga, Ge) are performed, with special empha- sis on the determination of the Gilbert damping and the interatomic exchange interactions. A study treatment of thesystemswithdifferentexchangecorrelationpotentials is also performed. The paper is organized as follows, in section II the computational methods used are presented. Then, in section III, magnetic moments and spectral properties are discussed. In section IV the results for the exchange stiffness parameter, the critical temperature obtained via MonteCarlosimulationsandmagnondispersionrelations are presented. Finally in section V, the calculated damp- ing parameter for the different Heusler is presented and2 discussed. II. COMPUTATIONAL METHODS The full Heusler alloys(X 2YZ) havea crystalstructure given by the space group Fm-3m with X occupying the Wyckoffposition 8c (1 4,1 4,1 4), while Ysits in the 4a(0,0,0) and Z in the 4b (1 2,1 2,1 2). To determine the properties of the systems first prin- ciples electronic structure calculations were performed. They were mainly done by means of the Korringa-Kohn- Rostocker Green’s function formalism as implemented in the SPR-KKRpackage16. The shape ofthe potential was considered by using both the Atomic Sphere Approxi- mation (ASA) and a full potential (FP) scheme. The calculations of exchange interactions were performed in scalar relativistic approximation while the full relativis- tic Dirac equation was used in the damping calculations. The exchange correlation functional was treated using both the Local Spin Density Approximation (LSDA), as considered by Vosko, Wilk, and Nusair (VWN)17, and the Generalized Gradient Approximation (GGA), as de- vised by Perdew, Burke and Ernzerhof (PBE)18. For cases in which substitutional disorder is considered, the Coherent Potential Approximation (CPA) is used19,20. Static correlation effects beyond LSDA or GGA are taken into account by using the LSDA+ Uapproach, wherethe Kohn-ShamHamiltonianissupplemented with an additional term describing local Hubbard interac- tions21, for thed-states of Co, Mn and Fe. The U-matrix describing this on-site interactions was parametrized through the Hubbard parameter Uand the Hund ex- changeJ, using values UCo=UMn=UFe= 3 eV and JCo=JMn=JFe= 0.8 eV, which are in the range of the values considered in previous theoretical studies13,22–24. This approach is used for the Heusler alloys families Co2MnZ and Co 2FeZ, as previous studies have shown that for systems such as Co 2FeSi it might be necessary to reproduce several experimental observations, although, this topic is still up for debate23. Since part of correla- tioneffectsofthe3 dorbitalsisalreadyincludedinLSDA, their contribution has to be subtracted before adding the +Uself-energy. This contribution to be removed is usu- ally called “double-counting”(DC) correction and there is no unique way of defining it (see e.g. Ref. 25). We have used two of the most widely used schemes for the DC, namely the Atomic Limit (AL), also known as Fully Localized Limit (FLL)26, and the Around Mean Field (AMF)27. The dependence of the results on this choice will be extensively discussed in the following sections. In order to shine some light on the importance of the dynamical correlations for the magnetic properties of the selected Heusler alloys, a series of calculations were performed in the framework of DFT plus Dynami- cal Mean Field Theory (DMFT)28,29, as implemented in the full-potential linear muffin-tin orbital (FP-LMTO) code RSPt30. As for LSDA+ U, the DMFT calculationsare performed for a selected set of metal 3 dorbitals on top of the LSDA solution in a fully charge self-consistent manner.31,32Theeffectiveimpurityproblem, whichisthe core of the DMFT, is solved through the spin-polarized T-matrix fluctuation-exchange (SPTF) solver33. This type of solver is perturbative and is appropriate for the systems with moderate correlationeffects, where U/W < 1 (Wdenotes the bandwidth).34Contrary to the prior DMFT studies35,36, we have performed the perturba- tion expansion of the Hartree-Fock-renormalizedGreen’s function ( GHF) and not of the bare one. Concerning the DC correction, we here use both the FLL approach, de- scribed above, as well as the so-called “Σ(0)”correction. In the latter case, the orbitally-averaged static part of the DMFT self-energy is removed, which is often a good choice for metals29,37. Finally, in order to extract infor- mationaboutthemagneticexcitationsin thesystems, we have performed a mapping onto an effective Heisenberg Hamiltonian ˆH=−/summationdisplay i/negationslash=jJij/vector ei/vector ej, (1) whereJijis anexchangeinteractionbetweenthe spinslo- cated at site iandj, while the /vector ei(/vector ej) representsthe unity vectoralongthe magnetizationdirectionatsite i (j). The exchange parameters then are computed by making use of the well established LKAG (Liechtenstein, Katsnel- son, Antropov, and Gubanov) formalism, which is based on the magnetic force theorem38–40. More specific de- tails about the implementation of the LKAG formalism in RSPt can be found in Ref. 41. We also note that the performance of the RSPt method was recently published in Ref.42and it was found that the accuracy was similar to that of augmented plane wave methods. From the exchange interactions between magnetic atoms, it is possible to obtain the spin wave stiffness, D, which, for cubic systems is written as43 D=2 3/summationdisplay i,jJij√mimj|rij|2exp/parenleftbigg −ηrij alat/parenrightbigg ,(2) where the mi’s are the magnetic moments of a given atom,rijisthedistancebetweenthetwoconsideredmag- neticmoments, alatisthelatticeparameter, ηisaconver- gence parameter used to ensure the convergence of Eq. 2, the value of Dis taken under the limit η→0. To ensure the convergence of the summation, it is also important to take into consideration long range interactions. Hence the exchange interactions are considered up to 6 lattice constants from the central atom. The obtained exchange interactions were then used to calculate the critical temperature by making use of the Bindercumulant, obtainedfromMonteCarlosimulations as implemented in the UppASD package44. This was calculated for three different number of cell repetitions (10x10x10, 15x15x15 and 20x20x20), with the intersec- tion point determining the critical temperature of the system45.3 The Gilbert damping, α, is calculated via linear re- sponse theory46. Temperature effects in the scattering process of electrons are taken into account by consider- ing an alloy analogy model within CPA with respect to the atomic displacements and thermal fluctuations of the spin moments47. Vertex corrections are also considered here, because they provide the “scattering in”term of the Boltzmann equation and it corrects significant error in the damping, whenever there is an appreciable s-p or s-d scattering in the system16,48. From the calculated exchange interactions, the adia- batic magnon spectra (AMS) can be determined by cal- culating the Fourier transform of the interatomic ex- change interactions49. This is determined for selected cases and is compared with the magnon dispersion re- lation obtained from the dynamical structure factor, Sk(q,ω), resulting fromspin dynamics calculations. The Sk(q,ω) is obtained from the Fourier transform of the time and spatially displaced spin-spin correlation func- tion,Ck(r−r′,t)50 Sk(q,ω) =1√ 2πN/summationdisplay r,r′eiq·(r−r′)/integraldisplay∞ −∞eiωtCk(r−r′,t)dt. (3) The advantage of using the dynamical structure factor over the adiabatic magnon spectra is the capability of studying temperature effects as well as the influence of the damping parameter determined from first principles calculations or from experimental measurements. III. ELECTRONIC STRUCTURE The calculated spin magnetic moments for the selected systems are reported in Table I. These values are ob- tained from SPR-KKR with various approximations of the exchange correlation potential and for different geo- metrical shapes of the potential itself. For the Co 2MnZ family, when Z = (Si ,Ge), the obtained spin mag- netic moments do not seem to be heavily influenced by the choice of exchange correlation potential or potential shape. However, for Z = (Al ,Ga) a large variation is observed in the spin moment when one includes the Hub- bard parameter U. For the Co 2FeZ systems, a pronounced difference can be observed in the magnetic moments between the LSDA and the experimental values for Z = (Si ,Ge). Previ- ous theoretical works13,22,24suggested that the inclusion of a +Uterm is necessary to obtain the expected spin magnetic moments, but such a conclusion has been re- cently questioned23. To estimate which double counting schemewould be most suitableto treatcorrelationeffects in this class of systems, an interpolation scheme between the FLL and AMF treatments was tested, as described in Ref. 59 and implemented in the FP-LAPW package Elk60. It was found that both Co 2MnSi and Co 2FeSi are better described with the AMF scheme, as indicatedby their small αUparameter of ∼0.1 for both materials (αU= 0denotes completeAMF and αU= 1FLL), which is in agreement with the recent work by Tsirogiannis and Galanakis61. To test whether a more sophisticated way to treat cor- relation effects improves the description of these mate- rials, electronic structure calculations for Co 2MnSi and Co2FeSi using the DMFT scheme were performed. The LSDA+DMFT[Σ(0)] calculations yielded total spin mo- ments of 5.00 µBand 5.34 µBfor respectively Co 2MnSi and Co 2FeSi. These values are almost equal to those ob- tained in LSDA, which is also the case in elemental tran- sition metals32. As mentioned above for LSDA+ U, the choice of the DC is crucial for these systems. The main reason why no significant differences are found between DMFT and LSDA values is that the employed “Σ(0)”DC almost entirely preserves the static part of the exchange splitting obtained in LSDA62. For instance, by using FLL DC, we obtained a total magnetization of 5.00 µB and 5.61 µBin Co2MnSi and Co 2FeSi, respectively. We note that the spin moment of Co 2FeSi still does not reach the value expected from the Slater-Pauling rule, but the DMFT modifies it in a right direction, if albeit to a smaller degree that the LSDA+ Uschemes. Another important aspect of the presently studied sys- tems is the fact that they are predicted to be half- metallic. In Fig. 1, the density of states (DOS) for both Co 2MnSi and Co 2FeSi is presented using LSDA and LSDA+U. For Co 2MnSi, the DOS at the Fermi energy is observed to exhibit a very clear gap in one of the spin channels, in agreement with previous theoretical works7. For Co 2FeSi, instead a small pseudo-gap region is ob- served in one of the spin channels, but the Fermi level is located just at the edge of the boundary as shown in previous works24. Panels a) and b) of Fig. 1 also show that some small differences arise depending on the ASA or FP treatment. In particular, the gap in the minority spin channel is slightly reduced in ASA. When correlation effects are considered within the LSDA+Umethod, the observed band gap for Co 2MnSi becomes larger, while the Fermi level is shifted and still remainsin the gap. When applyingLSDA+ Uto Co2FeSi in the FLL scheme, EFis shifted farther away from the edgeofthe gap, whichexplainswhythemoment becomes almostanintegerasexpected fromtheSlater-Paulingbe- haviour7,24,63. Moreover,onecanseethatinASAthegap in the spin down channel is much smaller in comparison to the results obtained in FP. When the dynamical correlation effects are considered via DMFT, the overall shape of DOS remains to be quite similartothatofbareLSDA,especiallyclosetotheFermi level, as seen in Fig. A.1 in the Appendix A. This is re- lated to the fact that we use a perturbative treatment of the many-body effects, which favours Fermi-liquid be- haviour. Similarly to LSDA+ U, the LSDA+DMFT cal- culations result in the increased spin-down gaps, but the producedshiftofthebandsisnotaslargeasinLSDA+ U. This is quite natural, since the inclusion ofthe dynamical4 TABLE I. Summary of the spin magnetic moments obtained using different approximations as obtained from SPR-KKR for the Co2MnZ and Co 2FeZ families with Z = (Al ,Si,Ga,Ge). Different exchange correlation potential approximati ons and shapes of the potential have been used. The symbol†signifies that the Fermi energy is located at a gap in one of the spin channels. Quantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe alat[˚A] 5.75515.77515.65525.743535.730515.737515.640235.75054 mASA LDA[µB] 4.04†4.09†4.99†4.94†4.86†4.93†5.09 5.29 mASA GGA[µB] 4.09†4.15†4.99†4.96†4.93†5.00†5.37 5.53 mASA LDA+UAMF [µB] 4.02†4.08 4.98†4.98†4.94†4.99†5.19 5.30 mASA LDA+UFLL [µB] 4.77 4.90 5.02†5.11 5.22 5.36 5.86†5.94† mFP LDA[µB] 4.02†4.08†4.98†4.98†4.91†4.97†5.28 5.42 mFP GGA[µB] 4.03†4.11 4.98†4.99†4.98†5.01†5.55 5.70 mFP LDA+UAMF [µB] 4.59 4.99 4.98†5.13 5.12 5.40 5.98†5.98† mFP LDA+UFLL [µB] 4.03†4.17 4.99†4.99†4.99†5.09 5.86†5.98† mexp[µB] 4.04554.09564.96574.84574.96555.15576.00245.7458 0369n↑tot[sts./eV] 0 3 6 9 -6 -3 0 3n↓tot[sts./eV] E-EF[eV]ASA FPa) 0369n↑tot[sts./eV] 0 3 6 9 -6 -3 0 3n↓tot[sts./eV] E-EF[eV]ASA FPb ) 0369n↑tot[sts./eV] 0 3 6 9 -6 -3 0 3n↓tot[sts./eV] E-EF[eV]FP [FLL] FP [AMF]c) 0369n↑tot[sts./eV] 0 3 6 9 -6 -3 0 3n↓tot[sts./eV] E-EFASA [AMF] FP [FLL] FP [AMF]d ) [eV] FIG. 1. (Color online) Total density of states for different e xchange correlation potentials with the dashed line indica ting the Fermi energy, sub-figures a) and b) when LSDA is used for Co 2MnSi and Co 2FeSi respectively. Sub-figures c) and d) show the DOS when the systems (Co 2MnSi and Co 2FeSi respectively) are treated with LSDA+ U. It can be seen that the half metalicity of the materials can be affected by the shape of the potential a nd the choice of exchange correlation potential chosen. correlations usually tends to screen the static contribu- tions coming from LSDA+ U. According to Ref. 35 taking into account dynami- cal correlations in Co 2MnSi results in the emergence of the non-quasiparticle states (NQS’s) inside the minority- spin gap, which at finite temperature tend to decrease the spin polarisation at the Fermi level. These NQS’s were first predicted theoretically for model systems64and stem from the electron-magnon interactions, which are accounted in DMFT (for review, see Ref. 2). Our LSDA+DMFT results for Co 2MnSi indeed show the ap- pearance of the NQS’s, as evident from the pronounced imaginary part of the self-energy at the bottom of the conduction minority-spin band (see Appendix B). An analysis of the orbital decomposition of the self-energy reveals that the largest contribution to the NQS’s comes5 from the Mn- TEgstates. However, in our calculations, where the temperature was set to 300K, the NQS’s ap- peared above Fermi level and did not contribute to the system’s depolarization, in agreementwith the recent ex- perimental study12. We note that a half-metallic state with a magnetic moment of around 6 µBfor Co 2FeSi was reported in a previous LSDA+DMFT[FLL] study by Chadov et al.36. In their calculations, both LSDA+ Uand LSDA+DMFT calculations resulted in practically the same positions of the unoccupied spin-down bands, shifted to the higher energies as compared to LSDA. This is due to techni- cal differences in the treatment of the Hartree-Fock con- tributions to the SPTF self-energy, which in Ref. 36 is done separately from the dynamical contributions, while in this study a unified approach is used. Overall, the improvements in computational accuracy with respect to previousimplementationscouldberesponsiblefortheob- tained qualitative disagreement with respect to Refs. 35 and 36. Moreover, given that the results qualitatively depend on the choice of the DC term, the description of the electronic structure of Co 2FeSi is not conclusive. The discrepancies in the magnetic moments presented in Table I with respect to the experimental values can in part be traced back to details of the density of the states around the Fermi energy. The studied Heusler alloys are thought to be half-metallic, which in turn lead to inte- ger moments following the Slater-Pauling rule7. There- fore, any approximation that destroys half-metallicity will have a profound effect on their magnetic properties7. For example, for Co 2FeAl when the potential is treated in LSDA+ U[FLL] with ASA the Fermi energy is located at a sharp peak close to the edge of the band gap, de- stroyingthehalf-metallicstate(Seesupplementarymate- rial Fig.1). A similar situation occurs in LSDA+ U[AMF] with a full potential scheme. It is also worth mention- ing that despite the fact that the Fermi energy for many of these alloys is located inside the pseudo-gap in one of the spin channels, this does not ensure a full spin po- larization, which is instead observed in systems as e.g. Co2MnSi. Another important factor is the fact that EF can be close to the edge of the gap as in Co 2MnGa when the shape of the potential is considered to be given by ASA and the exchange correlation potential is dictated by LSDA, hence the half-metallicity of these alloys could be destroyed due to temperature effects. The other Heusler family investigated here is the ferri- magnetic Mn 2VZ with Z = (Al ,Si,Ga,Ge). The lattice constants used in the simulations correspond to either experimental or previous theoretical works. These data are reported in Table II together with appropriate ref- erences. Table II also illustrates the magnetic moments calculated using different exchange correlation potentials and shapes of the potential. It can be seen that in gen- eral there is a good agreement with previous works, re- sulting in spin moments which obey the Slater-Pauling behaviour. For these systems, the Mn atoms align themselves inTABLE II. Lattice constants used for the electronic struc- ture calculations and summary of the magnetic properties fo r Mn2VZ with Z = (Al ,Si,Ga,Ge). As for the ferromagnetic families, different shapes of the potential and exchange cor - relations potential functionals were used. The magnetic mo - ments follow quite well the Slater-Pauling behavior with al l the studied exchange correlation potentials. The symbol† signifies that the Fermi energy is located at a gap in one of the spin channels. Quantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe alat[˚A] 5.687655.905666.06656.09567 mASA LDA[µB] 1.87 1.97†1.00†0.99† mASA GGA[µB] 1.99†2.04†1.01†1.00† mFP LDA[µB] 1.92 1.95†0.99†0.99 mFP GGA[µB] 1.98†2.02†0.99†0.99† mexp[µB] — 1.8666— — an anti-parallel orientation with respect to the V mo- ments, resulting in a ferrimagnetic ground state. As for the ferromagnetic compounds, the DOS shows a pseu- dogap in one of the spin channels (see supplementary material Fig.8-9) indicating that at T= 0 K these com- poundscouldbehalf-metallic. An importantfactoristhe fact that the spin polarization for these systems is usu- ally considered to be in the opposite spin channel than for the ferromagneticalloys presently studied, henceforth the total magnetic moment is usually assigned to a neg- ative sign such that it complies with the Slater-Pauling rule7,65. IV. EXCHANGE INTERACTIONS AND MAGNONS In this section, the effects that different exchange cor- relation potentials and geometrical shapes of the poten- tial haveoverthe exchangeinteractionswill be discussed. A. Ferromagnetic Co 2MnZ and Co 2FeZ with Z= (Al,Si,Ga,Ge) In Table III the calculated spin wave stiffness, D, is shown. In general there is a good agreement between the calculated values for the Co 2MnZ family, with the obtained values using LSDA or GGA being somewhat larger than the experimental measurements. This is in agreement with the observations in the previous section, in which the same exchange correlation potentials were found to be able to reproducethe magnetic moments and half-metallicbehaviourfortheCo 2MnZfamily. Inpartic- ular, for Co 2MnSi the ASA calculations are in agreement with experiments68,69and previous theoretical calcula- tions70. It is important to notice that the experimen- tal measurements are performed at room temperature, which can lead to softening of the magnon spectra, lead- ing to a reduced spin wave stiffness.6 However, for the Co 2FeZ family neither LSDA or GGA can consistently predict the spin wave stiffness, with Z=(Al, Ga) resulting in an overestimated value of D, while for Co 2FeSi the obtained value is severely underes- timated. However, for some materials in this family, e.g. Co2FeGathespinwavestiffnessagreeswith previousthe- oretical results70. These data reflect the influence that certain approximations have on the location of the Fermi level, which previously has been shown to have profound effects on the magnitude of the exchange interactions71. This can be observed in the half-metallic Co 2MnSi; when it is treated with LSDA+ U[FLL] in ASA the Fermi level is located at the edge of the gap (see Fig. 1c). Result- ing in a severely underestimated spin wave stiffness with respect to both the LSDA value and the experimental measurements (see Table III). The great importance of the location of the Fermi energy on the magnetic proper- ties can be seen in the cases of Co 2MnAl and Co 2MnGa. In LSDA+ U[FLL], these systems show non integer mo- ments which are overestimated with respect to the ex- perimental measurements (see Table I), but also results in the exchange interactions of the system preferring a ferrimagnetic alignment. Even more the exchange inter- actions can be severely suppressed when the Hubbard U isused. Forexample, forCo 2MnGe inASAthe dominant interaction is between the Co-Mn moments, in LSDA the obtainedvalueis0.79mRy, while inLSDA+ U[FLL]isre- duced to 0.34 mRy, also, the nearest neighbour Co 1-Co2 exchange interaction changes from ferromagnetic to anti- ferromagnetic when going from LSDA to LSDA+ U[FLL] which lead the low values obtainedfor the spin wavestiff- ness. As will be discussed below also for the low Tcfor some of these systems. It is important to notice, that the systems that exhibit the largest deviation from the experimental values, are usually those that under a certain exchange correlation potential and potential geometry loosetheir half-metallic character. Such effect are specially noticeable when one compares LSDA+ U[FLL] results in ASA and FP, where half-metallicity is more easily lost in ASA due to the fact that the pseudogap is much smaller under this ap- proximation than under FP (see Fig. 1). In general, it is important to notice that under ASA the geometry of the potential is imposed, that is non-spherical contribu- tions to the potential are neglected. While this has been shown to be very successful to describe many properties, it does introduce an additional approximation which can lead to anill treatment ofthe properties ofsome systems. Hence, care must be placed when one is considering an ASA treatment for the potential geometry, since it can lead to large variations of the exchange interactions and thus is one of the causes of the large spread on the values observed in Table III for the exchange stiffness and in Table IV for the Curie temperature. One of the key factors behind the small values of the spin stiffness for Co 2FeSi and Co 2FeGe, in comparison with the rest of the Co 2FeZ family, lies in the fact that in LSDA and GGA an antiferromagnetic long-range Fe-Fe interaction is present (see Fig. C.2 in Appendix C). As the magnitude of the Fe-Fe interaction decreases the exchange stiffness increases, e.g. as in LSDA+ U[AMF] with afull potential scheme. Theseexchangeinteractions are one of the factors behind the reduced value of the stiffness, this is evident when comparing with Co 2FeAl, which while having similar nearest neighbour Co-Fe ex- change interactions, overall displays a much larger spin wave stiffness for most of the studied exchange correla- tion potentials. Using LSDA+DMFT[Σ(0)] for Co 2MnSi and Co 2FeSi, the obtained stiffness is 580 meV ˚A2and 280 meV ˚A2re- spectively, whilst in LSDA+DMFT[FLL] for Co 2MnSi thestiffnessis630meV ˚A2andforCo 2FeSiis282meV ˚A2. As can be seen for Co 2MnSi there is a good agree- ment between the KKR LSDA+ U[FLL], the FP-LMTO LSDA+DMFT[FLL] and the experimental values. The agreement with experiments is particularly good when correlation effects are considered as in the LSDA+DMFT[Σ(0)] approach. On the other hand, for Co2FeSi the spin wave stiffness is severely underesti- mated which is once again consistent with what is shown in Table III. Using the calculated exchange interactions, the criti- cal temperature, Tc, for each system can be calculated. Using the ASA, the Tcof both the Co 2MnZ and Co 2FeZ systems is consistently underestimated with respect to experimental results, as shown in Table IV. The same underestimation has been observed in previous theo- retical studies78, for systems such as Co 2Fe(Al,Si) and Co2Mn(Al,Si). However, using a full potential scheme instead leads to Curie temperatures in better agreement with the experimental values, specially when the ex- change correlation potential is considered to be given by the GGA (see Table IV). Such observation is consistent with what was previouslymentioned, regardingthe effect ofthe ASA treatmentonthe spin wavestiffness andmag- netic moments, where in certain cases, ASA was found to not be the best treatment to reproduce the experimen- tal measurements. As mentioned above, this is strongly related to the fact that in general ASA yields a smaller pseudogapin the half-metallic materials, leading to mod- ification of the exchange interactions. Thus, in general, a fullpotentialapproachseemstobeabletobetterdescribe the magnetic properties in the present systems, since the pseudogaparoundthe Fermienergyisbetter describedin a FP approach for a given choice of exchange correlation potential. The inclusionofcorrelationeffects forthe Co 2FeZfam- ily, lead to an increase of the Curie temperature, as for the spin stiffness. This is related to the enhancement of the interatomic exchange interactions as exemplified in the case of Co 2FeSi. However, the choice of DC once more is shown to greatly influence the magnetic proper- ties. For the Co 2FeZ family, AMF results in much larger Tcthan the FLL scheme, whilst for Co 2MnZ the dif- ferences are smaller, with the exception of Z=Al. All these results showcase how important a proper descrip-7 TABLE III. Summary of the spin wave stiffness, Dfor Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge). For the Co 2MnZ family both LSDA and GGA exchange correlation potentials yield val ues close to the experimental measurements. However, for th e Co2FeZ family a larger data spread is observed. The symbol∗implies that the ground state for these systems was found to b e Ferri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic ground state. Quantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe DASA LDA[meV˚A2] 282 291 516 500 644 616 251 206 DASA GGA[meV˚A2] 269 268 538 515 675 415 267 257 DASA LDA+UFLL [meV ˚A2] 29∗487∗205 94 289 289 314 173 DASA LDA+UAMF [meV ˚A2] 259 318 443 417 553 588 235 214 DFP LDA[meV˚A2] 433 405 613 624 692 623 223 275 DFP GGA[meV˚A2] 483 452 691 694 740 730 323 344 DFP LDA+UFLL [meV ˚A2] 447 400 632 577 652 611 461 436 DFP LDA+UAMF [meV ˚A2] 216 348 583 579 771 690 557 563 Dexp[meV˚A2] 190722647357568-5346941374370754967671577— tion of the pseudogap region is in determining the mag- netic properties of the system. Another observation, is the fact that even if a given combination of exchange correlation potential and geo- metrical treatment of the potential can yield a value of Tcin agreementwith experiments, it does not necessarily means that the spin wave stiffness is correctly predicted (see Table III and Table IV). When considering the LSDA+DMFT[Σ(0)] scheme, critical temperatures of 688 K and 663 K are ob- tained for Co 2MnSi and Co 2FeSi, respectively. Thus, the values of the Tcare underestimated in compari- son with the LSDA+ Uor LSDA results. The reason for such behaviour becomes clear when one looks di- rectly on the Jij’s, computed with the different schemes, which are shown in Appendix C. These results sug- gest that taking into account the dynamical correlations (LSDA+DMFT[Σ(0)]) slightly suppresses most of the Jij’s as compared to the LSDA outcome. This is an expected result, since the employed choice of DC correc- tion preserves the exchange splitting obtained in LSDA, while the dynamical self-energy, entering the Green’s function, tends to lower its magnitude. Since these two quantities are the key ingredients defining the strength of the exchange couplings, the Jij’s obtained in DMFT are very similar to those of LSDA (see e.g. Refs. 41 and 81). The situation is a bit different if one employs FLL DC, since an additional static correction enhances the local exchange splitting.82For instance, in case of Co2MnSi the LSDA+DMFT[FLL] scheme provided a Tc of 764 K, which is closer to the experiment. The con- sistently better agreement of the LSDA+ U[FLL] and LSDA+DMFT[FLL] estimates of the Tcwith experimen- tal values might indicate that explicit account for static local correlations is important for the all considered sys- tems. Using the calculated exchange interactions, it is also possible to determine the adiabatic magnon spectra (AMS). In Fig. 2 is shown the effect that different ex- change correlation potentials have overthe description ofthe magnon dispersion relation of Co 2FeSi is shown. The most noticeable effect between different treatments of the exchange correlation potential is shifting the magnon spectra, while its overall shape seems to be conserved. This is a direct result from the enhancement of nearest neighbour interactions (see Fig. C.2). When comparing the AMS treatment with the dy- namical structure factor, S(q,ω), atT= 300 K and damping parameter αLSDA= 0.004, obtained from first principles calculations (details explained in section V), a good agreement at the long wavelength limit is found. However, a slight softening can be observed compared to the AMS. Such differences can be explained due to temperature effects included in the spin dynamics sim- ulations. Due to the fact that the critical temperature of the system is much larger than T= 300 K (see Ta- ble IV), temperature effects are quite small. The high energy optical branches are also softened and in general are much less visible. This is expected since the correla- tion was studied using only vectors in the first Brillouin zone and as has been shown in previous works50, a phase shift is sometimes necessary to properly reproduce the optical branches, implying the need of vectors outside the first Brillouin zone. Also, Stoner excitations dealing with electron-holeexcitations arenot included in this ap- proach,whichresultintheLandaudampingwhichaffects the intensity of the optical branches. Such effects are not captured by the present approach, but can be studied by other methods such as time dependent DFT83. The shape of the dispersion relationalong the path Γ −Xalso corresponds quite well with previous theoretical calcula- tions performed by K¨ ubler84. B. Ferrimagnetic Mn 2VZ with Z = (Al,Si,Ga,Ge) Asmentionedabove,theMnbasedMn 2VZfullHeusler family has a ferrimagnetic ground state, with the Mn atoms orienting parallel to each other and anti-parallel with respect to the V moments. For all the studied sys-8 TABLE IV. Summary of the critical temperature for Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge), with different exchange correlation potentials and shape of the potentials. The sym bol∗implies that the ground state for these systems was found to b e Ferri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic ground state. Quantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe TLDA cASA [K] 360 350 750 700 913 917 655 650 TGGA cASA [K] 350 300 763 700 975 973 800 750 TLDA+U cASAFLL[K] 50∗625∗125 225 575 550 994 475 TLDA+U cASAAMF[K] 325 425 650 600 950 950 650 625 TLDA cFP [K] 525 475 875 825 1050 975 750 750 TGGA cFP [K] 600 525 1000 925 1150 1100 900 875 TLDA+U cFPFLL[K] 525 475 950 875 1050 975 1050 1075 TLDA+U cFPAMF[K] 450 450 1000 875 1275 1225 1450 1350 Texp c[K] 69778694 98513905 10007910938011002498158 TABLE V. Summary of the spin wave stiffness, D, and the critical temperature for Mn 2VZ with Z = (Al ,Si,Ga,Ge) for different shapes of the potential and exchange correlation p o- tentials. Quantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe DASA LDA[meV˚A2] 314 114 147 DASA GGA[meV˚A2] 324 73 149 DFP LDA[meV˚A2] 421 206 191 DFP GGA[meV˚A2] 415 91 162 Dexp[meV˚A2] 53485— — — TLDA cASA [K] 275 350 150 147 TGGA cASA [K] 425 425 250 250 TLDA cFP [K] 425 450 200 200 TGGA cFP [K] 600 500 350 350 Texp c[K] 7688578366— — temstheMn-Mnnearestneighbourexchangeinteractions dominates. In Table V the obtained spin wave stiffness, D, and critical temperature Tcare shown. For Mn 2VAl, it can be seen that the spin wave stiffness is trend when compared to the experimental value. The same under- estimation can be observed in the critical temperature. For Mn 2VAl, one may notice that the best agreement with experiments is obtained for GGA in FP. An inter- esting aspect of the high Tcobserved in these materials is the fact that the magnetic order is stabilized due to the anti-ferromagnetic interaction between the Mn and V sublattices, since the Mn-Mn interaction is in general much smaller than the Co-Co, Co-Mn and Co-Fe inter- actions present in the previously studied ferromagnetic materials. For these systems it can be seen that in general the FP descriptionyields Tc’swhichareinbetter agreementwith experiment, albeit if the values are still underestimated. As for the Co based systems the full potential technique improves the description of the pseudogap, it is impor- tant to notice that for most systems both in ASA and FP the half-metallic characteris preserved. However, the density of states at the Fermi level changes which could lead to changes in the exchange interactions.As for the ferromagnetic systems one can calculate the magnon dispersion relation and it is reported in Fig. 3 for Mn 2VAl. A comparison with Fig. 2 illustrates some of the differences between the dispersion relation of a fer- romagnet and of a ferrimagnetic material. In Fig. 3 some overlap between the acoustic and optical branches is ob- served, as well as a quite flat dispersion relation for one of the optical branches. Such an effect is not observed in the studied ferromagnetic cases. In general the different exchange correlation potentials only tend to shift the en- ergy of the magnetic excitations, while the overall shape of the dispersion does not change noticeably, which is consistent with what was seen in the ferromagnetic case. The observed differences between the LSDA and GGA results in the small qlimit, corresponds quite well with whatisobservedinTableV, wherethe spinwavestiffness for GGA with the potential given by ASA is somewhat largerthan the LSDA case. This is directly related to the observation that the nearest neighbour Mn-Mn and Mn- V interactions are large in GGA than in LSDA. Again, such observation is tied to the DOS at the Fermi level, since Mn 2VAl is not half-metallic in LSDA, on the other hand in GGA the half-metallic state is obtained (see Ta- ble. II. V. GILBERT DAMPING The Gilbert damping is calculated for all the previ- ously studied systems using ASA and a fully relativistic treatment. In Fig. 4, the temperature dependence of the Gilbert damping for Co 2MnSi is reported for different exchange-correlationpotentials. Whencorrelationeffects are neglected or included via the LSDA+ U[AMF], the dampingincreaseswith temperature. Onthe otherhand, in the LSDA+ U[FLL] scheme, the damping decreases as a function of temperature, and its overall magnitude is much larger. Such observation can be explained from the fact that in this approximation a small amount of states exists at the Fermi energyin the pseudogapregion, hence resulting in a larger damping than in the half-metallic9 0100200300400500 Γ X W L ΓEnergy [meV]FP-LSDA DMFT[Σ(0)]a) FIG. 2. (Color online) a) Adiabatic magnon spectra for Co2FeSi for different exchange correlation potentials. In the case of FP-LSDA and LSDA+DMFT[Σ(0)] the larger devia- tionsareobservedinthecase ofhighenergies, withtheDMFT curve having a lower maximum than the LSDA results. In b) a comparison of the adiabatic magnon spectra (solid lines) with the dynamical structure factor S(q,ω) atT= 300 K, when the shape of the potential is considered to be given by the atomic sphere approximation and the exchange cor- relation potential to be given by LSDA, some softening can be observed due to temperature effects specially observed at higher q-points. cases(see Fig. 1c). In general the magnitude of the damping, αLSDA= 7.4×10−4, is underestimated with respect to older ex- perimental measurements at room temperature, which yielded values of α= [0.003−0.006]86andα∼0.025 for polycrystalline samples87, whilst it agrees with previ- ously performed theoretical calculations14. Such discrep- ancy between the experimental and theoretical results could stem from the fact that in the theoretical calcula- tions only the intrinsic damping is calculated, while in experimental measurements in addition extrinsic effects such as eddy currents and magnon-magnon scattering can affect the obtained values. It is also known that sam-FIG. 3. (Color online) Adiabatic magnon dispersion relatio n for Mn 2VAl when different exchange correlation potentials are considered. In general only a shift in energy is observed when considering LSDA or GGA with the overall shape being conserved. 00.511.522.533.54 50 100 150 200 250 300 350 400 450 500Gilbert damping (10-3) Temperature [K]LSDA GGA LSDA+U [FLL] LSDA+U [AMF] FIG.4. (Color online)TemperaturedependenceoftheGilber t damping for Co 2MnSi for different exchange correlation po- tentials. For LSDA, GGA and LSDA+ U[AMF] exchange cor- relation potentials the damping increases with temperatur e, whilst for LSDA+ U[FLL]thedampingdecreases as afunction of temperature. ple capping or sample termination, can have profound ef- fects over the half-metallicity of Co 2MnSi88. Recent ex- periments showed that ultra-low damping, α= 7×10−4, for Co 1.9Mn1.1Si can be measured when the capping is chosen such that the half-metallicity is preserved89, which is in very good agreement with the present theo- retical calculations. In Fig. 5, the Gilbert damping at T= 300 K for the different Heusler alloys as a function of the density of states at the Fermi level is presented. As expected, the increased density of states at the Fermi energy results in10 FIG. 5. (Color online) Gilbert damping for different Heusler alloys at T= 300 K as a function of density of states at the Fermi energy for LSDA exchange correlation potential. In general the damping increases as the density of states at the Fermi Energy increases (the dotted line is to guide the eyes) . an increased damping. Also it can be seen that in gen- eral, alloys belonging to a given family have quite similar damping parameter, except for Co 2FeSi and Co 2FeGe. Their anomalous behaviour, stems from the fact that in the LSDA approach both Co 2FeSi and Co 2FeGe are not half-metals. Such clear dependence on the density of states is expected, since the spin orbit coupling is small for these materials, meaning that the dominating con- tribution to the damping comes from the details of the density of states around the Fermi energy90,91. 1. Effects of substitutional disorder In order to investigate the possibility to influ- ence the damping, we performed calculations for the chemically disordered Heusler alloys Co 2Mn1−xFexSi, Co2MeAl1−xSixand Co 2MeGa 1−xGexwhere Me = (Mn,Fe). Due to the small difference between the lattice param- eters of Co 2MnSi and Co 2FeSi, the lattice constant is unchanged when varying the concentration of Fe. This is expected to play a minor role on the following results. When one considers only atomic displacement contribu- tions to the damping (see Fig. 6a), the obtained values are clearlyunderestimated in comparisonwith the exper- imental measurements at room temperature92. Under the LSDA, GGA and LSDA+ U[AMF] treatments, the damping is shown to increase with increasing concentra- tion of Fe. On the other hand, in LSDA+ U[FLL] the damping at low concentrations of Fe is much larger than inthe othercases, andit decreaseswith Feconcentration, until a minima is found at Fe concentration of x∼0.8. This increase can be related to the DOS at the Fermi energy, which is reported in Fig. 1c for Co 2MnSi. One00.511.522.533.544.55 0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3) Fe concentrationLSDA GGA LSDA+ U[FLL] LSDA+ U[AMF]a) 00.511.522.533.544.5 0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3) Fe concentrationLSDA GGA LSDA+ U[FLL] LSDA+ U[AMF]b) FIG. 6. (Color online) Gilbert damping for the random alloy Co2Mn1−xFexSi as a function of the Fe concentration at T= 300 K when a) only atomic deisplacements are considered and b) when both atomic displacements and spin fluctuations are considered. can observe a small amount of states at EF, which could lead to increased values of the damping in comparison with the ones obtained in traditional LSDA. As for the pure alloys, a general trend relating the variation of the DOS at the Fermi level and the damping with respect to the variation of Fe concentration can be obtained, anal- ogous to the results shown in Fig. 5. When spin fluctuations are considered in addition to the atomic displacements contribution, the magnitude of the damping increases considerably, as shown in Fig. 6b. This is specially noticeable at low concentrations of Fe. Mn rich alloys have a Tclower than the Fe rich ones, thus resulting in larger spin fluctuations at T= 300 K. The overall trend for LSDA and GGA is modified at low concentrations of Fe when spin fluctuations are consid- ered, whilst for LSDA+ U[FLL] the changes in the trends occur mostly at concentrations between x= [0.3−0.8]. An important aspect is the overall good agreement of11 LSDA, GGA and LSDA+ U[AMF]. Instead results ob- tained in LSDA+ U[FLL] stand out as different from the rest. This is is expected since as was previously men- tioned the FLL DC is not the most appropriate scheme to treat these systems. An example of such inadequacy can clearly be seen in Fig. 6b for Mn rich concentrations, where the damping is much larger with respect to the other curves. As mentioned above, this could result from the appearance of states at the Fermi level. Overall the magnitude of the intrinsic damping pre- sented here is smaller than the values reported in experi- ments92, whichreportvaluesforthedampingofCo 2MnSi ofα∼0.005 and α∼0.020 for Co 2FeSi, in comparison with the calculated values of αLSDA= 7.4×10−4and αLSDA= 4.1×10−3for Co 2MnSi and Co 2FeSi, respec- tively. In experiments also a minimum at the concentra- tion of Fe of x∼0.4 is present, while such minima is not seen in the present calculations. However, similar trends as those reported here (for LSDA and GGA) are seen in the work by Oogane and Mizukami15. A possible reason behind the discrepancy between theory and experiment, could stem from the fact that as the Fe concentration increases, correlation effects also increase in relative im- portance. Such a situation cannot be easily described through the computational techniques used in this work, andwill affectthe detailsofthe DOSatthe Fermienergy, which in turn could modify the damping. Another im- portant factor influencing the agreement between theory and experiments arise form the difficulties in separating extrinsic and intrinsic damping in experiments93. This, combined with the large spread in the values reported in various experimental studies87,94,95, points towards the need of improving both theoretical and experimental ap- proaches,ifoneintendstodeterminetheminimumdamp- ing attainable for these alloys with sufficient accuracy. Up until now in the present work, disorder effects have been considered at the Y site of the Heusler struc- ture. In the following chemical disorder will be consid- ered on the Z site instead. Hence, the chemical structure changes to the type Co 2MeZA 1−xZB x(Me=Fe,Mn). The alloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xare consid- ered. The lattice constant for the off stoichiometric com- positions is treated using Vegard’s law96, interpolating between the values given in Table I. In Fig. 7 the dependence of the damping on the con- centration of defects is reported, as obtained in LSDA. For Co 2FeGaxGe1−xas the concentration of defects in- creases the damping decreases. Such a behaviour can be understood by inspecting the density of states at the Fermi level which follows the same trend, it is important to notice that Co 2FeGa is a half-metallic system, while Co2FeGe is not (see table I). On the other hand, for Co2FeAlxSi1−x, the damping increases slightly with Al concentration, however, for the stoichiometric Co 2FeAl is reached the damping decreases suddenly, as in the pre- vious case. This is a direct consequence of the fact that Co2FeAl is a half metal and Co 2FeSi is not, hence when the half-metallic state is reached a sudden decrease ofthe damping is observed. For the Mn based systems, as the concentration of defects increases the damping in- creases, this stark difference with the Fe based systems. For Co 2MnAlxSi1−xthis is related to the fact that both Co2MnAl and Co 2MnSi are half-metals in LSDA, hence, the increase is only related to the fact that the damp- ing for Co 2MnAl is larger than the one of Co 2MnSi, it is also relevant to mention, that the trend obtained here corresponds quite well with what is observed in both ex- perimental and theoretical results in Ref.86. A similar explanation can be used for the Co 2MnGa xGe1−xalloys, as both are half-metallic in LSDA. As expected, the half metallic Heuslers have a lower Gilbert damping than the other ones, as shown in Fig. 7. 00.511.522.533.544.5 0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3) Concentration of defectsCo2FeAlxSi1-xCo2FeGaxGe1-xCo2MnAlxSi1-xCo2MnGaxGe1-x FIG. 7. (color online) Dependence of the Gilbert damping for the alloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xwith Me denoting Mn or Fe under the LSDA exchange correlation po- tential. VI. CONCLUSIONS The treatment of several families of half-metallic Heusler alloys has been systematically investigated us- ing several approximations for the exchange correlation potential, as well as for the shape of the potential. Spe- cial care has been paid to the calculation of their mag- netic properties, such as the Heisenberg exchange inter- actions and the Gilbert damping. Profound differences have been found in the description of the systems de- pending on the choice of exchange correlation potentials, speciallyforsystems in whichcorrelationeffects might be necessarytoproperlydescribethepresumedhalf-metallic nature of the studied alloy. In general, no single combination of exchange correla- tion potential and potential geometry was found to be able to reproduce all the experimentally measured mag- netic properties of a given system simultaneously. Two of the key contributing factors are the exchange correla-12 tion potential and the double counting scheme used to treat correlation effects. The destruction of the half- metallicity of any alloy within the study has profound effects on the critical temperature and spin wave stiff- ness. A clear indication of this fact is that even if the FLL double counting scheme may result in a correct de- scription of the magnetic moments of the system, the exchange interactions may be severely suppressed. For the systems studied with DMFT techniques either mi- nor improvement or results similar to the ones obtained from LSDA is observed. This is consistent with the in- clusion of local d−dscreening, which effectively dimin- ishes the strength of the effective Coulomb interaction with respect to LSDA+ U(for the same Hubbard param- eterU). In general, as expected, the more sophisticated treatment forthe geometricalshape ofthe potential, that is a full potential scheme, yields results closer to experi- ments, which in these systems, is intrinsically related to the description of the pseudogap region. Finally, the Gilbert damping is underestimated with respect to experimental measurements, but in good agreement with previous theoretical calculations. One of the possible reasons being the difficulty from the experi- mental point of view of separating intrinsic and extrinsic contributions to the damping, as well as the strong de- pendence of the damping on the crystalline structure. A clear correlation between the density of states at the Fermi level and the damping is also observed, which is related to the presence of a small spin orbit coupling in these systems. This highlights the importance that half-metallic materials, and their alloys, have in possible spintronic and magnonic applications due to their low in- trinsic damping, and tunable magnetodynamic variables. These results could spark interest from the experimental community due to the possibility of obtaining ultra-low damping in half-metallic Heusler alloys. VII. ACKNOWLEDGEMENTS The authors acknowledge valuable discussions with M.I. Katsnelsson and A.I. Lichtenstein. The work was financed through the VR (Swedish Research Council) and GGS (G¨ oran Gustafssons Foundation). O.E. ac- knowledges support form the KAW foundation (grants 2013.0020 and 2012.0031). O.E. and A.B acknowledge eSSENCE. L.B acknowledge support from the Swedish e-Science Research Centre (SeRC). The computer sim- ulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC) and High Performance Computing Center North (HPC2N). Appendix A: DOS from LSDA+DMFT Here we show the DOS in Co 2MnSi and Co 2FeSi ob- tained from LSDA and LSDA+DMFT calculations. The0369n↑tot[sts./eV] 0 3 6 9-6 -3 0 3n↓tot[sts./eV] E-EF[eV]LSDA 0369n↑tot[sts./eV] 0 3 6 9-6 -3 0 3n↓tot[sts./eV] E-EF[eV]LSDA DMFT[Σ(0)] DMFT[FLL] FIG. A.1. (color online) DOS in Co 2FeSi (top panel) and Co2MnSi (bottom panel) obtained in different computational setups. results shown in Fig. A.1 indicate that the DMFT in- creases the spin-down (pseudo-)gap in both Co 2FeSi and Co2MnSi. In the latter casethe shift ofthe bands is more pronounced. InCo 2FeSiitmanifestsitselfinanenhanced value of the total magnetization. For both studied sys- tems, the FLL DC results in relatively larger values of the gaps as compared with the “Σ(0)”estimates. How- ever, for the same choice of the DC this gap appears to be smaller in LSDA+DMFT than in LSDA+ U. Present conclusion is valid for both Co 2FeSi and Co 2MnSi (see Fig. 1 for comparison.) Appendix B: NQS in Co 2MnSi Here we show the calculated spectral functions in Co2MnSi obtained with LSDA+DMFT[Σ(0)] approach. As discussed in the main text, the overall shape of DOS is reminiscent of that obtained in LSDA. However, a cer- tain amount of the spectral weight appears above the minority-spin gap. An inspection of the imaginary part of the self-energy in minority-spin channel, shown in the bottom panel of Fig. B.1, suggests a strong increase of Mn spin-down contribution at the corresponding ener- gies, thus confirming the non-quasiparticle nature of the obtained states. We note that the use of FLL DC formu- lationresultsinanenhancedspin-downgapwhichpushes the NQS to appear at even higher energies above EF(see Appendix A).13 -30030PDOS [sts./Ry] -0.2-0.10Im [ Σ↑] Co Eg Co T2g -0.2 -0.1 0 0.1 0.2 E-EF [Ry]-0.2-0.10Im [ Σ↓] Mn Eg Mn T2g FIG. B.1. (color online) Top panel: DOS in Co 2MnSi pro- jected onto Mn and Co 3 dstates of different symmetry. Mid- dle and bottom panels: Orbital-resolved spin-up and spin- down imaginary parts of the self-energy. The results are shown for the “Σ(0)”DC. Appendix C: Impact of correlation effects on the Jij’s in Co 2MnSi and Co 2FeSi In this section we present a comparison of the ex- change parameters calculated in the framework of the LSDA+DMFT using different DC terms. The calculated Jij’s between different magnetic atoms within the first few coordination spheres are shown in Fig. C.1. One can see that the leading interactions which stabilize the fer- romagnetism in these systems are the nearest-neighbour intra-sublattice couplings between Co and Fe(Mn) atoms and, to a lower extend, the interaction between two Co atoms belonging to the different sublattices. This qual- itative behaviour is obtained independently of the em- ployed method for treating correlation effects and is in good agreement with prior DFT studies. As explained in the main text, the LSDA and LSDA+DMFT[Σ(0)] re- sults are more similar to each other, whereas most of the Jij’s extracted from LSDA+DMFT[FLL] are relatively enhanced due to inclusion of an additional static contri- bution to the exchange splitting. This is also reflected in both values of the spin stiffness and the Tc. In order to have a further insight into the details of the magnetic interactions in the system, we report here the orbital-resolved Jij’s between the nearest-neighbours obtained with LSDA. The results, shown in Table. C.1, reveal few interesting observations. First of all, all the0.6 1.2 1.8 2.4-0.0500.050.1Jij[mRy] 0.6 1.2 1.8 2.400.10.2 0.6 1.2 1.8 2.4 Rij/aalat00.10.20.30.4Jij[mRy] 0.6 1.2 1.8 2.4 Rij/aalat00.511.5 LSDA LSDA+DMFT [ Σ0] LSDA+DMFT [FLL]Co1-Co1Mn-Mn Co1-Co2Co-Mn FIG. C.1. (color online) The calculated exchange parameter s in Co 2MnSi within LSDA and LSDA+DMFT for different choice of DC. TABLEC.1. Orbital-resolved Jij’sbetweenthenearestneigh- bours in Co 2MnSi in mRy. In the case of Co 1-Co1, the second nearest neighbour value is given, due to smallness of the firs t one. The results were obtained with LSDA. TotalEg−EgT2g−T2gEg−T2gT2g−Eg Co1-Co10.070 0.077 -0.003 -0.002 -0.002 Co1-Co20.295 0.357 -0.058 -0.002 -0.002 Co-Mn 1.237 0.422 -0.079 0.700 0.194 Mn-Mn 0.124 -0.082 0.118 0.044 0.044 T2g-derived contributions are negligible for all the inter- actions involving Co atoms. This has to do with the fact that these orbitals are practicallyfilled and therefore can not participate in the exchange interactions. As to the most dominant Co-Mn interaction, the Eg−Egand Eg−T2gcontributions are both strong and contribute to the total ferromagnetic coupling. This is related to strong spin polarisation of the Mn- Egstates. 00.050.1 0.6 1.2 1.8 2.4Jij[mRy] -0.15-0.1-0.0500.050.1 0.61.21.82.4 00.20.40.6 0.61.21.82.4Jij[mRy] Rij/alat00.511.522.53 0.61.21.82.4 Rij/alatLSDA LSDA+U[FLL] LDA+U[AMF]Co1-Co1 Fe-Fe Co1-Co2 Co-Fe FIG. C.2. Exchange interactions for Co 2FeSi within LSDA and LSDA+ Uschemes and a full potential approach for dif- ferent DC choices.14 Correlationeffectsalsohaveprofoundeffectsontheex- change interactions of Co 2FeSi. In particular, the Fe-Fe interactions can be dramatically changed when consid- ering static correlation effects. 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2001.06217v1.Fermi_Level_Controlled_Ultrafast_Demagnetization_Mechanism_in_Half_Metallic_Heusler_Alloy.pdf

Fermi Level Controlled Ultrafast Demagnetization Mechanism in Half -Metallic Heusler Alloy Santanu Pan1, Takeshi Seki2,3, Koki Takanashi2,3,4, and Anjan Barman1,* 1Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India. 2Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan. 3Center for Spintronics Research Network, Tohoku University, Sendai 980 -8577, Japan. 4Center f or Science and Innovation in Spintronics, Core Research Cluster, Tohoku University, Sendai 980-8577, Japan . *E-mail: abarman@bose.res.in The electronic band structure -controlled ultrafast demagnetization mechanism in Co2FexMn 1- xSi Heusler alloy is underpinned by systematic variation of composition. We find the spin-flip scattering rate controlled by spin density of states at Fermi level is responsible for non- monotonic variation of ultrafast demagnetization time (τ M) with x with a maximum at x = 0.4 . Furthermore, Gilbert damping constant exhibits an inverse relationship with τM due to the dominance of inter -band scattering mechanism. This establishes a unified mechanism of ultrafast spin dynamics based on Fermi level position. The tremendous application potential of spin -polarized Heusler alloys in advanced spintronic s devices ignites immense interest to investigate the degree and sustainability of their spin- polarization under various conditions [1-4]. However, interpreting spin -polarization from the conventional methods such as photoemission, spin transport measurement, point contact Andreev reflection and spin-resolved positron annihilation are non -trivial [5-7]. In the quest of developing alternative methods, Zhang et al . demonstrated that all -optical ultrafast demagne tization measurement is a reliable technique for probing spin -polarization [8]. They observed a very large ultrafast demagnetization time as a signature of high spin -polarization in half-metallic CrO 2. However, Co -based half -metallic Heusler alloys exhibit a comparatively smaller ultrafast demagnetization time (~ 0.3 ps) which raised a serious debate on the perception of ultrafast demagnetization mechanism in Heusler alloys [9-11]. A smaller demagnetization time in Heusler alloys than in CrO 2 is explained d ue to the smaller effective band gap in the minority spin band and enhanced spin-flip scattering (SFS) rate [9]. However, further experimental evidence shows that the amount of band gap in minority spin band cannot be the only deciding factors for SFS medi ated ultrafast demagnetization efficiency [10]. Rather, one also has to consider the efficiency of optical excitation for majority and minority spin bands as well as the optical pump -induced hole dynamics below Fermi energy (EF). Consequently, a clear interpretation of spin -polarization from ultrafast demagnetization measurement requires a clear and thorough understanding of its underlying mechanism. Since its inception in 1996 [12], several theoretical models and experimental evi dences based on different microscopic mechanisms, e.g. spin -flip scattering (SFS) and super -diffusive spin current have been put forward to interpret ultrafast demagnetization [13-20]. However, the preceding proposals are complex and deterring to each othe r. This complexity increases even more in case of special class of material such as the Heusler alloys. The electronic band structure and the associated position of Fermi level can be greatly tuned by tuning the alloy composition of Heusler alloy [21,22]. By utilizing this tunability, h ere, we experimentally demonstrate that the ultrafast demagnetization mechanism relies on the spin density of states at Fermi level in case of half - metallic Heusler alloy system. We extracted the value of ultrafast demagnetiz ation time using three temperature modelling [23] and found its non -monotonic dependency on alloy composition ( x). We have further showed that the Gilbert damping and ultrafast demagnetization time are inversely proportional in CFMS Heusler alloys suggesti ng the inter - band scattering as the primary mechanism behind the Gilbert damping in CFMS Heusler alloys . Our work has established a unified theory of ultrafast spin dynamics. A series of Co 2FexMn 1-xSi (CFMS) thin films have been deposited using magnetron co - sputtering system for our investigation with x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00 . The thickness of the CFMS layer was fixed at 30 nm. It is imperative to study the crystalline phase which is the most crucial parameter that determines other magnetic properties of Heusler alloy. Prior to the magnetization dynamics measurement, we invest igate both the crystalline phase as well as growth quality of all the samples. Fig. 1A shows the ex-situ x-ray diffraction (XRD) pattern for all the samples. The well -defined diffraction peak of CFMS (400) at 2θ = 66.50º indicates that the samples are well crystalline having cubic symmetry. The intense superlattice peak at 2θ = 31.90º represents the formation of B2 phase. The presence of other crucial planes are investigated by tilting the sample x = 0.4 by 54.5º and 45.2º from the film plane to the normal direction, respectively and observed the presence of (111) superlattice peak along with the (220) fundamental peak as shown in Fig. 1B and 1C. The presence of (111) superlattice peak confirms the best atomic site ordering in the desired L2 1 ordered phase, whereas the (220) fundamental peak results from the cubic symmetry. The intensity ratios of the XRD peaks are analysed to obtain the microscopic atomic site ordering which remain same for the whole range of x (given in Supplemental Materials). The epitaxia l growth of the thin films is ensured by observing the in-situ reflection high -energy electron diffraction (RHEED) images. The square shaped hysteresis loops obtained using in -plane bias magnetic field shows the samples have in - plane magnetization. The nearly increasing trend of saturation magnetization with alloy composition ( x) follow the Slater -Pauling curve. In -depth details of sample deposition procedure, RHEED pattern and the hysteresis loops are provided in the Supplemental Materials [24]. The ultrafast demagnetization dynamics measurements using time-resolved magneto - optical Kerr effect (TRMOKE) magnetometer have been performed at a fixed probe fluence of 0.5 mJ/cm2, while the pump fluence have been varied over a large range . Details of the TRMOKE technique is provided in Supplemental Materials [24]. The experi mental data of variation of Kerr rotation corresponding to the ultrafast demagnetization measured for pump fluence = 9.5 mJ/cm2 is plotted in Fig. 2A for different values of x. The data points are then fitted with a phenomenological expression derived from the three temperature model -based coupled rate equations in order to extract the ultrafast demagnetization time ( Mτ) and fast relaxation ( Eτ) time [23], which is given below: ME/τ - /τ- 1 2 E 1 M E 1 2 k3 1/2 0 E M E MA (A τ -A τ ) τ (A -A )-Δ {[ - e - e ]H( ) A δ( )} G( )( / t 1) ( τ -τ ) (τ -τ )ttθ t t tt= + + (1) where A1 represents the magnetization amplitude after equilibrium between electron, spin and lattice is restored, A2 is proportional to the maximum rise in the electron temperature and A3 represents the state filling effects during pump -probe temporal overlap described by a Dirac delta function. H(t) and δ(t) are the Heaviside step and Dirac delta functions , and G(t) is a Gaussian function which corresponds to the laser pulse. The Mτ extracted from the fit s are plotted as a function of x in Fig. 2B, which shows a slight initial increment followed by a sharp decrement with x. In addition, the ultrafast demagnetization rate is found to be slower in the present Heusler alloys than in the 3d metals [9]. The theoretical calculation of electronic band structure of CFM S showed no discernible change in the amount of energy gap in minority spin band but a change in position of EF with x, which lies at the two extreme ends of the gap for x = 0 and x = 1. Thus, the variation of Mτ with x clearly indicates that the composition dependent EF position is somehow responsible for the variation in Mτ . This warrants the investigation of ultrafast demagnetization with continuously varying x values between 0 and 1. However, a majority of earlier investigations [10,11,2 5], being focused on exploring the ultrafast demagnetization only of Co 2MnSi ( x = 0) and Co 2FeSi ( x = 1), lack a convincing conclusion about the role of electronic band structure on ultrafast demagnetization mechanism . In case of 3d transition metal ferromagnets, Elliott -Yafet (EY) -based SFS mechanism is believed to be responsible for rapid rise in the spin temperature and ultrafast demagnetization [15]. In this theory it has been shown that a scattering event of an excited electron with a phonon changes the probability to find that electron in one of the spin states, namely the majority spin -up ( ) or minority spin -down ( ) state, thereby delivering angular momentum to the lattice from the electronic system. It arises from the band mixing of majority and minority spin states with similar energy value near the Fermi surface owing to the spin -orbit coupling (SOC). The spin mixing para meter (b2) from the EY theory [26,27] is given by: 2 k k k k b min ( ψ ψ , ψ ψ )= (2) where kψ represent the eigen -state of a single electron and the bar denotes a defined average over all electronic states involved in the EY scattering processes. This equation represents that the spin-mixing due to SFS between spin -up and spin -down states depend o n the number of spin-up ( ) and spin -down ( ) states at the Fermi level, which is already represented by D F. A compact differential equation regarding rate of ultrafast demagnetization dynamics as derived by Koopmans et al. [27], is given below: p C CeT TR (1 coth( ))TTm dmmdt=− (3) where m = M/MS, and Tp, TC, and Te denote the phonon temperature, Curie temperature and electronic temperature, respectively. R is a material specific scaling factor [28], which is calculated to be: 2 sf C ep 2 B D S8a T gRk T D= , (4) where asf, gep, DS represent the SFS probability, coupling between electron and phonon sub - system and magnetic moment divided by the Bohr -magneton ( B ), whereas TD is the Debye temperature and kB represents the Boltzmann constant. Further, the expression for gep is: 22 F P B D ep ep3πD D k T λg2= , where DP, and λep denote the number of polarization states of spins and electron -phonon coupling constant, respectively , and ℏ is the reduced Planck’s constant. Moreover, the ultrafast demagnetization time at low fluence limit can be derived under various approximations as: 0C M 22 F si B CC F( / T )τπD λ k TT= , (5) where C0 = 1/4, siλ is a factor scaling with impurity concentration, and F(T/TC) is a function solely dependent on ( T/TC) [29]. Earlier, it has been shown that a negligible DF in CrO 2 is responsible for large ultrafast demagnetization time. The theoretical calculation for CFMS by Oogane et al. shows that DF initially decreases and then increases with x [30] having a minima at x = 0.4. As DF decreases, the number of effective minority spin states become less, reducing both SOC strength, as shown by Mavropoulos et al. [31], and the effective spin -mixing paramet er is given by Eq. (2), and vice versa. This will result in a reduced SFS probability and rate of demagnetization. In addition, the decrease in DF makes gep weaker, which, in turn, reduces the value of R as evident from Eq. (4). As the value of R diminishes, it will slow down the rate of ultrafast demagnetization which is clear from Eq. (3). In essence , a lower value of DF indicates a lower value of R, i.e. slower demagnetization rate and larger ultrafast demagnetization time. Thus, demagnetization time is highest for x = 0.4. O n both sides of x = 0.4, the value of R will increase and ultrafast demagnetization time will decline continuously. Our experimental results, supported by the existing theoretical re sults for the CFMS samples with varying alloy composition, clearly show that the position of Fermi level is a crucial decisive factor for the rate of ultrafast demagnetization. This happens due to the continuous tunability of DF with x, which causes an ensuing variation in the number of scattering channels available for SFS. To capture the effect of pump fluence on the variation of Mτ, we have measured the ultrafast demagnetization curves for various applied pump fluences. All the flu ence dependent ultrafast demagnetization curves are fitted with Eq. (1) and the values of corresponding Mτ are extracted. The change in Mτ with fluence is shown in Fig. 2C. A slight change in Mτ with fluence is observed which is negligible in comparison to the change of Mτ with x. However, this increment can be explained using the enhanced spin fluctuations at much higher elevated temperature of the spin sy stem [28]. As the primary microscopic channel for spin angular momentum transfer is the same for both ultrafast demagnetization and magnetic damping, it is expected to find a correlation between them. We have measured the time -resolved Kerr rotation data corresponding to the magnetization precession at an applied in -plane bias magnetic field (Hb) of 3.5 kOe as shown in Fig. 3A. The macrospin modelling is employed to analyse the time dependent precessional data obtained by solving the Landau -Lifshitz -Gilbert equation [32] which is given below: effˆˆˆˆγ( ) α( )dm dmm H mdt dt=− + (6) where γ is the gyromagnetic ratio and is related to Lande g factor by /μg=γB . Heff is the total effective magnetic field consisting of Hb, exchange field ( Hex), dipolar field ( Hdip) and anisotropy field ( KH ). The experimental variation of precession frequency ( f) against Hb is fitted with the Kittel formula for uniform precession to extract HK values. The details of the fit are discussed in the Supplementa l Materials [24] . For evaluation of α, all the measured data representing single frequency oscillation are fitted with a general damped sine -wave equation superimposed on a bi -exponential decay function, which is given as: fast slow/τ /τ /τ 12 ( ) A B e B e (0)e sin( ω ζ)tt tM t M t−− −= + + + − , (7) where ζ is the initial phase of oscillation and τ is the precessional relaxation time . fastτ and slowτ are the fast and slow relaxation times, representing the rate of energy transfer in between different energy baths (electron, spin and lattice) following the ultrafast demagnetization and the energy transfer rate between the lattice and surrounding, respec tively. A, B1 and B2 are constant coefficients. The value of α is extracted by further analysing τ using ( )122α[γτ 2 cos( H H ]=− + +bHδφ (8) where 22 12 1S S S S2K 2K sin K (2 sin (2 ))4πMM M MφφH⊥ −= + − + and 12 2 SS2K cos(2 ) 2K cos(4 ) MMφφH=+ . Here and represent the angles of Hb and in -plane equilibrium M with respect to the CFMS [110] axis [33]. The uniaxial, biaxial and out -of-plane magnetic anisotropies are denoted as K1, K2 and K⊥, respectively. In our case K2 has a reasonably large value while K1 and K⊥ are negligibly small. Plugging in all parameters including the magnetic anisotropy constant K2 in Eq. (8), we have obtained the values of α to be 0.0041, 0.0035, 0.0046, 0.0055, 0.0061, and 0.0075 for x = 0.00, 0.40, 0.50, 0.60, 0.75, and 1.00, respectively. Figure 3B shows the variation of α with frequency for all the samples. For each sample, α remains constant with frequency, which rules out the presence of extrinsic mechanisms contributing to the α. Next, we focus on the variation of α with x. Our experimental results show a non -monotonic variation of α with x with a minima at x = 0.4 , which is exactly opposite to the variation of Mτ with x. On the basis of Kambersky’s SFS model [34], α is governed by the spin -orbit interaction and can be expressed as: 22 F Sγ (δg)αD4ΓM= (9) where gδ and 1− represent the deviation of g factor from free electron value (~2.0) and ordinary electron -phonon collision frequency. Eq. (9) suggests that α is directly proportional to DF and thus it become s minimum when DF is minimum [3 0]. This leads to the non -monotonic variation of α , which agrees well with earlier observation [30]. To eliminate the possible effects of γ and SM , we have plotted the variation of relaxation frequency, SMαγ=G with x which also exhibits similar variation as α (see the supplementary materials [24] ). Finally , to explore the correlation between α , Mτ and alloy composition, we have plotted these quantities against x as shown in Fig. 4A. We observe that Mτ and α varies in exactly opposite manner with x, having their respective maxima and minima at x = 0.4. Although Mτ and α refer to two different time scales, both of them follow the trend of variation of DF with x. This shows that the alloy composition -controlled Fermi level tunability and the ensuing SFS is responsible for both ultrafast demagnetization and Gilbert damping . Figure 4B represents the variation of Mτ with inverse of α, which establishes an inversely proportional relation between them . Initially under the assumption of two different magnetic fields, i.e. exchange field and total effective magnetic field, Koopmans et al. theoretically proposed that Gilbert damping parame ter and ultrafast demagnetization time are inversely proportional [29]. However, that raised intense debate and in 2010, Fahnle et al. showed that α can either be proportional or inversely proportional to Mτ depending upon the dominating microscopic contribution to the magnetic damping [32]. The linear relation sustains when the damping is dominated by conductivity -like contribution, whereas the resistivity -like contribution leads to an inverse relation. The basic difference between the conductivity -like and the resistivity -like contribution s lies in the angular momentum transfer mechanism via electron -hole ( e-h) pair generation. The generation of e-h pair in the same band, i.e. intra -band mechanism leads to t he conductivity -like contribution. On the contrary, when e-h pair is generated in different bands (inter -band mechanism), the contribution is dominated by resistivity. Our observation of the inversely proportional relation between α and Mτ clearly indicates that in case of the CFMS Heusler alloy systems, the damping is dominated by resistivity -like contribution arising from inter-band e-h pair generation. This is in contrast to the case of Co, Fe and Ni, where the conductivity contribution dominates [35]. Typical resistivity ( ρ ) values for Co 2MnSi ( x = 0) are 5 cm− at 5 K and 20 cm− at 300 K [36]. The room temperature value of ρ corresponds to an order of magnitude larger contribution of the inter -band e-h pair generation than the intra -band generation [36]. This is in strong agreement with our experimental results and its conclusion. This firmly establishes that unlike convention al transition metal ferromagnets, damping in CFMS Heusler alloys is dominated by resistivity -like contribution , which results in an inversely proportional relation between α and Mτ . In summary, we have investigated the ultrafast demagnetization and magnetic Gilbert damping in the CFMS Heusler alloy systems with varying alloy composition ( x), ranging from x = 0 (CMS) to x = 1 (CFS) and identified a strong correlation between Mτ and x, the latter controlling the position of Fermi level in the electronic band structure of the system. We have found that Mτ varies non -monotonically with x, having a maximum value of ~ 350 fs for x = 0.4 corresponding to the lowest DF and highest degree of spin -polarization. In -depth investigation has revealed that the ultrafast demagnetization process in CFMS is primarily governed by the composition -controlled variation in spin -flip scattering rate due to variable DF. Furthermore, we have systematically investigated the precessional dynamics with variation in x and extracted the value of α from there. Our results have led to a systematic correlation in between Mτ , α and x and we have found an inversely proportional relationship between Mτ and α . Our thorough investigation across the alloy composition ranging from CMS to CFS have firmly establishe d the fact that both ultrafast demagnetization and magnetic Gilbert damping in CFMS are strongly controlled by the spin density of states at Fermi level. Therefore, our study has enlighten ed a new path for qualitative understanding of spin -polarization from ultrafast demagnetization time as well as magnetic Gilbert dampin g and led a step forward for ultrafast magnetoelectronic device applications. Acknowledgements This work was funded by: S. N. Bose National Centre for Basic Sciences under Projects No. SNB/AB/12 -13/96 and No. SNB/AB/18 -19/211. References [1] T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami, T. Miyazaki, H. Naganuma, and Y. Ando, Appl. Phys. Lett. 94, 122504 (2009). [2] A. Hirohata , and K. Takanashi, J. Phys. D: Appl. Phys. 47, 193001 (2014). [3] R. J. Soulen et al., Science 282, 85 (1998). [4] I. I. Mazin, Phys. Rev. Lett. 83, 1427 (1999). [5] K. E. H. M. Hanssen, P. E. Mijnarends, L. P. L. M. Rabou, and K. H. J. Buschow, Phys. Rev. B 42, 1533 (1990). [6] L. Ritchie , Phys. Rev. B 68, 104430 (2003). [7] D. T. Pierce , and F. Meier, Phys. Rev. B 13, 5484 (1976). [8] Q. Zhang, A. V. Nurmikko, G. X. Miao, G. 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Mallick, M. M. Raja, A. K. Nigam, B. S. D. C. S. Varaprasad, Y. K. Takahashi, A. Alam, K. G. Suresh, and K. Hono, Phys. Rev. B 91, 104408 (2015). [23] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M. Swagten, and B. Koopmans, Nat. Phys. 4, 855 (2008). [24] See Supplementary Materials at [URL] for details analysis of XR D peak intensity ratio, RHEED images, hysteresis loops, variation of precession frequency with external bias magnetic field, and the variation of relaxation frequency as a function of alloy composition. [25] D. Steil et al., New J. of Phys. 16, 063068 (2014). [26] R. J. Elliott, Phys. Rev. 96, 266 (1954). [27] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fähnle, T. Roth, M. Cinchetti, and M. Aeschlimann, Nat. Mater. 9, 259 (2010). [28] K. C. Kuiper, T. Roth, A. J. Schellekens, O. Schmitt, B. Koopmans, M. Cinchetti, and M. Aeschlimann, Appl. Phy. Lett. 105, 202402 (2014). [29] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge, Phys. Rev. Lett. 95, 267207 (2005). [30] M. Oogane, T. Kubota, Y. Kota, S. Mizukami, H. Naganuma, A. Sakuma, and Y. Ando, Appl. Phys. Lett. 96, 252501 (2010). [31] P. Mavropoulos, K. Sato, R. Zeller, P. H. Dederichs, V. Popescu, and H. Ebert, Phys. Rev. B 69, 054424 (2004). [32] M. Fähnle, J. Seib, and C. Illg, Phys. Rev. B 82, 144405 (2010). [33] S. Qiao, S. Nie, J. Zhao, and X. Zhang, Appl. Phys. Lett. 105, 172406 (2014). [34] V. Kamberský, Can. J. of Phy. 48, 2906 (1970). [35] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). [36] C. Liu, C. K. A. Mewes, M. Chshiev, T. Mewes, and W. H. Butler, Appl. Phys. Lett. 95, 022509 (2009). Fig. 1. (A) X-ray diffraction (XRD) patterns of Co 2FexMn 1-xSi (CFMS) thin films for different alloy composition ( x) measured in conventional θ-2θ geometry. Both CFMS (200) superlattice and CFMS (400) fundamental peaks are marked along with Cr (200) peak. (B) The tilted XRD patterns reveal the CFMS (111) superlattice peak for L2 1 structure. (C) CFMS (220) fundamental peak together with Cr (110) peak. Fig. 2. (A) Ultrafast demagnetization curves for the samples with different alloy composition ( x) measured using TRMOKE. Scattered symbols are the experimental data and solid lines are fit using Eq. 3. (B) Evolution of Mτ with x at pump fluence of 9.5 mJ/cm2. Symbols are experimental results and dashed line is guide to eye. (C) Variation in Mτ with pump fluence. Fig. 3. (A) Time -resolved Kerr rotation data showing precessional dynamics for samples with different x values . Symbols are the experimental data and solid lines are fit with damped sine wave equation ( Eq. 6). The extracted α values are given below every curve. (B) Variation of α with precession frequency (f) for all samples as shown by symbols, while solid lines are linear fit. Fig. 4. (A) Variation of Mτ and α with x. Square and circular symbols denote the experimental results , and dashed , dotted lines are guide to eye. (B) Variation of Mτ with 1α− . Symbols represent the experimentally obtained values and solid line refers to linear fit. Supplementa l Material s I. Sample preparation method A series of MgO Substrate /Cr (20 nm)/ Co 2FexMn 1-xSi (30 nm)/Al -O (3 nm) sample stacks were deposited using an ultrahigh vacuum magnetron co -sputtering system. First a 20 -nm-thick Cr layer was deposited on top of a single crystal MgO (100) substrate at room temperature (RT) followed by annealing it at 600 ºC for 1 h. Next, a Co 2FexMn 1-xSi layer of 30 nm thickness was deposited on the Cr layer followed by an in -situ annealing process at 500 ºC for 1 h. Finally, each sample stack was capped with a 3 -nm-thick Al -O protective layer. A wide range of values of x is chosen, namely, x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00. To achieve the desired composition of Fe and Mn precisely, the samples were deposited using well controlled co-sputtering of Co 2FeSi and Co 2MnSi. Direct deposition of Co 2FexMn 1-xSi on top of MgO produces strain due to lattice mismatch in the Co 2FexMn 1-xSi layer which alters its intrinsic properties [1S]. Thus, Cr was used as a buffer layer to protect the intrinsic Co 2FexMn 1-xSi layer properties [2S]. II. Details of measurement techniques Using ex-situ x-ray diffraction ( XRD ) measurement we investigated the crystal structure and crystalline phase of the samples. The in-situ reflection high -energy electron diffraction (RHEED ) images were observed after the layer deposition without breaking the vacuum condition in order to investigate the epitaxial relation and surface morphology of Co 2FexMn 1- xSi layer. To quantify the values of M S and H C of the samples, we measured the magnetization vs. in -plane magnetic field (M-H) loops using a vibrating sample magnetometer ( VSM) at room temperature with H directed along the [110] direction of Co 2FexMn 1-xSi. The ultrafast magnetization dynamics for all the samples were measured by using a time-resolved magneto - optical Kerr effect ( TRMOKE ) magnetometer [ 3S]. This is a two -colour pump -probe experiment in non -collinear arrangement. The fundamental output (wavelength, λ = 800 nm, pulse -width, tσ ~ 40 fs) from an amplified laser system (LIBRA, Coherent) acts as probe and its second harmonic signal (λ = 400 nm, tσ ~ 50 fs) acts as pump beam. For investigating both ultrafast demagnetization within few hundreds of femtosecond s and p recessional magnetization dynamics in few hundreds of picosecond time scale, we collected the time - resolved Kerr signal in two different time regimes. The time resolution during the measurements was fixed at 50 fs in -0.5 To 3.5 ps and 5 ps in -0.1 ns to 1 .5 ns to trace both the phenomena precisely. The pump and probe beams were focused using suitable lenses on the sample surface with spot diameters of ~250 µm and ~100 µm, respectively. The reflected signal from the sample surface was collected and analysed using a polarized beam splitter and dual photo detector assembly to extract the Kerr rotation and reflectivity signals separately. A fixed in-plane external bias magnetic field ( Hb) of 1 kOe was applied to saturate the magnetization for measurement of ult rafast demagnetization dynamics, while it was varied over a wide range during precessional dynamics measurement. III. Analysis of XRD peaks To estimate the degree of Co atomic site ordering, one has to calculate the ratio of integrated intensity of (200) and (400) peak. Here, we fit the peaks with Lorentzian profile as shown in inset of Fig. 1S and extracted the integrated intensities as a parameter from the fit . The calculated ratio of I(200) and I(400) with re spect to alloy composition ( x) is sho wn in Fi g. 1S. We note that there is no significant change in the I(200)/I(400) ratio. This result indicates an overall good quality atomic site ordering in the broad range of samples used in our study . Fig. 1S. Variation of integrated intensity ratio I(200)/I(40 0) with x, obtained from XRD patterns. Inset shows the fit to the peaks with Lorentzian profile. IV. Analysis of RHEED pattern The growth quality of the CFMS thin films was experimentally investigated using in-situ RHEED technique. Figure 2S shows the RHEED images captured along the MgO [100] direction for all the samples. All the images contain main thick streak lines in between the thin streak lines , which are marked by the white arrows, suggesting the formation o f ordered phases. The presence of regularly -aligned streak lines confirms the epitaxial growth in all the films. Fig. 2S. In-situ RHEED images for all the Co 2FexMn 1-xSi films taken along the MgO [100] direction. White arrows mark the presence of thin streak lines originating from the L2 1 ordered phase. V. Analysis of magnetic hysteresis loops Figure 3SA represents the M-H loops measured at room temperature using VSM for all the samples. All the loops are square in nature, which indicates a very small saturation magnetic field. We have estimated the values of saturation magnetization ( MS) and coercive field ( HC) from the M-H loops. Figure 3SB represents MS as a function of x showing a nearly monotonic increasing trend, which is consistent with the Slater -Pauling rule for Heusler alloys [4S], i.e. the increment in MS due to the increase in the number of valence electrons. However, it deviates remarkably at x = 1.0. This deviation towards the Fe -rich region is probably due to the slight degradation in the film quality. Figure 3SC shows that HC remains almost constant with variation of x. Fig. 3S. (A) Variation of M with H for all the samples. (B) Variation of MS as a function of x. Symbols are experimentally obtained values and dashed line is a linear fit. (C) Variation of HC with x. VI. Anal ysis of frequency ( f) versus bias magnetic field ( Hb) from TRMOKE measurements We have experimentally investigated the precessional dynamics of all the samples using TRMOKE technique. By varying the external bias magnetic field ( Hb), various precessional dynamics have been measured. The post -processing of these data foll owed by fast Fourier transform (FFT) provides the precessional frequency (f) and this is plotted against Hb as shown in Fig. 4S . To determine the value of in-plane magnetic anisotropy constant , obtained f-Hb curves have been analysed with Kittel formula which is given below: 2 1 2 S S S S2K 2K 2K γ(4πM )( )2π M M Mbb f H H= + + + + (1S) where MS is saturation magnetization and γ denote the gyromagnetic ratio given by Bgμγ= while K1 and K2 represent the two -fold uniaxial and four -fold biaxial magnetic anisotropy constant, respectively. Fig. 4S. Variation of f as a function of Hb. Circular filled symbols represent the experimental data and solid lines are Kittel fit. We have found the values of several parameters from the fit including K1 and K2. K1 has a negligible value while K2 has reasonably large value in our samples. The e xtracted values of the parameters from the fit are tabulated as follows in Table 1S : Table 1S: The extracted values of Lande g factor and the four -fold biaxial magnetic anisotropy constant K 2 for different values of x. x g K2 (erg/cm3) 0.00 2.20 3.1×104 0.40 2.20 2.6×104 0.50 2.20 3.0×104 0.60 2.20 2.5×104 0.75 2.20 2.6×104 1.00 2.20 3.4×104 VII. Variation of r elaxation frequency with alloy composition We have estimated the damping coefficient (α) and presented its variation with alloy composition ( x) in the main manuscript. According to the Slater -Pauling rule, M S increases when the valence electron number systematically increases. As in our case the valence electron number changes with x, one may expect a marginal effect of M S on the estimation of damping. Thus, to rule out any such possibilit ies, we have calculated the variation of relaxation frequency , S GαγM= with x, which is represented in Fig. 5S. It can be clearly observed from Fig. 5S that relaxation frequen cy exactly follows the trend of α . This rules out any possible spurious contribution of M S in magnetic damping. Fig. 5S. Non-monotonic v ariation of G with x for all the samples. References: [1S] S. Pan, S. Mondal, T. Seki, K. Takanashi, , and A. Barman, Influence of the thickness -dependent structural evolution on ultrafast magnetization dynamics in Co 2Fe0.4Mn 0.6Si Heusler alloy thin films. Phys. Rev. B 94, 184417 (2016). [2S] S. Pan, T. Seki, K. Takanashi, and A. Barman, Role of the Cr buffer layer in the thickness - dependent ultrafast magnetization dynamics of Co 2Fe0.4Mn 0.6Si Heusler alloy thin films. Phys. Rev. Appl. 7, 064012 (2017). [3S] S. Panda, S. Mondal, J. Sinha, S. Choudhury, and A. Barman, All-optical det ection of interfacial spin transparency from spin pumping in β -Ta/CoFeB thin films. Science Adv. 5, eaav7200 (2019). [4S] I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Slater -Pauling behavior and origin of half - metallicity of the full Hesuler alloys. Phys. Rev. B 66, 174429 (2002).

1308.0192v1.Inverse_Spin_Hall_Effect_in_nanometer_thick_YIG_Pt_system.pdf

1 Inverse Spin Hall Effect in nanometer -thick YIG/Pt system O. d’Allivy Kelly1, A. Anane1*, R. Bernard1, J. Ben Youssef2, C. Hahn3, A-H. Molpeceres1, C. Carrétéro1, E. Jacquet1, C. Deranlot1, P. Bortolotti1, R. Lebourgois4, J-C. Mage1, G. de Loubens3, O. Klein3, V. Cros1 and A. Fert1 1Unité Mixte de Physique CNRS/Thales and Université Paris -Sud, 1 avenue Augustin Fresnel, Palaiseau, France 2Université de Br etagne Occidentale, LMB -CNRS, Brest, France 3Service de Physique de l’Etat Condensé, C EA/CNRS, Gif-sur-Yvette, France 4Thales Research and Technology, 1 avenue Augustin Fresnel, Palaiseau, France Key words: YIG, FMR, Inverse Spin Hall Effect, spin waves Abstract: High quality nanometer -thick (20 nm , 7 nm and 4 nm) epitaxial YIG films have been grown on GGG substrates using pulsed laser deposition. The Gilbert damping coefficient for the 20 nm thick films is 2.3 x 10-4 which is the lowest value reported for sub -micrometric thick films. We demonstrate Inverse spin Hall effect (ISHE) detection of propagating spin waves using Pt. The amplitu de and the lineshape of the ISHE voltage correlate well to the increase of the Gilbert damping when decreasing thickness of YIG. Spin Hall effect based loss -compensation experiments have been conducted but no change in the magnetization dynamics could be d etected. * Contact author : Abdel madjid Anane abdelmadjid.anane@thalesgroup.com 2 Among all magnetic materials, Yttrium Iron Garnet Y 3Fe5O12 (YIG) has been the one that had the most prominent role in understanding high frequency magnetization dynamics. Because of its unique properties, bulk YIG crystal was the prototypal material for ferromagnetic resonance (FMR) studies in the mid -twentieth ce ntury. The attractive properties of YIG include: high Curie temperature , ultra low damping (the lowest among all materials at room temperature), electrical insulation, high chemical stability and easy synthesis in single crystalline form. Micrometer thick films of YIG were first grown using liquid phase epitaxy (LPE)1, and paved the way for the emergence of a large variety of microwave devices for high -end analogue electronic applic ations throughout the 1970’s 2. More recently, the interest in emerging large -scale integrated circuit technologies for beyond CMOS applications has fostered new paradigms for data processing . Many of them are based on state variables other than the electron charge and may eventually allow for unforeseen functionalities. Coding the information in a spin wave (SW) is among the most promising routes under investigation and has been referred to as magnonics 3. Exciting and detecting spin waves has been ma inly achieved through inductive coupling with radiofrequenc y (rf) anten nas but this technology remains incompatible with large scale integration4. Disruptive solution s merging magnonics and spintronics have been recently proposed where spin transfer torque (STT) and magnetoresistive effects would be used to couple to the SW s. For instance, using a STT -nano -oscillator in a nanocontact geometry, coherent SWs emission in Ni81Fe19 (Py) thin metallic layer has been recently demonstrated and probed by micro -focused Brilloui n light scattering (BLS) 5. YIG is often considered as the best medium for SW propagation because of its very small Gilbert damping coefficient (2x10-5 for bulk YIG) . Being an electrical insulator, electron mediated angular momentum transfer can only occur at the interface between YIG and a metallic layer . In that context , metals with large Spin Orbit Coupling (SOC) like Pt where a pure spin current can be generated through Spin Hall Effect (SHE) 6 have been used to excite7 or amplify8,9 propagating SWs through loss compensation in YIG . Moreover, d etection of SW can be achie ved using the Inverse Spin Hall Effect (ISHE) . In ISHE , the flow of a pure spin current from the YIG into the large SOC metal generates a dc voltage . The ISHE voltage is proportional to the Spin Hall angle () and the effective spin mixing conductance ( ) that is in play in the physics of spin pumping . As for the SOC materials, u p to recently, mainly Pt has been used , however it can be observed that other 5 d heavy metals such as Ta 10, W 11 or CuBi 12 are also very promising . As the amount of angular momentum transferred from (to) the YIG magnetic film per unit volume scales with 1/ (where t is the YIG thickness) , it is necessary to reduce the YIG thickness as much as possible while keeping its magnetic properties . Indeed , the threshold current density in the SOC metal for the macrospin mode excitation is expressed as13 : (Eq. 1) where is the spin Hall angle, the electron charge, the gyromagnetic ratio , the FMR frequency and the YIG’s 3 saturation magnetization . Furthermore, a better understanding of the physics involved in spin momentum transfer at the YIG/metal interface would be achieved by reducing the YIG film thickness below the exchange length (~ 10 nm) 14 . Up to now, sub micrometer -thick YIG films have been mainly grown by LPE but the ultimate thickn ess are around 200 nm15. To further reduce the thickness, other growth methods are to be considered . Pulsed laser deposition (PLD) is the most versatile technique for oxide films epitax y. Several groups have worked on PLD grown YIG16,17,18,19 but it is only recently that the films quality is approaching that of LPE20,21. In this letter , we present PLD growth of ultrathin YIG films with various thickness (20 nm, 7 nm and 4 nm) on Gadolinium Gallium Garnet (GGG) (111) substrates . Structural and magnetic characterizations and FMR measurements demonstrate the high quality of our nanometer -thick YIG films comparable to state of the art LPE films . The growth has been performed using a frequency tripled ( = 355 nm) Nd:YAG laser and a stochiometric polycrystalline YIG pellet . The pulse rate was 2.5 Hz and the substrate -target distance was 4 4 mm. Prior to the YIG deposition, the GGG substrate is annealed at 700°C under an oxygen pressure of 0.4 mbar. Growth temperature is then set to be 650 °C, and oxygen pressure to 0.25 mbar. After the film deposition , samples are cooled down to room temperat ure under 300 mbar of O 2. The YIG thickness is measured for each sample using X -ray reflectometry which yield a precision better than 0.3 nm. The surface morphology and roughness have been studied by atomic force microscopy ( AFM ). RMS roughness has been measured over 1 µm2 ranges between 0.2 nm and 0.3 nm for all films (Fig 1a) . As often with PLD growth, d roplets are present on the film surface, here the ir lateral sizes are below 100 nm and their density is very low (~ 0.1 µm-2). X-Ray Diffraction (XRD) spectr a using Cu K 1 radiation show that the growth is along the (111) direction . Only peaks characteristic of YIG and the GGG substrate are observed ( see Fig 1b ). The YIG lattice parameter is very close to that of the substrate and can only be resolved ev entually at large diffraction angle s. For the 20 nm YIG film (Fig 1b ,1a), refinement using EVA software on the 888 reflection yields a cubic lattice parameter of 1.2459 nm, to be compared to 1.2376 nm for the bulk YIG22. For thinner films (between 4 and 15 nm) , it was not possible to distinguish the YIG peaks from those of the substrate (Fig 1d, 1e). This sharp variation of the XRD spectra with respect to the film thickness tends to point towards a critical thickness for strain relaxation. It is however worth noting that the cubic lattice parameter of the 20 nm thick film is larger than the bulk lattice parameter but also of that to the GGG substrate (1.2383 nm) . A slight o ff-stoichiometry (either oxygen vaca ncies or cation interstitials) is probably at the origin of this observation . Pole figure measurements have been performed to gain insight s into the in -plane crystal structure, but it was not possible to resolve , at this stage, the film s peaks from the substrate peaks from w hich we infer that the growth is epitaxial and the film single crystalline . From SQUID magnetometry with in -plane magnetic field , we measure a magnetization of 4 Ms = 2100 G 50 G at room temperature for both the 20 and 7 nm films . This value is independently 4 confirmed by out -of plan e FMR resonance while the tabulated bulk value for YIG is 4 Ms = 1760 G. A similar increase of the PLD grown YIG magnetization have been reported and attributed to an off- stoichiome try19. The coercitive fields are extremely small , about 0.2 Oe (which is the experimental resolution ) and the saturation field is 5 Oe. There is no evidence for in -plane magnetic anisotropy . The overall magnetic signature is that of an ultra -soft material . Note that for the thinnest films (4 nm) we measure a decrease of the saturation magnetization to roughly 1700 G . Finally, we emphasize that the structural and magnetic properties of the samples are well reproducible with respect to the elaboration conditions. FMR fields and linewidths were measured at frequencies in the range 1 -40 GHz using high sensitive wideband r esonance spectrometer with a nonresonant microstrip transmission line. The FMR is measured via the derivative of microwave power absorption using a small rf exciting field. Resonance spectra were recorded with the applied static magnetic field oriented in plane . During the magnetic field sweeps , the amplitude of the modulation field was appreciably smaller than the FMR linewidth. The amplitude of the excit ation field h rf is about 1 mOe , which corresponds to the linear response regime. A phase -sensitive detector with lock -in detection was used. The field derivative of the absorbed power is proportional to the field derivative of the imaginary part of the rf susceptibility: (where and ″ is the imaginary part of the susceptibility of the uniform mode) . Typical resonance curve s are plotted in Fig. 2a 2b . In Fig. 2c, we show the frequency dependence of the peak -to-peak linewidth for three different YIG thicknesses , i.e., 20, 7 and 4 nm . As for the 20 nm YIG film, we find a linear dependence of the FMR linewidth with rf frequency while for the thinnest films , we do find an almost linear increase in the low frequency range (< 12 GHz) and then a saturation of the linewidth with frequency . Such qualitative difference depending on the thickness is reminiscent of the qualitative difference discussed earlier in the X -ray diffraction data (Fig 1). The linear dependence of the resonance linewidth is expected with in the frame of the Landau - Lifshitz Gilbert equation and allow for a straightforward calculation of the intrinsic Gilbert damping coefficient ( for the 20 nm thick film). The zero frequency intercept of the fitting line , usually referred to a s the extrinsic lin ewidth 23,24, is found to be H0 =1.4 Oe. We emphasize that our value for the intrinsic damping on the 20 nm thick film is among the best ever reported independ ently of the growth technique and is only outperformed by the 1.3 µm film used by Y. Kajiwara et al. 7. As for the extrinsic damping, our value s are still a bit larger than those obtained for 200 nm thick films grown by LPE ( H0= 0.4 Oe) 10. The saturation of the linewidth with increasing excitation frequency observed for thinnest films ( t = 7 and 4 nm) is usually ascribed to two -magnon s scattering due to the interfaces 25. An estimation of the intrinsic damping in such thin films is thus not correct. Nevertheless , and only for the sake of comparison, considering frequencies under 6 GHz , we can rough ly estimate the low frequency Gilbert damping to be 1.610-3 for the 7 nm and 3.810-3 the 4nm YIG films . However, it is worth mentioning that for those two thinnest films; samples sliced from the 5 same substrate can give different linewidths ( up to a factor of 3) with for some of them up to 2 absorption s lines. This observation point s to a slight lateral non-homogeneity in the chemical composition24. The data presented in figure 2 are tho se of the best samples showing a single absorption line. For the 20 nm thick films , all samples have only one resonance line and the dispersion of linewidths is within 5%. In order to characterize the conversion of propagating SWs in YIG into a charge current in a normal adjacent metal with large SOC , we perform ISHE detection of SW with the strip geometry used by Chumak et al.26. In our sample design (see Fig 3a) , SWs excitation is achieved using a patterned 100 µm wide Au stripline antenna whereas the ISHE voltage is measured on a 13 nm thick Pt strip ( 0.2 mm x 5 mm ) located at 100 µm away from the Au stripe and parallel to it . The metallic Pt strip is deposited using dc magnetron sputtering and lift -off. Prior to the Pt deposition , an in -situ O2/Ar- plasma is used to remove the photo -resist residue s and increase the ISHE voltage. This cleaning step has been shown recently to improve the ISHE signal by one order of magnitude27,28. Measurement s of the ISHE signal is performed either using a lock -in (with a 5 kHz TTL modulation of the rf power ) or a nano -voltmeter. In order to increase the output ISHE signa l, we chose a specific configuration with a magnetic field at 45° from to the SW propagation direction (cf Fig. 3a). This configuration has the advantage of provid ing a good coupling of the YIG film to the rf field under the antenna while still having a sig nificant spin polarization () that is orthogonal to the measured ISHE electrical field . We should point out that our choice of magnetic field direction implies that the propagating SWs are neither Damon -Eshbach modes where k M nor backward volume modes where k // M. In Fig 3b, we display the ISHE voltage (without any geometrical correction) as a function of the in plane magnetic field measured on a 20 nm thick YIG under a 10 mW rf excitation at 1 GHz. The sign of voltage peak (occurring at magnetic fields that resonantly excite the magnetization) reverses when the applied field is reversed as expected from the ISHE symmetry 29 : where is the pure spin current that flows through the YIG/Pt interface , is the spin polarization vector parallel to the dc magnetization direction and is the unit vector parallel to the Pt strip. A close -up on the ISHE signal lineshape for the different thickness es is plotted in Fig . 3c 3d 3e. In accordance with FMR study, the linewidth increases with decreasing YIG thickness. The spectral lineshap es are almost symmetrical confirming previous reports on YIG films grown using LPE 30. The ISHE maximum voltage decreases when the film thickness is decrease d. Going from 20 nm to 4 nm this decrease is as large as two order s of magn itudes and correlates to the increase of the Gilbert damping . A decrease of the ISHE signal when increasing damping is expected. Indeed, a s we are in the weak excitation regime, the ISHE voltage is expected to scale with 1/ 2, see for instance supplementary mate rials of Ref [7]. If we consider the Gilbert damping obt ained from FMR on the bar e YIG samples, a more dramatic decrease of the I SHE signal than the one observed is expected. In fact here, one should consider the effective damping of the Pt/YIG stack as spin -pumping will increase the YIG 6 effective damping under the Pt strip. We should ag ain point out that our measurement geometry relies on propagating SWs and therefor e they are subject to an exponential decay with distance . The length scale of this exponential decay is different for the three thicknesses owing to the difference in the damping parameter ; hence a quantitative interpretation of the voltage amplitude is to be avoided here. We thus succeeded to grow high quality ultrathin YIG film and demonstrated that a n efficient spin angular momentum transfer from the YIG film toward the Pt layer . The nex t objective is to induce an modification on the effective YIG damping coefficient via interfacial spin injection using SHE , as it has been reported in Py/Pt31,32 system . Using YIG/Pt, Kajiwara et al. have shown that even without rf excitation, spin injection induced by a dc current in Pt can generate propagating SWs ; the threshold current density has been estimated to be 4.4 x 108 A.m-2. Such unexpectedly low value has been attributed to the presence of an easy –axis surface anisotropy13. Hence, w e have performed experiments where a large dc bias current is applied on the Pt electrode while pumping spin waves in the 20 nm thick YIG film with TTL modulated rf excitation field . The VISHE linewidth is measured at the TTL modulation frequency using a lock -in. We expected to increase or decrease this linewidth depending on the dc current polarity but w e have not been able to see any sizable effect even for current densities as large as 6 x 109 Am-2, within a 0.2 Oe resolution, the measured linewidth remains absolutely unchanged . Theory predicts that th e threshold dc current for the onset of magnetic excitations scales with the Gilbert damping and the thickness of the ferromagnetic insulator (Eq. 1), the smaller the product the lower the threshold current for SW excitation is. In Kajiwara et al. ’s experiment nm; in our case nm (considering the spin pumping contribution to the damping ). We therefore applie d a dc current that is roughly 2 orders of magnitude larger than the expected threshold current . One possible explanation is that in our films , surface anisotropy is absent and therefore the pumped angular momentum is spread over many excitations modes33. Further investigations are under progress to clarify this point. In summary, we have fabricated PLD grown YIG on GGG (111) substrates with t hickness as low as 4 nm. We present here a comparative study on three different thicknesses : 4, 7 and 20 nm . The Gilbert damping coefficient for the 20 nm thick films is 2.3 x 10-4 which is the lowest value reported for sub - micrometric thick films. We demonstrate ISHE detection of SWs for ultra thin YIG film. The amplitudes of the ISHE voltage correlate s well to the increase of the Gilbert damping when decreasing thickness. Owing to extremely low product of the 20 nm film that is almost 10 time s smaller than the one reported by Kajiwara et al. we expected to observe compensation of the damping by spin current injection through the SHE but our preliminary results on the VISHE linewidth did not reveal any effect on the magnetization dynamics. Acknowledgement: This work has been supported by ANR -12-ASTR -0023 Trinidad 7 References: 1 J. E. Mee, J. L. Archer, R.H. Meade, and T.N. Hamilton, Applied Physics Letters 10 (1967). 2 J. P. Castera, Journal of Applied Physics 55 (6), 2506 (1984). 3 A. Khitun, M. Q. Bao, and K. L. Wang, Journal of Physics D -Applied Physics 43 (26), 10 (2010). 4 V. Vlaminck and M. Bailleul, Physical Review B 81 (1) (2010). 5 Vladislav E. Demidov, Sergei Urazhdin, and Sergej O. Demokritov, Nature Materials 9 (12), 984 (2010). 6 J. E. Hirsch, Physical Review Letters 83 (9), 1834 (1999). 7 Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464 (7286), 262 (2010). 8 Zihui Wang, Yiyan Sun, Mingzhong Wu, Vasil Tiberkevich, and Andrei Slavin, Physical Review Letters 107 (14) (2011). 9 E. Padron -Hernandez, A. Azevedo, and S. M. Rezende, Applied Physics Letters 99 (19), 3 (2011). 10 C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Physical Review B 87 (17) (2013). 11 C. F. Pai, L. Q. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Applied Physics Letters 101 (12) (2012). 12 Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deran lot, H. X. Yang, M. Chshiev, T. Valet, A. Fert, and Y. Otani, Physical Review Letters 109 (15) (2012). 13 J. Xiao and G. E. W. Bauer, Physical Review Letters 108 (21), 4 (2012). 14 Hujun Jiao and Gerrit E. W. Bauer, Physical review letters 110 (21), 217602 (2013). 15 V. Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, Physical Review B 86 (13), 6 (2012). 16 P. C. Dorsey, S. E. Bushnell, R. G. Seed, and C. Vittoria, Journal of Applied Physics 74 (2), 1242 (1993). 17 Y. Dumont, N. Keller, O. Popova, D. S. Schmool, F. Gendron, M. Tessier, and M. Guyot, Journal of Magnetism and Magnetic Materials 272, E869 (2004). 8 18 Y. Krockenberger, H. Matsui, T. Hasegawa, M. Kawasaki, and Y. Tokura, Applied Physics Letters 93 (9), 3 (2008). 19 S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, Journal of Applied Physics 106 (12) (2009). 20 Y. Y. Sun, Y. Y. Song, H. C. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Z. Wu, H. Schultheiss, and A. Hoffmann, Applied Physics Letters 101 (15), 5 (2012). 21 M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Geprags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J. M. Schmalhorst, G. Reiss, L. M. Shen, A. Gupta, Y. T. Chen, G. E. W. Bauer, E. Sait oh, and S. T. B. Goennenwein, Physical Review B 87 (22), 15 (2013). 22 M. A. Gilleo and S. Geller, Physical Review 110 (1), 73 (1958). 23 D. L. Mills and S. M. Rezende, Spin Dynamics in Confined Magnetic Structures Ii 87, 27 (2003). 24 B. Heinrich, C. B urrowes, E. Montoya, B. Kardasz, E. Girt, Young -Yeal Song, Yiyan Sun, and Mingzhong Wu, Physical Review Letters 107 (6) (2011). 25 R. Arias and D. L. Mills, Physical Review B 60 (10), 7395 (1999). 26 A. V. Chumak, A. A. Serga, M. B. Jungfleisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich, and B. Hillebrands, Applied Physics Letters 100 (8), 3 (2012). 27 C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y. Y. Song, and M. Z. Wu, Applied Physics Letters 100 (9), 4 (2012). 28 M. B. Jungfle isch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands, (arXiv:1302.6697 [cond -mat.mes -hall], 2013). 29 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Applied Physics Letters 88 (18) (2006). 30 T. Tashiro, R. Takahashi, Y. Kajiwara, K. Ando, H. Nakaya ma, T. Yoshino, D. Kikuchi, and E. Saitoh, presented at the Conference on Spintronics V, San Diego, CA, 2012 (unpublished). 31 K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Physical Review Letters 101 (3), 4 (2008). 32 V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and S. O. Demokritov, Nature Materials 11 (12), 1028 (2012). 33 M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti, G. Gubbiotti, F. B. Mancoff, M. A. Yar, a nd J. Akerman, Nature Nanotechnology 6 (10), 635 (2011). 9 Figure Captions : Figure 1 : (color online). (a) 1 µm x 1µm AFM surface topography of a 20nm YIG film on GGG (111) (RMS roughness=0.23 nm). (b) XRD : -2 scan of the same film using Cu -K 1 radiation. This spectrum shows that the growth is along the (111) direction with no evidence of parasitic phases. The YIG peak are best resolved at high diffraction angle, a zoom around the 888 GGG peak is shown in panel (c). (c) (d) (e) XRD diffraction s pectrum centered on GGG (888) reflection for different YIG thicknesses: 20, 7 and 4 nm. For the 2 thinnest films, the YIG peak is masked by the substrate peak . The cubic lattice parameter of the 20 nm thick YIG film (obtained from 888 peak) is slightly lar ger than that of the substrate ( aYIG=1.2459 nm , aGGG=1.2383 nm) Figure 2 : (color online) ( a) (b). FMR absorption derivative spectra of 20 and 4 nm thick YIG films at an excitation frequency of 6 GHz. (c) rf excitation frequency dependence of FMR absorption linewidth measured on different YIG film thicknesses with an in -plane oriented static field. The black continuous line is a linear fit on the 20 nm thick film from w hich a Gilbert damping coefficient of 2. 3 x 10-4 can be inferred ( . The damping of the 7 nm and 4 nm films is significantly larger but must off all the frequency dependence is not linear (see text for discussion). Figure 3 : (a) Schematic illustration of the experimental setup: spin waves are excited through the YIG waveguide with a microstrip antenna. The detection is performed by measuring ISHE voltage dc signal on a ~200µm wide Pt stripe. (b) External dc magnetic field dependen ce of ISHE voltage measured on Pt electrode showing the polarity inversion when the magnetic field is reversed. The peaks occur at the FMR conditions as verified through S 11 spectroscopy on the Au antenna (not shown here). (c). (d). (e). ISHE voltage for d ifferent YIG film thicknesses around the resonance magnetic field; microwave frequency is f=3GHz. The peaks can be fitted to a lorentzian shape and the extracted linewidth are respectively: 19.2 , 13.2 and 4.6 Oe. The dramatic increase of the ISHE signal w ith thickness is discussed in the text. 10 Fig 1 11 Fig 2 12 Fig 3

2202.06154v1.Generalization_of_the_Landau_Lifshitz_Gilbert_equation_by_multi_body_contributions_to_Gilbert_damping_for_non_collinear_magnets.pdf

Generalization of the Landau-Lifshitz-Gilbert equation by multi-body contributions to Gilbert damping for non-collinear magnets Sascha Brinker,1Manuel dos Santos Dias,2, 1,and Samir Lounis1, 2,y 1Peter Gr unberg Institut and Institute for Advanced Simulation, Forschungszentrum J ulich & JARA, 52425 J ulich, Germany 2Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg, Germany (Dated: February 15, 2022) We propose a systematic and sequential expansion of the Landau-Lifshitz-Gilbert equation utilizing the dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises from multi-body scattering processes. The tensor consists of a damping-like term and a correction to the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend on e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: ei
ej, (nij
ei)(nij
ej),nij
(eiej), (ei
ej)2,ei
(ejek)..., where some terms are subjected to the spin-orbit eld nijin rst and second order. We explore the magnitude of the dierent contributions using both the Alexander-Anderson model and time-dependent density functional theory in magnetic adatoms and dimers deposited on Au(111) surface.arXiv:2202.06154v1 [cond-mat.mtrl-sci] 12 Feb 20222 I. INTRODUCTION In the last decades non-collinear magnetic textures have been at the forefront in the eld of spintronics due to the promising applications and perspectives tied to them1,2. Highly non-collinear particle-like topological swirls, like skyrmions3,4and hopons5, but also domain walls6can potentially be utilized in data storage and processing devices with superior properties compared to conventional devices. Any manipulation, writing and nucleation of these various magnetic states involve magnetization dynamical processes, which are crucial to understand for the design of future spintronic devices. In this context, the Landau-Lifshitz-Gilbert (LLG) model7,8is widely used to describe spin dynamics of materials ranging from 3-dimensional bulk magnets down to the 0-dimensional case of single atoms, see e.g. Refs.9{12. The LLG model has two important ingredients: (i) the Gilbert damping being in general a tensorial quantity13, which can originate from the presence of spin-orbit coupling (SOC)14and/or from spin currents pumped into a reservoir15,16; (ii) the eective magnetic eld acting on a given magnetic moment and rising from internal and external interactions. Often a generalized Heisenberg model, including magnetic anisotropies and magnetic exchange interactions, is utilized to explore the ground state and magnetization dynamics characterizing a material of interest. Instead of the con- ventional bilinear form, the magnetic interactions can eventually be of higher-order type, see e.g.17{23. Similarly to magnetic interactions, the Gilbert damping, as we demonstrate in this paper, can host higher-order non-local contri- butions. Previously, signatures of giant anisotropic damping were found24, while chiral damping and renormalization of the gyromagnetic ratio were revealed through measurements executed on chiral domain wall creep motion24{28. Most rst-principles studies of the Gilbert damping were either focusing on collinear systems or were case-by-case studies on specic non-collinear structures lacking a general understanding of the fundamental behaviour of the Gilbert damping as function of the non-collinear state of the system. In this paper, we discuss the Gilbert damping tensor and its dependencies on the alignment of spin moments as they occur in arbitrary non-collinear state. Utilizing linear response theory, we extract the dynamical magnetic susceptibility and identify the Gilbert damping tensor pertaining to the generalized LLG equation that we map to that obtained from electronic structure models such as the single orbital Alexander-Anderson model29or time-dependent density functional theory applied to realistic systems10,30,31. Applying systematic perturbative expansions, we nd the allowed dependencies of the Gilbert damping tensor on the direction of the magnetic moments. We identify terms that are aected by SOC in rst and second order. We generalize the LLG equation by a simple form where the Gilbert damping tensor is amended with terms proportional to scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments, i.e. terms like ei
ej, (nij
ei)(nij
ej), nij
(eiej), (ei
ej)2,ei
(ejek)..., where we use unit vectors, ei=mi=jmij, to describe the directional dependence of the damping parameters and nijrepresents the spin-orbit eld. The knowledge gained from the Alexander-Anderson model is applied to realistic systems obtained from rst-principles calculations. As prototypical test system we use 3 dtransition metal adatoms and dimers deposited on the Au(111) surface. Besides the intra-site contribution to the Gilbert damping, we also shed light on the inter-site contribution, usually referred to as the non-local contribution. II. MAPPING THE GILBERT DAMPING FROM THE DYNAMICAL MAGNETIC SUSCEPTIBILITY Here we extract the dynamical transverse magnetic response of a magnetic moment from both the Landau-Lifshitz- Gilbert model and electronic structure theory in order to identify the Gilbert damping tensor Gij10,11,32,33. In linear response theory, the response of the magnetization mat siteito a transverse magnetic eld bapplied at sites jand oscillating at frequency !reads m i(!) =X j ij(!)b j(!); (1) with the magnetic susceptibility  ij(!) and;are thex;ycoordinates dened in the local spin frame of reference pertaining to sites iandj. In a general form13the LLG equation is given by dmi dt= mi0 @Be i+X jGij
dmj dt1 A; (2)3 Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f) Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7 MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)°2°101234Energy [U]°1.00°0.75°0.50°0.250.000.250.500.751.00DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8 °2°101234Energy [U]°1.0°0.50.00.51.0DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8dI/dVVexcitationa)b) abc FIG. 1. Illustration of the Landau-Lifshitz-Gilbert model and local density of states within the Alexander-Anderson model. (a) A magnetic moment (red arrow) precesses in the the presence of an external eld. The blue arrow indicates the direction of a damping term, while the green arrow shows the direction of the precession term. (b) Density of states for dierent magnetizations in the range from 0 :2 to 0:8. Density of states of dimers described within the Alexander-Anderson model for dierent magnetizations in the range from 0 :2 to 0:8. Shown is the ferromagnetic reference state. The magnetizations are self-consistently constrained using a longitudinal magnetic eld, which is shown in the inset. Model parameters: U= 1:0 eV, Ed= 1:0 eV; t= 0:2 eV; = 0:2 eV; 'R= 0 °. where = 2 is the gyromagnetic ratio, Be i=dHspin=dmiis the eective magnetic eld containing the contributions from an external magnetic eld Bext i, as well as internal magnetic elds originating from the interaction of the moment with its surrounding. In an atomistic spin model described by e.g. the generalized Heisenberg hamiltonian, Hspin=P imiKimi+1 2P ijmiJijmj, containing the on-site magnetic anisotropy Kiand the exchange tensor Jij, the eective eld is given by Be i=Bext iKimiP jJijmj(green arrow in Fig. 1a). The Gilbert damping tensor can be separated into two contributions { a damping-like term, which is the symmetric part of the tensor, S, (blue arrow in Fig. 1a), and a precession-like term A, which is the anti-symmetric part of the tensor. In Appendix A we show how the antisymmetric intra-site part of the tensor contributes to a renormalization of the gyromagnetic ratio. To extract the magnetic susceptibility, we express the magnetic moments in their respective local spin frame of references and use Rotation matrices that ensure rotation from local to globa spin frame of reference (see Appendix B). The magnetic moment is assumed to be perturbed around its equilibrium value Mi,mloc i=Miez i+mx iex i+my iey i, where e iis the unit vector in direction in the local frame of site i. Using the ground-state condition of vanishing magnetic torques, Miez i Bext i+Bint i
= 0 and the inverse of the transverse magnetic susceptibility can be identied as 1 ij(!) =ij Be iz Mi+i! Mi +1 MiMj(RiJijRT j)+ i!(RiGijRT j); (3) from which it follows that the Gilbert damping is directly related to the linear in frequency imaginary part of the inverse susceptibility d d!=[1] ij=ij1 Mi + (RiGijRT j): (4) Note thatRiandRjare rotation matrices rotating to the local frames of site iandj, respectively, which dene the coordinates ;=fx;yg(see Appendix B). Based on electronic structure theory, the transverse dynamical susceptibility can be extracted from a Dyson-like equation:1(!) =1 0(!)U, where0is the susceptibility of non-interacting electron while Uis a many-body interaction Kernel, called exchange-correlation Kernel in the context of time-dependent density functional theory30. The Kernel is generally assumed to be adiabatic, which enables the evaluation of the Gilbert damping directly from4 the non-interacting susceptibility. Obviously:d d!1(!) =d d!1 0(!). For small frequencies !,0has a simple !-dependence11: 0(!)<0(0) +i!=d d!0j!=0 (5) and as shown in Ref.33 d d!1 0(!)[<0(0)]2=d d!0j!=0: (6) Starting from the electronic Hamiltonian Hand the corresponding Green functions G(Ei) = (EHi)1, one can show that the non-interacting magnetic susceptibility can be dened via  0;ij(!+ i) =1 TrZEF dE Gij(E+!+ i)ImGji(E) +ImGij(E)Gji(E!i) ;(7) withbeing the vector of Pauli matrices. Obviously to identify the Gilbert damping and how it reacts to magnetic non-collinearity, we have to inspect the dependence of the susceptibility, and therefore the Green function, on the misalignment of the magnetic moments. III. MULTI-SITE EXPANSION OF THE GILBERT DAMPING Assuming the hamiltonian Hconsisting of an on-site contribution H0and an inter-site term encoded in a hopping termt, which can be spin-dependent, one can proceed with a perturbative expansion of the corresponding Green function utilizing the Dyson equation Gij=G0 iij+G0 itijG0 j+G0 itikG0 ktkjG0 j+::: : (8) Within the Alexander-Anderson single-orbital impurity model29,H0 i=Edi Uimi
Bi
, whereEd is the energy of the localized orbitals, is the hybridization in the wide band limit, Uiis the local interaction responsible for the formation of a magnetic moment and Biis an constraining or external magnetic eld. SOC can be incorportated as tsoc ij=iijnij
, whereijandnij=njirepresent respectively the strength and direction of the anisotropy eld. It can be parameterized as a spin-dependent hopping using the Rashba-like spin-momentum locking tij=t(cos'R0i sin'Rnij
)34. Depending on whether the considered Green function is an on-site Green function Giior an inter-site Green function Gijdierent orders in the hopping are relevant. On-site Green functions require an even number of hopping processes, while inter-site Green functions require at least one hopping process. The on-site Green function G0 ican be separated into a spin-less part Niand a spin dependent part Mi, G0 i=Ni0+Mi
 ; (9) where the spin dependent part is parallel to the magnetic moment of site i,Mikmi(note that SOC is added later on to the hoppings). Using the perturbative expansion, eq. (8), and the separated Green function, eq. (9), to calculate the magnetic susceptibility, eq. (7), one can systematically classify the allowed dependencies of the susceptibility with respect to the directions of the magnetic moments, e.g. by using diagrammatic techniques as shown in Ref.18for a related model in the context of higher-order magnetic exchange interactions. Since our interest is in the form of the Gilbert damping, and therefore also in the form of the magnetic susceptibility, the perturbative expansion can be applied to the magnetic susceptibility. The general form of the magnetic susceptibility in terms of the Green function, eq. (7), depends on a combination of two Green functions with dierent energy arguments, which are labeled as !and 0 in the following. The relevant structure is then identied as33,  ij(!)Tr iGij(!) jGji(0): (10) The sake of the perturbative expansion is to gather insights in the possible forms and dependencies on the magnetic moments of the Gilbert damping, and not to calculate explicitly the strength of the Gilbert damping from this5 expansion. Therefore, we focus on the structure of eq. (10), even though the susceptibility has more ingredients, which are of a similar form. Instead of writing all the perturbations explicitly, we set up a diagrammatic approach, which has the following ingredients and rules: 1. Each diagram contains the operators NandM, which are andfor the magnetic susceptibility. The operators are represented by a white circle with the site and spin index: i 2. Hoppings are represented by grey circles indicating the hopping from site itoj:ij. The vertex corresponds totij. 3. SOC is described as a spin-dependent hopping from site itojand represented by: ij;. The vertex corresponds to tsoc ij=iij^n ij. 4. The bare spin-independent (on-site) Green functions are represented by directional lines with an energy at- tributed to it: !. The Green function connects operators and hoppings. The line corresponds to Ni(!). 5. The spin-dependent part of the bare Green function is represented by: !; .indicates the spin direction. The direction ensures the right order within the trace (due to the Pauli matrices, the dierent objects in the diagram do not commute). The line corresponds to Mi(!)m i. Note that the diagrammatic rules might be counter-intuitive, since local quantities (the Green function) are represented by lines, while non-local quantities (the hopping from itoj) are represented by vertices. However, these diagrammatic rules allow a much simplied description and identication of all the possible forms of the Gilbert damping, without having to write lengthy perturbative expansions. Spin-orbit coupling independent contributions. To get a feeling for the diagrammatic approach, we start with the simplest example: the on-site susceptibility without any hoppings to a dierent site, which describes both the single atom and the lowest order term for interacting atoms. The possible forms are, ii (!)/ !0 i i+ !; 0 i i + !0; i i+ !;0; i i; (11)6 which evaluate to, !0 i i= TrNi!)Ni(0) =Ni(!)Ni(0) (12) !; 0 i i= Tr Mi(!)Ni(0)m i= i Mi(!)Ni(0)m i (13) !0; i i= Tr Ni(!)Mi(0)m i= i Mi(!)Mi(0)m i (14) !;0; i i= Tr Mi(!)Mi(0)m im i = ( +  )Mi(!)Mi(0)m im i: (15) The rst diagram yields an isotropic contribution, the second and third diagrams yield an anti-symmetric contribution, which is linear in the magnetic moment, and the last diagram yields a symmetric contribution being quadratic in the magnetic moment. Note that the energy dependence of the Green functions is crucial, since otherwise the sum of eqs. (13) and (14) vanishes. In particular this means that the static susceptibility has no dependence linear in the magnetic moment, while the the slope of the susceptibility with respect to energy can have a dependence linear in the magnetic moment. The static part of the susceptibility maps to the magnetic exchange interactions, which are known to be even in the magnetic moment due to time reversal symmetry. Combining all the functional forms of the diagrams, we nd the following possible dependencies of the on-site Gilbert damping on the magnetic moments, G ii(fmg)/f; m i;m im ig: (16) Since we work in the local frames, mi= (0;0;mz i), the last dependence is a purely longitudinal term, which is not relevant for the transversal dynamics discussed in this work. If we still focus on the on-site term, but allow for two hoppings to another atom and back, we nd the following new7 diagrams, !00 0 i iij ji + !; 00 0 i iij ji +:::+ !; 0;0 0 i iij ji +::: + !; 0;0; 0 i iij ji +:::+ !; 0;0; 0; i iij ji : (17) The dashed line in the second diagram can be inserted in any of the four sides of the square, with the other possibilities omitted. Likewise for the diagrams with two or three dashed lines, the dierent possible assignments have to be considered. The additional hopping to the site jyields a dependence of the on-site magnetic susceptibility and therefore also the on-site Gilbert damping tensor on the magnetic moment of site j. Another contribution to the Gilbert damping originates from the inter-site part, thus encoding the dependence of the moment site ion the dynamics of the moment of site jviaGij. This contribution is often neglected in the literature, since for many systems it is believed to have no signicant impact. Using the microscopic model, a dierent class of diagrams is responsible for the inter-site damping. In the lowest order in t=Um the diagrams contain already two hopping events, ! !0 0 i j ijij + !; !0 0 i j ijij +:::+ !; !;0 0 i j ijij +::: + !; !;0; 0 i j ijij +:::+ !; !;0; 0; i j ijij : (18) In total, we nd that the spin-orbit independent intra-site and inter-site Gilbert damping tensors can be respectively written as Gii= Si+Sij;(1) i (ei
ej) +Sij;(2) i (ei
ej)2 I + Ai+Aij i(ei
ej) E(ei);(19)8 and G ij= Sij+Sdot ij(ei
ej)
 + Aij+Adot ij(ei
ej)
(E(ei) +E(ej)) +Scross ij(eiej)(eiej)+Sba ije ie j; (20) where as mentioned earlier SandArepresent symmetric and asymmetric contributions, Iis the 33 identity while E(ei) =0 @0ez iey i ez i0ex i ey iex i01 A. Remarkably, we nd that both the symmetric and anti-symmetric parts of the Gilbert damping tensor have a rich dependence with the opening angle of the magnetic moments. We identify, for example, the dot and the square of the dot products of the magnetic moments to possibly play a crucial role in modifying the damping, similarly to bilinear and biquadratic magnetic interactions. It is worth noting that even though the intra-site Gilbert damping can explicitly depend on other magnetic moments, its meaning remains unchanged. The anti-symmetric precession-like term describes a precession of the moment around its own eective magnetic eld, while the diagonal damping-like term describes a damping towards its own eective magnetic eld. The dependence on other magnetic moments renormalizes the intensity of those two processes. The inter-site Gilbert damping describes similar processes, but with respect to the eective eld of the other involved magnetic moment. On the basis of the LLG equation, eq. (2), it can be shown that the term related to Sba ijwith a functional form of e ie jdescribes a precession of the i-th moment around thej-th moment with a time- and directional-dependent amplitude, @tmi/(mimj) (mi
@tmj). The double cross product term yields a time dependence of @tmi/(mi(mimj)) ((mimj)
@tmj). Both contributions are neither pure precession-like nor pure damping-like, but show complex time- and directional-dependent dynamics. Spin-orbit coupling contributions. The spin-orbit interaction gives rise to new possible dependencies of the damping on the magnetic structure. In particular, the so-called chiral damping, which in general is the dierence of the damping between a right-handed and a left-handed opening, rises from SOC and broken inversion symmetry. Using our perturbative model, we can identify all possible dependencies up to second order in SOC and third order in the magnetic moments. In the diagramms SOC is added by replacing one spin-independent hopping vertex by a spin-dependent one, !00 0 i iij ji ! !00 0 i iij ij : (21) Up to rst-order in SOC, we nd the the following dependencies were found for the on-site Gilbert damping Gii(fmg)/f ^n ij;^n ij^n ji;^n ijm i;^n ijm i;(^nij
mi);(^nij
mj); ^n ijm j;^n ijm j;m i(^nijmi);m i(^nijmi); ^nij
(mimj);m i(^nijmj);m i(^nijmj);(^nij
mj) m i; m im i(^nij
mj);(m im jm im j)(^nij
mj);^n ijm i(mi
mj);^n ijm i(mi
mj)g: (22) We identied the following contributions for the on-site and intersite damping to be the most relevant one after the numerical evaluation discussed in the next sections: Gsoc ii=Ssoc;ij i nij
(eiej)I +Ssoc;ij;(2) i (nij
ei)(nij
ej)I +Asoc;ij i nij
(eiej)E(ei) +Asoc;ij;(2) i (nij
ej)E(nij); (23)9 and Gsoc; ij =Ssoc ijnij
(eiej)+Ssoc;ba ijn ij(eiej) +Asoc ijE(nij): (24) The contributions being rst-order in SOC are obviously chiral since they depend on the cross product, eiej. Thus, similar to the magnetic Dzyaloshinskii-Moriya interaction, SOC gives rise to a dependence of the Gilbert damping on the vector chirality, eiej. The term chiral damping used in literature refers to the dependence of the Gilbert damping on the chirality, but to our knowledge it was not shown so far how this dependence evolves from a microscopic model, and how it looks like in an atomistic model. Extension to three sites. Including three dierent sites i,j, andkin the expansions allows for a ring exchange i!j!k!iinvolving three hopping processes, which gives rise to new dependencies of the Gilbert damping on the directions of the moments. An example of a diagram showing up for the on-site Gilbert damping is given below for the on-site Gilbert damping the diagram, !00 0; 0 i iijjk ki(25) Apart from the natural extensions of the previously discussed 2-site quantities, the intra-site Gilbert damping of site i can depend on the angle between the sites jandk,ej
ek, or in higher-order on the product of the angles between site iandjwithiandk, (ei
ej)(ei
ek). In sixth-order in the magnetic moments the term ( ei
ej)(ej
ek)(ek
ei) yields to a dependence on the square of the scalar spin chirality of the three sites, [ ei
(ejek)]2. Including SOC, there are two interesting dependencies on the scalar spin chirality. In rst-order one nds similarly to the recently discovered chiral biquadratic interaction18and its 3-site generalization19, e.g. ( nij
ei) (ei
(ejek)), while in second order a direct dependence on the scalar spin chirality is allowed, e.g. n ijn ki(ei
(ejek)). The scalar spin chirality directly relates to the topological orbital moment35{37and therefore the physical origin of those dependencies lies in the topological orbital moment. Even though these terms might not be the most important ones in our model, for specic non-collinear congurations or for some realistic elements with a large topological orbital moment, e.g. MnGe20, they might be important and even dominant yielding interesting new physics. IV. APPLICATION TO THE ALEXANDER-ANDERSON MODEL Magnetic dimers. Based on a 2-site Alexander-Anderson model, we investigated the dependence of the Gilbert damping on the directions of the magnetic moments using the previously discussed possible terms (see more details on the method in Appendix C). The spin splitting Udenes the energy scale and all other parameters. The energy of orbitals is set to Ed= 1:0. The magnetization is self-consistently constrained in a range of m= 0:2 tom= 0:8 using magnetic constraining elds. The corresponding spin-resolved local density of states is illustrated in Fig. 1b, where the inter-site hopping is set to t= 0:2 and the hybridization to = 0 :2. We performed two sets of calculations: one without spin-dependent hopping, 'R= 0 °, and one with a spin-dependent hopping, 'R= 20 °. The dierent damping parameters are shown in Fig. 2 as function of the magnetization. They are obtained from a least-squares t to several non-collinear congurations based on a Lebedev mesh for `= 238. The damping, which is independent of the relative orientation of the two sites, is shown in Fig. 2a. The symmetric damping-like intra-site contributionSidominates the damping tensor for most magnetizations and has a maximum at m= 0:3. The anti- symmetric intra-site contribtuion Ai, which renormalizes the gyromagnetic ratio, approximately changes sign when the Fermi level passes the peak of the minority spin channel at m0:5 and has a signicantly larger amplitude for small magnetizations. Both contributions depend mainly on the broadening , which mimics the coupling to an10 Magnetization°1.0°0.50.00.51.0Intra-site damping [U]a MagnetizationcGsiGasi 0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b 0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a MagnetizationcGsiGasi 0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b 0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a MagnetizationcGsiGasi 0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b 0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a MagnetizationcGsiGasi 0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b 0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a MagnetizationcGsiGasi 0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b 0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f) Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7 MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f) Magnetization°1.0°0.50.00.51.0Intra-site damping [U]a MagnetizationcGsiGasi 0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b 0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7 MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7 MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7 MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)Magnetization−1.0−0.50.00.51.0 Magnetization Magnetization−0.20.00.2 Magnetization 0.30.50.7−0.2−0.10.00.10.2 0.30.50.7 MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a) b) c)d) e) f)abc ab FIG. 2. Damping parameters as function of the magnetization for the dimers described within the Alexander-Anderson model including spin orbit coupling. A longitudinal magnetic eld is used to self-consistently constrain the magnetization. The parameters are extracted from tting to the inverse of the transversal susceptibility for several non-collinear congurations based on a Lebedev mesh. Model parameters in units of U:Ed= 1:0; t= 0:2; = 0:2; 'R= 20 °. electron bath and is responsible for the absorption of spin currents, which in turn are responsible for the damping of the magnetization dynamics15,16. The directional dependencies of the intra-site damping are shown in Fig. 2b. With our choice of parameters, the correction to the damping-like symmetric Gilbert damping can reach half of the direction-independent term. This means that the damping can vary between 0:41:0 for a ferromagnetic and an antiferromagnetic state at m= 0:4. Also for the renormalization of the gyromagnetic ratio a signicant correction is found, which in the ferromagnetic case always lowers and in the antiferromagnetic case enhances the amplitude. The most dominant contribution induced by SOC is the chiral one, which depends on the cross product of the moments iandj, which in terms of amplitude is comparable to the isotropic dot product terms. Interestingly, while the inter-site damping term is in general known to be less relevant than the intra-site damping, we nd that this does not hold for the directional dependence of the damping. The inter-site damping is shown in Fig. 2c. Even though the directional-independent term, Sij, is nearly one order of magnitude smaller than the equivalent intra-site contribution, this is not necessarily the case for the directional-dependent terms, which are comparable to the intra-site equivalents. V. APPLICATION TO FIRST-PRINCIPLES SIMULATIONS To investigate the importance of non-collinear eects for the Gilbert damping in realistic systems, we use DFT and time-dependent DFT to explore the prototypical example of monoatomic 3 dtransition metal adatoms and dimers deposited on a heavy metal surface hosting large SOC (see Fig. 3a for an illustration of the conguration). We consider a Cr, Mn, Fe and Co atoms deposited on the fcc-Au(111) surface (details of the simulations are described in Appendix D). The parameters and the corresponding functional forms are tted to our rst-principles data using 196 non-collinear states based on a Lebedev mesh for `= 238. Adatoms on Au(111). To illustrate the dierent eects on the Gilbert damping, we start by exploring magnetic adatoms in the uniaxial symmetry of the Au(111) surface. For the adatoms no non-local eects can contribute to the Gilbert damping. The Gilbert damping tensor of a single adatom without SOC has the form shown in relation to eq. (16), G0 i=SiI+AiE(ei): (26) Note that SOC can induce additional anisotropies, as shown in eq. (22). The most important ones for the case of a single adatom are f ^n ij;^n ij^n jig, which in the C3vsymmetry result in Gi=G0 i+Ssoc i0 @0 0 0 0 0 0 0 0 11 A+Asoc i0 @0 1 0 1 0 0 0 0 01 A; (27)11 DampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters Si 0:083 0 :014 0 :242 0 :472 Ai 0:204 0 :100 0 :200 0 :024 Ssoc i 0:000 0 :000 0 :116 0 :010 Asoc i 0:000 0 :0000:022 0 :012 renorm x=y 1:42 1 :67 1 :43 1 :91 renorm z 1:42 1 :67 1 :48 1 :87 TABLE I. Gilbert damping parameters of Cr, Mn, Fe and Co adatoms deposited on the Au(111) surface as parametrized in eqs. (26) and (27). The SOC eld points in the z-direction due to the C3vsymmetry. The renormalized gyromagnetic ratio renormis calculated according to eqs. (28) for an in-plane magnetic moment and an out-of-plane magnetic moment. since the sum of all SOC vectors points in the out-of-plane direction with ^nij!ez. Thus, the Gilbert damping tensor of adatoms deposited on the Au(111) surface can be described by the four parameters shown in eqs. (26) and (27), which are reported in Table I for Cr, Mn, Fe and Co adatoms. Cr and Mn, being nearly half-lled, are characterized by a small damping-like contribution Si, while Fe and Co having states at the Fermi level show a signicant damping of up to 0:47 in the case of Co. The antisymmetric part Aiof the Gilbert damping tensor results in an eective renormalization of the gyromagnetic ratio , as shown in relation to eq. (A5), which using the full LLG equation, eq. (2), and approximating mi
dmi dt= 0 is given by, renorm= 1 1 + (ei
Ai); (28) where Aidescribes the vector Ai= Ai;Ai;Ai+Asoc i
. For Cr and Fe there is a signicant renormalization of the gyromagnetic ratio resulting in approximately 1 :4. In contrast, Co shows only a weak renormalization with 1 :9 being close to the gyromagnetic ratio of 2. The SOC eects are negigible for most adatoms except for Fe, which shows a small anisotropy in the renormalized gyromagentic ratio ( 10 %) and a large anisotropy in the damping-like term of nearly 50 %. Dimers on Au(111). In contrast to single adatoms, dimers can show non-local contributions and dependencies on the relative orientation of the magnetic moments carried by the atoms. All quantities depending on the SOC vector are assumed to lie in the y-z-plane due to the mirror symmetry of the system. A sketch of the dimer and its nearest neighboring substrate atoms together with adatoms' local density of states are presented in Fig. 3. The density of states originates mainly from the d-states of the dimer atoms. It can be seen that the dimers exhibit a much more complicated hybridization pattern than the Alexander-Anderson model. In addition the crystal eld splits the dierent d-states resulting in a rich and high complexity than assumed in the model. However, the main features are comparable: For all dimers there is either a fully occupied majority channel (Mn, Fe, and Co) or a fully unoccupied minority channel (Cr). The other spin channel determines the magnetic moments of the dimer atoms f4:04;4:48;3:42;2:20gBfor respectively Cr, Mn, Fe and Co. Using the maximal spin moment, which is according to Hund's rule 5 B, the rst-principles results can be converted to the single-orbital Alexander-Anderson model corresponding to approximately m=f0:81;0:90;0:68;0:44gBfor the aforementioned sequence of atoms. Thus by this comparison, we expect large non-collinear contributions for Fe and Co, while Cr and Mn should show only weak non-local dependencies. The obtained parametrization is given in Table II. The Cr and Mn dimers show a weak or nearly no directional dependence. While the overall damping for both nanostructures is rather small, there is a signicant correction to the gyromagnetic ratio. In contrast, the Fe and Co dimers are characterized by a very strong directional dependence. Originating from the isotropic dependencies of the damping-like contributions, the damping of the Fe dimer can vary between 0 :21 in the ferromagnetic state and 0 :99 in the antiferromagnetic state. For the Co dimer the inter-site damping is even dominated by the bilinear and biquadratic term, while the constant damping is negligible. In total, there is a very good qualitative agreement between the expectations derived from studying the Alexander-Anderson model and the rst-principles results.12 DampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters Si 0:0911 0 :0210 0 :2307 0 :5235 Sij;(1) i 0:0376 0 :00060:39240:2662 Sij;(2) i 0:01330:0006 0 :3707 0 :3119 Ai 0:2135 0 :1158 0 :1472 0 :0915 Aij i 0:0521 0 :00280:07100:0305 Sij0:0356 0 :0028 0 :2932 0 :0929 Sdot ij0:03440:00180:33960:4056 Sdot;(2) ij 0:0100 0 :0001 0 :1579 0 :2468 Aij0:02810:0044 0 :0103 0 :0011 Adot ij0:0175 0 :00000:02340:0402 Scross ij 0:0288 0 :00020:28570:0895 Sba ij 0:0331 0 :0036 0 :2181 0 :2651 Ssoc;ij;y i 0:0034 0 :0000 0 :01430:0225 Ssoc;ij;z i 0:0011 0 :00000:0104 0 :0156 Asoc;ij;y i 0:00240:00010:0036 0 :0022 Asoc;ij;z i 0:00180:0005 0 :00390:0144 Ssoc;y ij 0:0004 0 :0001 0 :0307 0 :0159 Ssoc;z ij0:0011 0 :00000:0233 0 :0206 Sba,soc ;y ij0:0027 0 :00000:01840:0270 Sba,soc ;z ij 0:00050:0001 0 :01160:0411 TABLE II. Damping parameters of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The possible forms of the damping are taken from the analytic model. The SOC eld is assumed to lie in the y-zplane and inverts under permutation of the two dimer atoms. 3210123EEF[eV]6303DOS [#states/eV]Cr dimerMn dimerFe dimerCo dimersurface ab FIG. 3. aIllustration of a non-collinear magnetic dimer (red spheres) deposited on the (111) facets of Au (grey spheres). From the initial C3vspatial symmetry of the surface the dimers preserve the mirror plane (indicated grey) in the y-zplane. b Local density of states of the Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The grey background indicates the surface density of states. The dimers are collinear in the z-direction. VI. CONCLUSIONS In this article, we presented a comprehensive analysis of magnetization dynamics in non-collinear system with a special focus on the Gilbert damping tensor and its dependencies on the non-collinearity. Using a perturbative expansion of the two-site Alexander-Anderson model, we could identify that both, the intra-site and the inter-site part of the Gilbert damping, depend isotropically on the environment via the eective angle between the two magnetic moments, ei
ej. SOC was identied as the source of a chiral contribution to the Gilbert damping, which similarly to the Dzyaloshinskii-Moriya and chiral biquadratic interactions depends linearly on the vector spin chirality, eiej. We unveiled dependencies that are proportional to the three-spin scalar chirality ei
(ejek), i.e. to the chiral or topological moment, and to its square. Using the Alexander-Anderson model, we investigated the importance of the13 dierent contributions in terms of their magnitude as function of the magnetization. Using the prototypical test system of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface, we extracted the eects of the non-collinearity on the Gilbert damping using time-dependent DFT. Overall, the rst-principles results agree qualitatively well with the Alexander-Anderson model, showing no dependence for the nearly half-lled systems Cr and Mn and a strong dependence on the non-collinearity for Fe and Co having a half-lled minority spin-channel. The realistic systems indicate an even stronger dependence on the magnetic texture than the model with the used parameters. The Fe and the Co dimer show signicant isotropic terms up to the biquadratic term, while the chiral contributions originating from SOC have only a weak impact on the total Gilbert damping. However, the chiral contributions can play the deciding role for systems which are degenerate in the isotropic terms, like e.g. spin spirals of opposite chirality. We expect the dependencies of the Gilbert damping on the magnetic texture to have a signicant and non-trivial impact on the spin dynamics of complex magnetic structures. Our ndings are readily implementable in the LLG model, which can trivially be amended with the angular dependencies provided in the manuscript. Utilizing multiscale mapping approaches, it is rather straightforward to generalize the presented forms for an implementation of the micromagnetic LLG and Thiele equations. The impact of the dierent contributions to the Gilbert damping, e.g. the vector (and/or scalar) chiral and the isotropic contributions, can be analyzed on the basis of either free parameters or sophisticated parametrizations obtained from rst principles as discussed in this manuscript. It remains to be explored how the newly found dependencies of the Gilbert damping aect the excitations and motion of a plethora of highly non-collinear magnetic quasi-particles such as magnetic skyrmions, bobbers, hopons, domain walls and spin spirals. Future studies using atomistic spin dynamics simulations could shed some light on this aspect and help for the design of future devices based on spintronics. ACKNOWLEDGMENTS This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC-consolidator grant 681405 { DYNASORE) and from Deutsche Forschungsgemeinschaft (DFG) through SPP 2137 \Skyrmionics" (Project LO 1659/8-1). The authors gratefully acknowledge the computing time granted through JARA-HPC on the supercomputer JURECA at the Forschungszentrum J ulich39. VII. METHODS Appendix A: Analysis of the Gilbert damping tensor The Gilbert damping tensor Gcan be decomposed into a symmetric part Sand an anti-symmetric part A, A=GGT 2andS=G+GT 2: (A1) While the symmetric contribution can be referred to as the damping-like contribution including potential anisotropies, the anti-symmetric Atypically renormalizes the gyromagnetic ratio as can be seen as follows: The three independent components of an anti-symmetric tensor can be encoded in a vector Ayielding A= A ; (A2) where is the Levi-Cevita symbol. Inserting this into the LLG equation yields dmi dt= mi0 @Be i+X jAijdmj dt1 A (A3)  mi0 @Be i X jAij mjBe j
1 A: (A4) The last term can be rewritten as (Aij
mj)Be j Aij
Be j
mj: (A5)14 For the local contribution, Aii, the correction is kmiandkBe iyielding a renormalization of miBe i. However, the non-local parts of the anti-symmetric Gilbert damping tensor can be damping-like. Appendix B: Relation between the LLG and the magnetic susceptibility The Fourier transform of the LLG equation is given by i!mi= mi0 @Bext iX jJijmji!X jGijmj1 A: (B1) Transforming this equation to the local frames of site iandjusing the rotation matrices RiandRjyields i! Mimloc i=mloc i Mi0 @RiBext iX jRiJijRT jmloc ji!X jRiGijRT jmloc j1 A; (B2) where mloc i=Rimiandmloc j=Rjmj. The rotation matrices are written as R(#i;'i) = cos(#i=2)0+ i sin(#i=2) sin('i)xcos('i)y
, with (#i;'i) being the polar and azimuthal angle pertaining to the moment mi. In the ground state the magnetic torque vanishes. Thus, denoting mloc i= (mx i; my i; Mi), wheremx=y iare perturbations to the ground states, yields for the ground state 0 @(RiBext i)xP j(RiJijRT jMjez)x (RiBext i)yP j(RiJijRT jMjez)y (RiBext i)zP j(RiJijRT jMjez)z1 A=0 @0 0 (RiBe i)z1 A: (B3) Linearizing the LLG and using the previous result and limiting our expansion to transveral excitations yield i! Mimx i=my i(RiBe i)z Mi(RiBext i)y+X j(RiJijRT jmloc j)y+ i!X j(RiGijRT jmloc j)y(B4) i! Mimy i=mx i(RiBe i)z Mi+ (RiBext i)xX j(RiJijRT jmloc j)xi!X j(RiGijRT jmloc j)x; (B5) which in a compact form gives X j =x;y0 @ij (RiBe i)z Mi+i! Mi +X j(RiJijRT j)+ i!X j(RiGijRT j)1 Am j= (RiBext i); (B6) and can be related to the inverse of the magnetic susceptibility X j =x;y1 i;j(!)m j= (RiBext i): (B7) Thus, the magnetic susceptibility in the local frames of site iandjis given by 1 i;j(!) =ij (RiBe i)z Mi+i! Mi +X j(RiJijRT j)+ i!X j(RiGijRT j)(B8) Appendix C: Alexander-Anderson model{more details We use a single orbital Alexander-Anderson model, H=X ij[ij(Edi Uimi
Bi
)(1ij)tij]; (C1)15 whereiandjsum over all n-sites,Edis the energy of the localized orbitals, is the hybridization in the wide band limit,Uiis the local interaction responsible for the formation of a magnetic moment, miis the magnetic moment of site i,Biis an constraining or external magnetic eld, are the Pauli matrices, and tijis the hopping parameter between siteiandj, which can be in general spin-dependent. SOC is added as spin-dependent hopping using a Rashba-like spin-momentum locking tij=t(cos'R0i sin'Rnij
), where the spin-dependent hopping is characterized by its strength dened by 'Rand its direction nij=nji34. The eigenenergies and eigenstates of the model are given by, Hjni= (Eni )jni: (C2) The single particle Green function can be dened using the eigensystem, G(E+ i) =X njnihnj EEn+ i; (C3) whereis an innitesimal parameter dening the retarded ( !0+) and advanced ( !0) Green function. The magnitude of the magnetic moment is determined self-consistently using mi=1 Im TrZ dEGii(E); (C4) whereGii(E) is the local Green function of site idepending on the magnetic moment. Using the magnetic torque exerted on the moment of site i, dH d^ei=miBe i; (C5) magnetic constraining elds can be dened ensuring the stability of an arbitrary non-collinear conguration, Bconstr=Pm ?mi jmijBe i) Hconstr=Bconstr
 ; (C6) wherePm ?is the projection on the plane perpendicular to the moment m. The constraining elds are added to the hamiltonian, eq. (C1), and determined self-consistently. Appendix D: Density functional theory{details The density functional theory calculations were performed with the Korringa-Kohn-Rostoker (KKR) Green function method. We assume the atomic sphere approximation for the the potential and include full charge density in the self-consistent scheme40. Exchange and correlation eects are treated in the local spin density approximation (LSDA) as parametrized by Vosko, Wilk and Nusair41, and SOC is added to the scalar-relativistic approximation in a self- consistent fashion42. We model the pristine surfaces utilizing a slab of 40 layers with the experimental lattice constant of Au assuming open boundary conditions in the stacking direction, and surrounded by two vacuum regions. No relaxation of the surface layer is considered, as it was shown to be negligible43. We use 450450k-points in the two-dimensional Brillouin zone, and the angular momentum expansions for the scattering problem are carried out up to`max= 3. Each adatom is placed in the fcc-stacking position on the surface, using the embedding KKR method. Previously reported relaxations towards the surface of 3 dadatoms deposited on the Au(111) surface44indicate a weak dependence of the relaxation on the chemical nature of the element. Therefore, we use a relaxation towards the surface of 20 % of the inter-layer distance for all the considered dimers. The embedding region consists of a spherical cluster around each magnetic adatom, including the nearest-neighbor surface atoms. The magnetic susceptibility is eciently evaluated by utilizing a minimal spdf basis built out of regular scattering solutions evaluated at two or more energies, by orthogonalizing their overlap matrix10. We restrict ourselves to the transversal part of the susceptibility using only the adiabatic exchange-correlation kernel and treat the susceptibility in the local frames of sites iandj. To investigate the dependence of the magnetic excitations on the non-collinarity of the system, we use all possible non-collinear states based on a Lebedev mesh for `= 238.16 REFERENCES m.dos.santos.dias@fz-juelich.de ys.lounis@fz-juelich.de 1Fert A, Cros V and Sampaio J 2013 Nat. Nanotech. 8152{156 ISSN 1748-3387 2Fert A, Reyren N and Cros V 2017 Nature Reviews Materials 217031 ISSN 2058-8437 3Bogdanov A and Hubert A 1994 Journal of Magnetism and Magnetic Materials 138255 { 269 ISSN 0304-8853 4R ossler U K, Bogdanov A N and P eiderer C 2006 Nature 442797{801 5Tai J S B and Smalyukh I I 2018 Phys. Rev. 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1502.02699v1.Large_amplitude_oscillation_of_magnetization_in_spin_torque_oscillator_stabilized_by_field_like_torque.pdf

arXiv:1502.02699v1 [cond-mat.mes-hall] 9 Feb 2015Large amplitude oscillation of magnetization in spin-torq ue oscillator stabilized by field-like torque Tomohiro Taniguchi1, Sumito Tsunegi2, Hitoshi Kubota1, and Hiroshi Imamura1 1National Institute of Advanced Industrial Science and Tech nology (AIST), Spintronics Research Center, Tsukuba 305-8568, Japan, 2Unit´ e Mixte de Physique CNRS/Thales and Universit´ e Paris Sud 11, 1 av. A. Fresnel, Palaiseau, France. (Dated: July 8, 2021) Oscillation frequency of spin torque oscillator with a perp endicularly magnetized free layer and an in-plane magnetized pinned layer is theoretically inves tigated by taking into account the field-like torque. It is shown that the field-like torque plays an import ant role in finding the balance between the energy supplied by the spin torque and the dissipation du e to the damping, which results in a steady precession. The validity of the developed theory is confirmed by performing numerical simulations based on the Landau-Lifshitz-Gilbert equatio n. Spin torque oscillator (STO) has attracted much at- tention as a future nanocommunication device because it can produce a large emission power ( >1µW), a high quality factor ( >103), a high oscillation frequency ( >1 GHz), a wide frequency tunability ( >3 GHz), and a nar- rowlinewidth ( <102kHz) [1–9]. In particular,STOwith a perpendicularly magnetized free layer and an in-plane magnetizedpinnedlayerhasbeendevelopedafterthedis- covery of an enhancement of perpendicular anisotropy of CoFeB free layer by attaching MgO capping layer [10– 12]. In the following, we focus on this type of STO. We have investigated the oscillation properties of this STO both experimentally [6, 13] and theoretically [14, 15]. An important conclusion derived in these studies was that field-like torque is necessary to excite the self-oscillation in the absence of an external field, nevertheless the field- like torque is typically one to two orders of magnitude smaller than the spin torque [16–18]. We showed this conclusion by performing numerical simulations based on the Landau-Lifshitz-Gilbert (LLG) equation [15]. This paper theoretically proves the reason why the field-like torque is necessary to excite the oscillation by using the energy balance equation [19–27]. An effective energy including the effect of the field-like torque is in- troduced. It is shown that introducing field-like torque is crucial in finding the energy balance between the spin torque and the damping, and as a result to stabilize a steady precession. A good agreement with the LLG sim- ulation on the current dependence of the oscillation fre- quency shows the validity of the presented theory. Thesystemunderconsiderationisschematicallyshown in Fig. 1 (a). The unit vectorspointing in the magnetiza- tion directions of the free and pinned layers are denoted asmandp, respectively. The z-axis is normal to the film-plane, whereas the x-axis is parallel to the pinned layer magnetization. The current Iis positive when elec- trons flow from the free layer to the pinned layer. The LLG equation of the free layer magnetization mis dm dt=−γm×H−γHsm×(p×m) −γβHsm×p+αm×dm dt,(1)pxz+ - m(a) (b) mxmy 1 -1 001 -1 FIG. 1: (a) Schematic view of the system. (b) Schematic views of the contour plot of the effective energy map (dotted) , Eq. (2), and precession trajectory in a steady state with I= 1.6 mA (solid). whereγis the gyromagnetic ratio. Since the external field is assumed to be zero throughout this paper, the magnetic field H= (HK−4πM)mzezconsists of the per- pendicular anisotropy field only, where HKand 4πMare the crystalline and shape anisotropy fields, respectively. Sinceweareinterestedintheperpendicularlymagnetized free layer, HKshould be larger than 4 πM. The second and third terms on the right-hand-side of Eq. (1) are the spin torque and field-like torque, respectively. The spin torque strength, Hs=/planckover2pi1ηI/[2e(1+λm·p)MV], includes the saturation magnetization Mand volume Vof the free layer. The spin polarization of the current and the2 dependence of the spin torque strength on the relative angle of the magnetizations are characterized in respec- tive byηandλ[14]. According to Ref. [15], βshould be negative to stabilize the self-oscillation. The values of the parameters used in the following calculations are M= 1448 emu/c.c., HK= 20.0 kOe,V=π×60×60×2 nm3,η= 0.54,λ=η2,β=−0.2,γ= 1.732×107 rad/(Oe·s), andα= 0.005, respectively [6, 15]. The crit- ical current of the magnetization dynamics for β= 0 is Ic= [4αeMV/(/planckover2pi1ηλ)](HK−4πM)≃1.2 mA, where Ref. [15] shows that the effect of βon the critical current is negligible. Whenthecurrentmagnitudeisbelowthecrit- ical current, the magnetization is stabilized at mz= 1. In the oscillation state, the energy supplied by the spin torquebalancesthedissipationdue tothedamping. Usu- ally, the energy is the magnetic energy density defined as E=−M/integraltextdm·H[28], which includes the perpendic- ular anisotropy energy only, −M(HK−4πM)m2 z/2, in the present model. The first term on the right-hand-side of Eq. (1) can be expressed as −γm×[−∂E/∂(Mm)]. However, Eq. (1) indicates that an effective energy den- sity, Eeff=−M(HK−4πM) 2m2 z−β/planckover2pi1ηI 2eλVlog(1+λm·p), (2) should be introduced because the first and third terms on the right-hand-side of Eq. (1) can be summarized as −γm×[−∂Eeff/∂(Mm)]. Here, we introduce aneffective magnetic field H=−∂Eeff/∂(Mm) = (β/planckover2pi1ηI/[2e(1 + λmx)MV],0,(HK−4πM)mz). Dotted line in Fig. 1 (b) schematically shows the contour plot of the effective en- ergy density Eeffprojected to the xy-plane, where the constant energy curves slightly shift along the x-axis be- cause the second term in Eq. (2) breaks the axial sym- metry of E. Solid line in Fig. 1 (b) shows the preces- sion trajectory of the magnetization in a steady state withI= 1.6 mA obtained from the LLG equation. As shown, the magnetization steadily precesses practically on a constant energy curve of Eeff. Under a given cur- rentI, the effective energy density Eeffdetermining the constant energycurve of the stable precessionis obtained by the energy balance equation [27] αMα(Eeff)−Ms(Eeff) = 0. (3) In this equation, MαandMs, which are proportional to the dissipation due to the damping and energy supplied by the spin torque during a precession on the constant energy curve, are defined as [14, 25–27] Mα=γ2/contintegraldisplay dt/bracketleftBig H2−(m·H)2/bracketrightBig , (4) Ms=γ2/contintegraldisplay dtHs[p·H−(m·p)(m·H)−αp·(m×H)]. (5) The oscillation frequency on the constant energy curve(a) 00.010.02 -0.01 -0.020 0.2 0.4 0.6 0.8 1.0 mzMs, -αM α, Ms-αM αMs -αM αMs-αM α (b) 00.010.02 -0.01 -0.030 0.2 0.4 0.6 0.8 1.0 mzMs, -αM α, Ms-αM αMs -αM αMs-αM α -0.02β=0 β=-0.2 FIG. 2: Dependences of Ms,−αMα, and their difference Ms−Mαnormalized by γ(HK−4πM) onmz(0≤mz<1) for (a)β= 0, and (b) β=−0.2, where I= 1.6 mA. determined by Eq. (3) is given by f= 1/slashbig/contintegraldisplay dt. (6) Since we are interested in zero-field oscillation, and from the fact that the cross section of STO in experiment [6] is circle, we neglect external field Hextor with in-plane anisotropy field Hin−plane Kmxex. However, the above formula can be expanded to system with such effects by adding these fields to Hand terms −MHext·m− MHin−plane Km2 x/2 to the effective energy. In the absence of the field-like torque ( β= 0), i.e., Eeff=E, thereisone-to-onecorrespondencebetween the energy density Eandmz. Because an experimentally measurable quantity is the magnetoresistance propor- tional to ( RAP−RP)max[m·p]∝max[mx] =/radicalbig 1−m2z, it is suitable to calculate Eq. (3) as a function of mz, in- stead ofE, whereRP(AP)is the resistance of STO in the (anti)parallel alignment of the magnetizations. Figure 2 (a) shows dependences of Ms,−αMα, and their differ- enceMs−αMαonmz(0≤mz<1)forβ= 0, where Ms andMαare normalized by γ(HK−4πM). The current is set asI= 1.6 mA (> Ic). We also show Ms,−αMα, and their difference Ms−αMαforβ=−0.2 in Fig. 2 (b), wheremxis set as mx=−/radicalbig 1−m2z. Because −αMα is proportional to the dissipation due to the damping, −αMαis always −αMα≤0. The implications of Figs. 2 (a) and (b) are as follows. In Fig. 2 (a), Ms−αMαis always positive. This means that the energy supplied by3 current (mA)frequency (GHz) 1.2 1.4 1.6 1.8 2.012 0345 : Eq. (6): Eq. (1) FIG. 3: Current dependences of peak frequency of |mx(f)| obtained from Eq. (1) (red circle), and the oscillation fre- quency estimated by using (6) (solid line). the spin torque is always larger than the dissipation due to the damping, and thus, the net energy absorbed in the free layer is positive. Then, starting from the initial equilibrium state ( mz= 1), the free layer magnetization moves to the in-plane mz= 0, as shown in Ref. [14]. On the other hand, in Fig. 2 (b), Ms−αMαis positive from mz= 1to acertain m′ z, whereasit is negativefrom m′ zto mz= 0 (m′ z≃0.4 in the case of Fig. 2 (b)). This means that, starting from mz= 1, the magnetization can move to a point m′ zbecause the net energy absorbed by the free layer is positive, which drives the magnetization dy- namics. However, the magnetization cannot move to the film plane ( mz= 0) because the dissipation overcomes the energy supplied by the spin torque from mz=m′ ztomz= 0. Then, a stable and large amplitude precession is realized on a constant energy curve. We confirm the accuracy of the above formula by com- paring the oscillation frequency estimated by Eq. (6) withthenumericalsolutionoftheLLGequation, Eq. (1). In Fig. 3, we summarize the peak frequency of |mx(f)| forI= 1.2−2.0 mA (solid line), where mx(f) is the Fourier transformation of mx(t). We also show the oscil- lation frequency estimated from Eq. (6) by the dots. A quantitatively good agreement is obtained, guaranteeing the validity of Eq. (6). In conclusion, we developed a theoretical formula to evaluate the zero-field oscillation frequency of STO in the presence of the field-like torque. Our approach was basedon the energybalance equationbetween the energy suppliedbythe spintorqueandthe dissipationdue tothe damping. An effective energy density was introduced to take into account the effect of the field-like torque. We discussed that introducing field-like torque is necessary to find the energy balance between the spin torque and the damping, which as a result stabilizes a steady preces- sion. The validity of the developed theory was confirmed by performing the numerical simulation, showing a good agreement with the present theory. The authors would like to acknowledge T. Yorozu, H. Maehara, H. Tomita, T. Nozaki, K. Yakushiji, A. Fukushima, K. Ando, and S. Yuasa. This work was sup- ported by JSPS KAKENHI Number 23226001. [1] S. I. Kiselev, J. 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[8] S. Tsunegi, H. Kubota, K. Yakushiji, M. Konoto, S. Tamaru, A. Fukushima, H. Arai, H. Imamura, E. Grimaldi, R. Lebrun, et al., Appl. Phys. Express 7, 063009 (2014). [9] A. Dussaux, E. Grimaldi, B. R. Salles, A. S. Jenkins, A. V. Khavalkovskiy, P. Bortolotti, J. Grollier, H. Kub-ota, A. Fukushima, K. Yakushiji, et al., Appl. Phys. Lett. 105, 022404 (2014). [10] S. Yakata, H. Kubota, Y. Suzuki, K. Yakushiji, A. Fukushima, S. Yuasa, and K. Ando, J. Appl. Phys. 105, 07D131 (2009). [11] 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). [12] H. Kubota, S. Ishibashi, T. Saruya, T. Nozaki, A. Fukushima, K. Yakushiji, K. Ando, Y. Suzuki, and S. Yuasa, J. Appl. Phys. 111, 07C723 (2012). [13] S. Tsunegi, T. Taniguchi, H. Kubota, H. Imamura, S. Tamaru, M. Konoto, K. Yakushiji, A. Fukushima, and S. Yuasa, Jpn. J. Appl. Phys. 53, 060307 (2014). [14] T. 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0804.0820v2.Inhomogeneous_Gilbert_damping_from_impurities_and_electron_electron_interactions.pdf

arXiv:0804.0820v2 [cond-mat.mes-hall] 9 Aug 2008Inhomogeneous Gilbert damping from impurities and electro n-electron interactions E. M. Hankiewicz,1,2,∗G. Vignale,2and Y. Tserkovnyak3 1Department of Physics, Fordham University, Bronx, New York 10458, USA 2Department of Physics and Astronomy, University of Missour i, Columbia, Missouri 65211, USA 3Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA (Dated: October 30, 2018) We present a unified theory of magnetic damping in itinerant e lectron ferromagnets at order q2 including electron-electron interactions and disorder sc attering. We show that the Gilbert damping coefficient can be expressed in terms of the spin conductivity , leading to a Matthiessen-type formula in which disorder and interaction contributions are additi ve. Inaweak ferromagnet regime, electron- electron interactions lead to a strong enhancement of the Gi lbert damping. PACS numbers: 76.50.+g,75.45.+j,75.30.Ds Introduction – In spite of much effort, a complete theoretical description of the damping of ferromagnetic spin waves in itinerant electron ferromagnets is not yet available.1Recent measurements of the dispersion and damping of spin-wave excitations driven by a direct spin- polarized current prove that the theoretical picture is in- complete, particularly when it comes to calculating the linewidth of these excitations.2One of the most impor- tant parameters of the theory is the so-called Gilbert damping parameter α,3which controls the damping rate and thermal noise and is often assumed to be indepen- dent of the wave vector of the excitations. This assump- tion is justified for excitations of very long wavelength (e.g., a homogeneous precession of the magnetization), whereαcanoriginateinarelativelyweakspin-orbit(SO) interaction4. But it becomes dubious as the wave vector qof the excitations grows. Indeed, both electron-electron (e-e) and electron-impurity interactions can cause an in- homogeneous magnetization to decay into spin-flipped electron-hole pairs, giving rise to a q2contribution to the Gilbert damping. In practice, the presence of this contri- bution means that the Landau-Lifshitz-Gilbert equation contains a term proportional to −m×∇2∂tm(wherem is the magnetization) and requires neither spin-orbit nor magnetic disorder scattering. By contrast, the homoge- neous damping term is of the form m×∂tmand vanishes in the absence of SO or magnetic disorder scattering. The influence of disorder on the linewidth of spin waves in itinerant electron ferromagnets was discussed in Refs. 5,6,7, and the role of e-e interactions in spin-wave damping was studied in Refs. 8,9 for spin-polarized liq- uid He3and in Refs. 10,11fortwo-and three-dimensional electron liquids, respectively. In this paper, we present a unified semiphenomenological approach, which enables us to calculate on equal footing the contributions of dis- order and e-e interactions to the Gilbert damping pa- rameter to order q2. The main idea is to apply to the transverse spin fluctuations of a ferromagnet the method first introduced by Mermin12for treating the effect of disorder on the dynamics of charge density fluctuations in metals.13Following this approach, we will show that theq2contribution to the damping in itinerant electron ferromagnets can be expressed in terms of the transversespin conductivity, which in turn separates into a sum of disorder and e-e terms. A major technical advantage of this approach is that the ladder vertex corrections to the transverse spin- conductivity vanish in the absence of SO interactions, making the diagrammatic calculation of this quantity a straightforwardtask. Thusweareabletoprovideexplicit analytic expressions for the disorder and interaction con- tribution to the q2Gilbert damping to the lowest order in the strength of the interactions. Our paper connects and unifies different approaches and gives a rather com- plete and simple theory of q2damping. In particular, we find that for weak metallic ferromagnets the q2damping can be strongly enhanced by e-e interactions, resulting in a value comparable to or larger than typical in the case of homogeneous damping. Therefore, we believe that the inclusionofadampingtermproportionalto q2inthephe- nomenologicalLandau-Lifshitzequationofmotionforthe magnetization14is a potentially important modification of the theory in strongly inhomogeneous situations, such as current-driven nanomagnets2and the ferromagnetic domain-wall motion15.17 Phenomenological approach – In Ref. 12, Mermin con- structed the density-density response function of an elec- tron gas in the presence of impurities through the use of a local drift-diffusion equation, whereby the gradient of the external potential is cancelled, in equilibrium, by an opposite gradient of the local chemical potential. In diagrammatic language, the effect of the local chemical potential corresponds to the inclusion of the vertex cor- rection in the calculation of the density-density response function. Here, we use a similar approach to obtain the transverse spin susceptibility of an itinerant electron fer- romagnet, modeled as an electron gas whose equilibrium magnetization is along the zaxis. Before proceeding we need to clarify a delicate point. The homogeneous electron gas is not spontaneously fer- romagnetic at the densities that are relevant for ordinary magneticsystems.13Inordertoproducethe desired equi- librium magnetization, we must therefore impose a static fictitious field B0. Physically, B0is the “exchange” field Bexplus any external/applied magnetic field Bapp 0which maybeadditionallypresent. Therefore,inordertocalcu-2 late the transverse spin susceptibility we must take into account the fact that the exchange field associated with a uniform magnetization is parallel to the magnetization and changes direction when the latter does. As a result, the actual susceptibility χab(q,ω) differs from the sus- ceptibility calculated at constant B0, which we denote by ˜χab(q,ω), according to the well-known relation:11 χ−1 ab(q,ω) = ˜χ−1 ab(q,ω)−ωex M0δab. (1) Here,M0is the equilibrium magnetization (assumed to point along the zaxis) and ωex=γBex(whereγis the gyromagnetic ratio) is the precession frequency associ- ated with the exchange field. δabis the Kronecker delta. The indices aandbdenote directions ( xory) perpen- dicular to the equilibrium magnetization and qandω are the wave vector and the frequency of the external perturbation. Here we focus solely on the calculation of the response function ˜ χbecause term ωexδab/M0does not contribute to Gilbert damping. We do not include the effects of exchange and external fields on the orbital motion of the electrons. The generalized continuity equation for the Fourier component of the transverse spin density Main the di- rectiona(xory) at wave vector qand frequency ωis −iωMa(q,ω) =−iγq·ja(q,ω)−ω0ǫabMb(q,ω) +γM0ǫabBapp b(q,ω), (2) whereBapp a(q,ω)isthetransverseexternalmagneticfield driving the magnetization and ω0is the precessional fre- quency associated with a static magnetic field B0(in- cluding exchange contribution) in the zdirection. jais theath component of the transverse spin-current density tensor and we put /planckover2pi1= 1 throughout. The transverse Levi-Civita tensor ǫabhas components ǫxx=ǫyy= 0, ǫxy=−ǫyx= 1, and the summation over repeated in- dices is always implied. The transverse spin current is proportional to the gra- dient of the effective magnetic field, which plays the role analogousto the electrochemicalpotential, and the equa- tion that expressesthis proportionalityis the analogueof the drift-diffusion equation of the ordinary charge trans- port theory: ja(q,ω) =iqσ⊥/bracketleftbigg γBapp a(q,ω)−Ma(q,ω) ˜χ⊥/bracketrightbigg ,(3) whereσ⊥(=σxxorσyy) is the transverse dc (i.e., ω= 0) spin-conductivity and ˜ χ⊥=M0/ω0is the static trans- verse spin susceptibility in the q→0 limit.18Just as in the ordinary drift-diffusion theory, the first term on the right-hand side of Eq. (3) is a “drift current,” and the second is a “diffusion current,” with the two canceling out exactly in the static limit (for q→0), due to the relationMa(0,0) =γ˜χ⊥Bapp a(0,0). Combining Eqs. (2) and (3) gives the following equation for the transversemagnetization dynamics: /parenleftbigg −iωδab+γσ⊥q2 ˜χ⊥δab+ω0ǫab/parenrightbigg Mb= /parenleftbig M0ǫab+γσ⊥q2δab/parenrightbig γBapp b,(4) which is most easily solved by transforming to the circularly-polarized components M±=Mx±iMy, in which the Levi-Civita tensor becomes diagonal, with eigenvalues ±i. Solving in the “+” channel, we get M+=γ˜χ+−Bapp +=M0−iγσ⊥q2 ω0−ω−iγσ⊥q2ω0/M0γBapp +, (5) from which we obtain to the leading order in ωandq2 ˜χ+−(q,ω)≃M0 ω0/parenleftbigg 1+ω ω0/parenrightbigg +iωγσ⊥q2 ω2 0.(6) The higher-orderterms in this expansion cannot be legit- imately retained within the accuracy of the present ap- proximation. We also disregard the q2correction to the static susceptibility, since in making the Mermin ansatz (3) we are omitting the equilibrium spin currents respon- sible for the latter. Eq. (6), however, is perfectly ade- quate for our purpose, since it allows us to identify the q2contribution to the Gilbert damping: α=ω2 0 M0lim ω→0ℑm˜χ+−(q,ω) ω=γσ⊥q2 M0.(7) Therefore, the Gilbert damping can be calculated from the dc transverse spin conductivity σ⊥, which in turn can be computed from the zero-frequency limit of the transverse spin-current—spin-current response function: σ⊥=−1 m2∗Vlim ω→0ℑm/angb∇acketleft/angb∇acketleft/summationtextN i=1ˆSiaˆpia;/summationtextN i=1ˆSiaˆpia/angb∇acket∇ight/angb∇acket∇ightω ω,(8) whereˆSiaisthexorycomponentofspinoperatorforthe ith electron, ˆ piais the corresponding component of the momentum operator, m∗is the effective electron mass, V isthe systemvolume, Nisthe totalelectronnumber, and /angb∇acketleft/angb∇acketleftˆA;ˆB/angb∇acket∇ight/angb∇acket∇ightωrepresents the retarded linear response func- tion for the expectation value of an observable ˆAunder the action of a field that couples linearly to an observable ˆB. Both disorder and e-e interaction contributions can be systematically included in the calculation of the spin- current—spin-current response function. In the absence of spin-orbit and e-e interactions, the ladder vertex cor- rections to the conductivity are absent and calculation ofσ⊥reduces to the calculation of a single bubble with Green’s functions G↑,↓(p,ω) =1 ω−εp+εF±ω0/2+i/2τ↑,↓,(9) where the scattering time τsin general depends on the spin band index s=↑,↓. In the Born approximation,3 the scattering rate is proportional to the electron den- sity of states, and we can write τ↑,↓=τν/ν↑,↓, whereνs is the spin- sdensity of states and ν= (ν↑+ν↓)/2.τ parametrizes the strength of the disorder scattering. A standard calculation then leads to the following result: σdis ⊥=υ2 F↑+υ2 F↓ 6(ν−1 ↓+ν−1 ↑)1 ω2 0τ. (10) This, inserted in Eq. (7), gives a Gilbert damping pa- rameter in full agreement with what we have also calcu- lated from a direct diagrammatic evaluation of the trans- verse spin susceptibility, i.e., spin-density—spin-density correlation function. From now on, we shall simplify the notation by introducing a transversespin relaxation time 1 τdis ⊥=4(EF↑+EF↓) 3n(ν−1 ↓+ν−1 ↑)1 τ, (11) whereEFs=m∗υ2 Fs/2istheFermienergyforspin- selec- trons and nis the total electron density. In this notation, the dc transverse spin-conductivity takes the form σdis ⊥=n 4m∗ω2 01 τdis ⊥. (12) Electron-electron interactions – One of the attractive fea- tures of the approach based on Eq. (8) is the ease with which e-e interactions can be included. In the weak cou- pling limit, the contributions of disorder and e-e inter- actions to the transverse spin conductivity are simply additive. We can see this by using twice the equation of motion for the spin-current—spin-current response func- tion. This leads to an expression for the transverse spin-conductivity (8) in terms of the low-frequency spin- force—spin-force response function: σ⊥=−1 m2∗ω2 0Vlim ω→0ℑm/angb∇acketleft/angb∇acketleft/summationtext iˆSiaˆFia;/summationtext iˆSiaˆFia/angb∇acket∇ight/angb∇acket∇ightω ω.(13) Here,ˆFia=˙ˆpiais the time derivative of the momentum operator, i.e., the operator of the force on the ith elec- tron. The total force is the sum of electron-impurity and e-e interaction forces. Each of them, separately, gives a contribution of order |vei|2and|vee|2, whereveiandvee are matrix elements of the electron-impurity and e-e in- teractions, respectively, while cross terms are of higher order, e.g., vee|vei|2. Thus, the two interactions give ad- ditive contributions to the conductivity. In Ref.16, a phe- nomenological equation of motion was used to find the spin current in a system with disorder and longitudinal spin-Coulomb drag coefficient. We can use a similar ap- proach to obtain transversespin currents with transverse spin-Coulomb drag coefficient 1 /τee ⊥. In the circularly- polarized basis, i(ω∓ω0)j±=−nE 4m∗+j± τdis ⊥+j± τee ⊥,(14)and correspondingly the spin-conductivities are σ±=n 4m∗1 −(ω∓ω0)i+1/τdis ⊥+1/τee ⊥.(15) In the dc limit, this gives σ⊥(0) =σ++σ− 2=n 4m∗1/τdis ⊥+1/τee ⊥ ω2 0+/parenleftbig 1/τdis ⊥+1/τee ⊥/parenrightbig2.(16) Using Eq. (16), an identification of the e-e contribution is possible in a perturbative regime where 1 /τee ⊥,1/τdis ⊥≪ ω0, leading to the following formula: σ⊥=n 4m∗ω2 0/parenleftbigg1 τdis ⊥+1 τee ⊥/parenrightbigg . (17) Comparison with Eq. (13) enables us to immediately identify the microscopic expressions for the two scatter- ing rates. For the disorder contribution, we recover what we already knew, i.e., Eq. (11). For the e-e interaction contribution, we obtain 1 τee ⊥=−4 nm∗Vlim ω→0ℑm/angb∇acketleft/angb∇acketleft/summationtext iˆSiaˆFC ia;/summationtext iˆSiaˆFC ia/angb∇acket∇ight/angb∇acket∇ightω ω,(18) whereFCis just the Coulomb force, and the force-force correlation function is evaluated in the absence of disor- der. The correlation function in Eq. (18) is proportional to the function F+−(ω) which appeared in Ref. 11 [Eqs. (18) and (19)] in a direct calculation of the transverse spin susceptibility. Making use of the analytic result for ℑmF+−(ω)presentedinEq. (21)ofthatpaperweobtain 1 τee ⊥= Γ(p)8α0 27T2r4 sm∗a2 ∗k2 B (1+p)1/3, (19) /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48/s49/s48/s45/s54/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49 /s112/s61/s48/s46/s57/s57/s40/s110/s111/s32/s101/s45/s101/s32/s105/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s41 /s112/s61/s48/s46/s53/s112/s61/s48/s46/s49/s112/s61/s48/s46/s49 /s32/s32 /s49/s47 /s32/s91/s49/s47/s110/s115/s93 FIG. 1: (Color online) The Gilbert damping αas a function of the disorder scattering rate 1 /τ. Red (solid) line shows the Gilbertdampingfor polarization p= 0.1inthepresenceofthe e-e and disorder scattering, while dashed line does not incl ude thee-escattering. Blue(dotted)andblack(dash-dotted)l ines show Gilbert damping for p= 0.5 andp= 0.99, respectively. We took q= 0.1kF,T= 54K,ω0=EF[(1+p)2/3−(1−p)2/3], M0=γpn/2,m∗=me,n= 1.4×1021cm−3,rs= 5,a∗= 2a04 whereTis the temperature, p= (n↑−n↑)/nis the degree of spin polarization, a∗is the effective Bohr radius, rsis the dimensionless Wigner-Seitz radius, α0= (4/9π)1/3 and Γ(p) – a dimensionless function of the polarization p– is defined by Eq. (23) of Ref. 11. This result is valid to second order in the Coulomb interaction. Collecting our results, we finally obtain a full expression for the q2 Gilbert damping parameter: α=γnq2 4m∗M01/τdis ⊥+1/τee ⊥ ω2 0+/parenleftbig 1/τdis ⊥+1/τee ⊥/parenrightbig2.(20) One of the salient features of Eq. (20) is that it scales as the total scattering ratein the weak disorder and e-e interactions limit, while it scales as the scattering timein the opposite limit. The approximate formula for the Gilbert damping in the more interesting weak- scattering/strong-ferromagnet regime is α=γnq2 4m∗ω2 0M0/parenleftbigg1 τdis ⊥+1 τee ⊥/parenrightbigg , (21) while in the opposite limit, i.e. for ω0≪1/τdis ⊥,1/τee ⊥: α=γnq2 4m∗M0/parenleftbigg1 τdis ⊥+1 τee ⊥/parenrightbigg−1 . (22) Our Eq. (20) agrees with the result of Singh and Teˇ sanovi´ c6on the spin-wave linewidth as a function of the disorder strength and ω0. However, Eq. (20) also describes the influence of e-e correlations on the Gilbert damping. A comparison of the scattering rates originat- ing from disorder and e-e interactions shows that the lat- ter is important and can be comparable or even greater than the disorder contribution for high-mobility and/or low density 3D metallic samples. Fig. 1 shows the be- havior of the Gilbert damping as a function of the dis- order scattering rate. One can see that the e-e scatter- ing strongly enhances the Gilbert damping for small po- larizations/weak ferromagnets, see the red (solid) line. This stems from the fact that 1 /τdis ⊥is proportional to 1/τand independent of polarization for small polar- izations, while 1 /τee ⊥is enhanced by a large prefactorΓ(p) = 2λ/(1−λ2) + (1/2)ln[(1 + λ)/(1−λ)], where λ= (1−p)1/3/(1+p)1/3. On the other hand, for strong polarizations(dotted anddash-dottedlinesinFig.1), the disorder dominates in a broad range of 1 /τand the inho- mogenous contribution to the Gilbert damping is rather small. Finally, we note that our calculation of the e-e in- teractioncontributiontothe Gilbertdampingisvalidun- der the assumption of /planckover2pi1ω≪kBT(which is certainly the case ifω= 0). More generally, as follows from Eqs. (21) and (22) of Ref. 11, a finite frequency ωcan be included through the replacement (2 πkBT)2→(2πkBT)2+(/planckover2pi1ω)2 in Eq. (19). Thus 1 /τee ⊥is proportional to the scattering rateofquasiparticlesnearthe Fermi level, andour damp- ing constant in the clean limit becomes qualitatively sim- ilar to the damping parameter obtained by Mineev9for ωcorresponding to the spin-wave resonance condition in some external magnetic field (which in practice is much smaller than the ferromagnetic exchange splitting ω0). Summary – We have presented a unified theory of the Gilbert damping in itinerant electron ferromagnets at the order q2, including e-e interactions and disorder on equal footing. For the inhomogeneous dynamics ( q/negationslash= 0), these processes add to a q= 0 damping contribution that is governed by magnetic disorder and/or spin-orbit interactions. We have shown that the calculation of the Gilbertdampingcanbe formulatedinthe languageofthe spin conductivity, which takes an intuitive Matthiessen form with the disorder and interaction contributions be- ing simply additive. It is still a common practice, e.g., in the micromagnetic calculations of spin-wave dispersions and linewidths, to use a Gilbert damping parameter in- dependent of q. However, such calculations are often at odds with experiments on the quantitative side, particu- larly where the linewidth is concerned.2We suggest that the inclusion of the q2damping (as well as the associ- ated magnetic noise) may help in reconciling theoretical calculations with experiments. Acknowledgements – This work was supported in part by NSF Grants Nos. DMR-0313681 and DMR-0705460 as well as Fordham Research Grant. Y. T. thanks A. Brataas and G. E. W. Bauer for useful discussions. ∗Electronic address: hankiewicz@fordham.edu 1Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 2I. N. Krivorotov et al., Phys. Rev. B 76, 024418 (2007). 3T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 4E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B75, 174434 (2007). 5A. Singh, Phys. Rev. B 39, 505 (1989). 6A. Singh and Z. Tesanovic, Phys. Rev. B 39, 7284 (1989). 7V.L.SafonovandH.N.Bertram, Phys.Rev.B 61, R14893 (2000). 8V. P. Silin, Sov. Phys. JETP 6, 945 (1958).9V. P. Mineev, Phys. Rev. B 69, 144429 (2004). 10Y. Takahashi, K. Shizume, and N. Masuhara, Phys. Rev. B60, 4856 (1999). 11Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 56404 (2002). 12N. D. Mermin, Phys. Rev. B 1, 2362 (1970). 13G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, UK, 2005). 14E.M.Lifshitz andL.P.Pitaevskii, Statistical Physics, Part 2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox- ford, 1980), 3rd ed. 15Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn. Magn. Mater. 320, 1282 (2008), and reference therein.5 16I. D’Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000). 17In ferromagnets whose nonuniformities are beyond the linearized spin waves, there is a nonlinear q2contribu- tion to damping, (see J. Foros and A. Brataas and Y. Tserkovnyak, and G. E. W. Bauer, arXiv:0803.2175) which has a different physical origin, related to the longitudinalspin-current fluctuations. 18Although both σ⊥and ˜χ⊥are in principle tensors in trans- verse spin space, they are proportional to δabin axially- symmetric systems—hence we use scalar notation.

1309.4897v1.Van_der_Waals_Coefficients_for_the_Alkali_metal_Atoms_in_the_Material_Mediums.pdf

arXiv:1309.4897v1 [physics.atom-ph] 19 Sep 2013Van der Waals Coefficients for the Alkali-metal Atoms in the Ma terial Mediums aBindiya Arora∗andbB. K. Sahoo† aDepartment of Physics, Guru Nanak Dev University, Amritsar , Punjab-143005, India, bTheoretical Physics Division, Physical Research Laborato ry, Navrangpura, Ahmedabad-380009, India (Dated: Received date; Accepted date) The damping coefficients for the alkali atoms are determined v ery accurately by taking into account the optical properties of the atoms and three distin ct types of trapping materials such as Au (metal), Si (semi-conductor) and vitreous SiO 2(dielectric). Dynamic dipole polarizabilities are calculated precisely for the alkali atoms that reproduc e the damping coefficients in the perfect conducting medium within 0.2% accuracy. Upon the considera tion of the available optical data of the above wall materials, the damping coefficients are found t o be substantially different than those of the ideal conductor. We also evaluated dispersion coeffici ents for the alkali dimers and compared them with the previously reported values. These coefficients are fitted into a ready-to-use functional form to aid the experimentalists the interaction potential s only with the knowledge of distances. PACS numbers: 34.35.+a, 34.20.Cf, 31.50.Bc, 31.15.ap Accurate information on the long-range interactions such as dispersion (van der Waals) and retarded (Casimir-Polder) potentials between two atoms and be- tween an atom and surface of the trapping material are necessary for the investigation of the underlying physics of atomic collisions especially in the ultracold atomic ex- periments [1–4]. Presence of atom-surface interactions lead to a shift in the oscillation frequency of the trap which alters the trapping frequency as well as magic wavelengths for state-insensitive trapping of the trapped condensate. Moreover, this effect has also gained inter- est in generating novel atom optical devices known as the “atom chips”. In addition, the knowledge of dis- persion coefficients is required in experiments of photo- association, fluorescence spectroscopy, determination of scattering lengths, analysis of feshbach resonances, de- termination of stability of Bose-Einstein condensates (BECs), probingextra dimensionsto accommodate New- tonian gravity in quantum mechanics etc. [5–10]. Therehavebeenmanyexperimentalevidencesofanat- tractiveforcebetweenneutralatomsandbetweenneutral atoms with trapping surfaces but their precise determi- nations are relatively difficult. In the past two decades, several groups have evaluated dispersion coefficients C3 defining interaction between an atom and a wall using various approaches [11–13] without rigorous estimate of uncertainties. More importantly, they are evaluated for a perfect conducting wall which are quite different from an actual trapping wall. Since these coefficients depend on thedielectricconstantsofthematerialsofthewall, there- fore it is worth determining them precisely for trapping materials with varying dielectric constants (for good con- ducting, semi conducting, and dielectric mediums) as has been attempted in [14, 15]. Casimir and Polder [2] had estimated that at intermediately largeseparationsthe re- ∗Email: arorabindiya@gmail.com †Email: bijaya@prl.res.intardation effects of the virtual photons passing between the atom and its image weakens the attractive atom- wall force and the force scales with a different power law (given in details below). In this paper, we carefully ex- amine these retardation or damping effects which have not been extensively studied earlier. We also parameter- ized our damping coefficients into a readily usable form to be used in experiments. The atom-surface interaction potential resulting from the fluctuating dipole moment of an atom interacting with its image in the surface is formulated by [1, 14] Ua(R) =−α3 fs 2π/integraldisplay∞ 0dωω3α(ιω)/integraldisplay∞ 1dξe−2αfsξωRH(ξ,ǫ(ιω)), (1) whereαfsis the fine structure constant, ǫ(ω) is the fre- quencydependentdielectricconstantofthesolid, Risthe distance between the atom and the surface and α(ιω) is the ground state dynamic polarizability with imaginary argument. The function H(ξ,ǫ(ιω)) is given by H(ξ,ǫ) = (1−2ξ2)/radicalbig ξ2+ǫ−1−ǫξ/radicalbig ξ2+ǫ−1+ǫξ+/radicalbig ξ2+ǫ−1−ξ/radicalbig ξ2+ǫ−1+ξ with the Matsubara frequencies denoted by ξ. In asymptotic regimes, the Matsubara integration is dominated by its first term and the potential can be ap- proximated to Ua(R) =−C3T R3withC3T=α(0) 4(ǫ(0)− 1)/(ǫ(0)+1). The potential form can be described more accurately at the retardation distances as Ua(R) =−C4 R4 and at the non-retarded region as Ua(R) =−C3 R3[2]. To express the potential in the intermediate region, these approximations are usually modified either to Ua(R) = −C4 (R+λ)R3or toUa(R) =−C3 R3f3(R) whereλandf3(R) are respectively known as the reduced wavelength and damping function. It would be interesting to testify the validity of both the approximations by evaluating C3,C4 andf3(R) coefficients togetherfor different atomsin con- ducting, semi-conducting and dielectric materials. Since the knowledgeofmagneticpermeability ofthe materialis required to evaluate C4coefficients, hence we determine2 0 50 100 150 200 250 300 α(ιω)(a.u.)(a)Li Na K Rb 0 5 10 15 0 0.2 0.4 0.6 0.8 1ε(ιω)(a.u.) Frequency (a.u.)(b) Au Si SiO2 FIG. 1: Dynamic polarizabilities of the Li, Na, K and Rb atoms and dielectric permittivity of the Au, Si and SiO 2sur- faces along the imaginary axis as functions of frequencies. only the C3andf3(R) coefficients. With the knowledge ofC3andf3values, the atom-surface interaction poten- tialscanbeeasilyreproducedandtheycanbegeneralized to other surfaces. In general, the C3coefficient is given by C3≈1 4π/integraldisplay∞ 0dωα(ιω)ǫ(ιω)−1 ǫ(ιω)+1. (2) Foraperfectconductor ǫ→ ∞,ǫ(ιω)−1 ǫ(ιω)+1→1andforother materialswith their refractiveindices n=√ǫvaryingbe- tween 1 and 2,ǫ(ιω)−1 ǫ(ιω)+1≈ǫ(0)−1 ǫ(0)+1is nearly a constant and can be approximated to 0.77. For more preciseness, it is necessary to consider the actual frequency dependencies ofǫs in the materials. In the present work, three distinct materials such as Au, Si and SiO 2belonging to conduct- ing, semi-conducting and dielectric objects respectively, are taken into account to find out f3(R) functions and compared against a perfect conducting wall for which case we express [16] f3(R) =1 4πC3/integraldisplay∞ 0dωα(ιω)e−2αfsωRQ(αfsωR),(3) withQ(x) = 2x2+2x+1. To find out f3(R) for the other surfaces, we evaluate Ua(R) by substituting their ǫ(ιω) values in Eq. (1). Similarly, the leading term in the long-range interac- tion between two atoms denoted by aandbis approxi- mated by Uab(R) =−Cab 6 R6, wherethe Cab 6is knownasthe van der Waals coefficient and Ris the distance between two atoms. If retardation effects are included then it is modified to Uab(R) =−Cab 6 R6fab 6(R). The dispersion coef- ficientCab 6and the damping coefficient fab 6(R) betweenTABLE I: Calculated C3coefficients along with their uncer- tainties for the alkali-metal atoms and their comparison wi th other reported values. Classification of various contribut ions are in accordance with [11]a, [12]band [16]c. Li Na K Rb Perfect Conductor Core 0.074 0.332 0.989 1.513 Valence 1.387 1.566 2.115 2.254 Core-Valence ∼0 ∼0−0.016−0.028 Tail 0.055 0.005 0.003 0.003 Total 1.516(2) 1.904(2) 3.090(4) 3.742(5) Others 1.5178a1.8858b2.860b3.362b 1.889c Metal: Au Core 0.010 0.051 0.263 0.419 Valence 1.160 1.285 1.804 1.927 Core-Valence ∼0 ∼0 -0.005 -0.010 Tail 0.029 0.002 0.001 0.002 Total 1.199(2) 1.338(1) 2.062(4) 2.338(4) Others [14] 1.210 1.356 2.058 2.79 Semi-conductor: Si Core 0.006 0.033 0.184 0.299 Valence 0.993 1.099 1.543 1.649 Core-Valence ∼0 ∼0 -0.004 -0.008 Tail 0.023 0.002 0.001 0.001 Total 1.022(2) 1.134(1) 1.724(3) 1.942(4) Dielectric: SiO 2 Core 0.004 0.022 0.116 0.184 Valence 0.468 0.519 0.726 0.775 Core-Valence ∼0 ∼0 -0.002 -0.004 Tail 0.012 0.001 0.001 0.001 Total 0.4844(8) 0.5424(5) 0.839(1) 0.956(2) the atoms can be estimated using the expressions [16] Cab 6=3 π/integraldisplay∞ 0dωαa(ιω)αb(ιω),and fab 6=1 πCab 6/integraldisplay∞ 0dωαa(ιω)αb(ιω)e−2αfsωRP(αfsωR), whereP(x) =x4+2x3+5x2+6x+3. Using our previously reported E1 matrix elements [17, 18] and experimental energies, we plot the dy- namic polarizabilities of the ground states in Fig. 1 of the considered alkali atoms. The static polarizabil- ities corresponding to ω= 0 come out to be 164.1(7), 162.3(2), 289.7(6) and 318.5(8), as given in [17, 18], against the experimental values 164.2(11) [19], 162.4(2) [20], 290.58(1.42) [21] and 318.79(1.42) [21] in atomic unit (a.u.) for Li, Na, K and Rb atoms respectively. It clearly indicates the preciseness of our estimated results. The main reason for achieving such high accuracies in the estimated static polarizabilities is due to the use of E1 matrix elements extracted from the precise lifetime measurements of few excited states and by fitting our3 0 0.2 0.4 0.6 0.8 1f3 (a) Li (b) NaPerfect conductor Au Si SiO2 0 0.2 0.4 0.6 0.8 02000400060008000f3 R(a.u.)(c) K 0200040006000800010000 R(a.u.)(d) Rb FIG. 2: The retardation coefficient f3(R) (dimensionless) for Li, Na, K and Rb as a function of atom-wall distance R. TABLE II: Fitting parameters aandbforf3coefficients with a perfectly conducting wall, Au, Si, and SiO 2surfaces. Li Na K Rb Perfect Conductor a 0.9843 1.0802 1.1845 1.2598 b 0.0676 0.0866 0.0808 0.0907 Metal: Au a 0.9775 0.9846 1.0248 1.0437 b 0.0675 0.0614 0.0532 0.0558 Semi-conductor: Si a 0.9436 0.9436 0.9749 0.9869 b 0.0638 0.0718 0.0622 0.0647 Dielectric: SiO 2 a 0.9754 0.9789 1.0238 1.0423 b 0.0650 0.0746 0.0649 0.0685 E1 results obtained from the relativistic coupled-cluster calculation at the singles, doubles and partial triples ex- citation level (CCSD(T) method) to the measurements of the static polarizabilities of the excited states. Substituting the dynamic polarizabilities in Eq. (2), we evaluatethe C3coefficients fora perfect conductor(to compare with previous studies), for a real metal Au, for a semi conductor object Si and for a dielectric substance of glassy structure SiO 2. These values are given in Table I with break down from various individual contributions andestimateduncertaintiesarequotedintheparentheses afterignoringerrorsfromthe usedexperimentaldata. To achieve the claimed accuracy in our results it was neces- saryto use the complete tabulated data for the refraction indices of Au, Si, and SiO 2to calculate their dielectric permittivities at all the imaginary frequencies [22]. We evaluate the imaginary parts of the dielectric constants using the relation Im( ǫ(ω)) = 2n(ω)κ(ω), where n and κ are the real and imaginaryparts of the refractiveindex ofa material. The available data for Si and SiO 2are suffi- ciently extended to lower frequencies. However, they are extended to the lower frequencies for Au with the help of the Drude dielectric function [15] ǫ(ω) = 1−ω2 p ω(ω+ιγ), (4) with relaxation frequency γ= 0.035 eV and plasma frequency ωp= 9.02 eV. The corresponding real val- ues at imaginary frequencies are obtained by using the Kramers-Kronig formula Re(ǫ(ιω)) = 1+2 π/integraldisplay∞ 0dω′ω′Im(ǫ(ω′)) ω2+ω′2.(5) In bottom part of Fig. 1, the ǫ(ιω) values as a function of imaginary frequency are plotted for Au, Si, and SiO 2. The behavior of ǫ(ιω) for various materials is obtained as expected and they match well with the graphical repre- sentations given by Caride and co-workers [15]. As shown in Table I, C3coefficients increase with the increase in atomic mass. First we present our results for theC3coefficients for the interaction of these atoms with a perfectly conducting wall. The dominant contribution to theC3coefficients is from the valence part of the po- larizability. We also observed that the core contribution to theC3coefficients increases with the increasing num- ber of electrons in the atom which is in agreement with the prediction made in Ref. [12]. Our results are also in good agreement with the results reported by Kharchenko et al.[16] for Na. Therefore, our results obtained for other materials seem to be reliable enough. We noticed that the C3coefficients for a perfect conductor were ap- proximately 1.5, 2, and 3.5 times larger than the C3co- efficients for Au, Si, and SiO 2respectively. The decrease in the coefficient values for the considered mediums can be attributed to the fact that in case of dielectric ma- terial the theory is modified for non-unity reflection and for different origin ofthe transmitted wavesfrom the sur- face. In addition to this, for Si and SiO 2there are addi- tionalinteractionsduetochargedanglingbonds specially at shorter separations. The recent estimations with Au medium carried out by Lach et al.[14] are in agreement with our results since the polarizability database they have used is taken from Ref. [12]. These calculations seem to be sensitive on the choice of grids used for the numerical integration. An exponential grid yield the re- sults more accurately and it is insensitive to choice of the size of the grid in contrast to a linear grid. In fact with the use of a linear grid having a spacing 0 .1, we observed a 3-5% fall in C3coefficients for the considered atoms. The reason being that most of the contributions to the evaluation of these coefficients come from the lower fre- quencies which yield inaccuracy in the results for large grid size. Fig. 2 showsa comparisonof the f3(R) values obtained for Li, Na, K, and Rb atoms as a function of atom- wall separationdistance R for the four different materials4 TABLE III: C6coefficients with fitting parameters for the alkali dimers. Co ntributions from the valence, core and valence-core polarizabilities alone are labeled as Cv 6,Cc 6andCvc 6, respectively and Cct 6corresponds to contributions from the remaining cross terms. References:a[23],b[24],c[25],d[26],e[27],f[28],g[29]. Dimer Cv 6Cc 6Cvc 6Cct 6C6(Total) Others Exp a b Li-Li 1351 0.07 ∼0 39 1390(4) 1389(2)a,1388b,1394.6c,1473d0.8592 0.0230 Li-Na 1428 0.32 ∼0 37 1465(3) 1467(2)a0.8592 0.0245 Li-K 2201 1.27 ∼0 119 2321(6) 2322(5)a0.8640 0.0217 Li-Rb 2368 1.94 ∼0 179 2550(6) 2545(7)a0.8666 0.0262 Na-Na 1515 1.51 ∼0 33 1550(3) 1556(4)a, 1472b,1561c0.8591 0.0262 Na-K 2316 6.24 ∼0 118 2441(5) 2447(6)a2519e0.8555 0.0231 Na-Rb 2490 9.60 ∼0 184 2684(6) 2683(7)a0.8686 0.0232 K-K 3604 29.89 0.01 261 3895(15) 3897(15)a, 3813b,3905c3921f0.8738 0.0207 K-Rb 3880 46.91 0.02 465 4384(12) 4274(13)a0.8738 0.0207 Rb-Rb 4178 73.96 0.4 465 4717(19) 4691(23)a, 4426b,4635c4698g0.8779 0.0207 studied in this work. As seen in the figure, the retarda- tion coefficients are the smallest for an ideal metal. At very short separation distances the results for a perfectly conductingmaterialdiffersfromtheresultsofAu, Si, and SiO2by less than 4%. As the atom-surface distance in- creases, the deviations of f3results for various materials from the results of an ideal metal are considerable and vary as 18%, 15% and 6% for Li; 33%, 14% and 18% for Na; 40%, 13% and 26% for K; and 50%, 13% and 33% for Rb in Au, Si, and SiO 2surfaces respectively. The deviation of results between an ideal metal and other di- electric surfaces is smallest for the Li atom and increases appreciably for the Rb atom. We use the functional form todescribeaccuratelytheatom-wallinteractionpotential at the separate distance R as f3(R) =1 a+b(αfsR). (6) By extrapolating data from the above figure, we list the extracted aandbvalues for the considered atoms in all the materials in Table II. In Table III, we present our calculated results for the C6coefficients for the alkali dimers. In columns II, III and IV, we give individual contributions from the va- lence, core and valence-core polarizabilities to C6eval- uation and column V represents contributions from the cross terms which are found to be crucial for obtaining accurate results. As can be seen from Table I, the trends are almost similar to C3evaluation. A comparison of our C6values with other recent calculations and available experimental results is also presented in the same table. Using the similar fitting procedure as for f3, we obtained fitting parameters aandbforf6from Fig. 3 which are quoted in the last two columns of the above table. To summarize, we haveinvestigatedthe dispersion and damping coefficients for the atom-wall and atom-atominteractions for the Li, Na, K, and Rb atoms and their dimers in this work. The interaction potentials of the al- kali atomsare studied with Au, Si, and SiO 2surfacesand 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2000 4000 6000 8000 10000f6 R(a.u.)Li-Li Li-Na Li-K Li-Rb Na-Na Na-K Na-Rb K-K K-Rb Rb-Rb FIG. 3: The retardation coefficient f6(R) (dimensionless) for the alkali dimers as a function of atom-atom distance R. found to be very different than a perfect conductor. It is also shown that the interaction of the atoms in these sur- faces is considerably distinct from each other. A readily usable functional form of the retardation coefficients for the interaction between two alkali atoms and alkali atom with the above mediums is provided. Our fit explains more than 99% of total variation in data about average. The results are compared with the other theoretical and experimental values. The work of B.A. is supported by the CSIR, India (Grant no. 3649/NS-EMRII). We thank Dr. G. Klim- chitskaya and Dr. G. Lach for some useful discussions. B.A.alsothanksMr. S.Sokhalforhishelpin somecalcu- lations. Computations were carried out using 3TFLOP HPC Cluster at Physical Research Laboratory, Ahmed- abad.5 [1] E. M. Lifshitz and L. P Pitaevskii, Statistical Physics , Pergamon Press, Oxford, London (1980). [2] H. B. G. Casimir and D. Polder, Phys. Rev. 73, 4 (1948). [3] F. London, Z. Physik 63, 245 (1930). [4] J. Israelachvili, Intermolecular and Surface Forces, A ca- demic Press, San Diego (1992). [5] J. L. Roberts et al., Phys. Rev, Lett. 81, 5109 (1998). [6] C. Amiot and J. Verges, J. Chem. Phys. 112, 7068 (2000). [7] P. J. Leo, C. J. Williams and P. S. Julienne, Phys. Rev. Lett.85, 2721 (2000). [8] D. M. Harber et al., J. Low Temp. Phys. 133, 229 (2003). [9] A. E. Leanhardt et al., Phys. Rev. Lett. 90, 100404 (2003). [10] Y. Lin et al., Phys. Rev. Lett. 92, 050404 (2004). [11] A. Derevianko, W. R. Johnson and S. Fritzsche, Phys. Rev. A57, 2629 (1998). [12] A. Derevianko et al., Phys. Rev. Lett. 82, 3589 (1999). [13] J. Jiang, Y. Cheng and J. Mitroy, J. Phys. B 46, 125004 (2013). [14] G. Lach, M. Dekieviet and U. D. Jentschura, Int. J. Mod. Phys. A 25, 2337 (2010). [15] A. O. Caride et al., Phys. Rev. A 71, 042901 (2005).[16] P. Kharchenko, J. F. Babb and A. Dalgarno, Phys. Rev. A55, 3566 (1997). [17] B. Arora and B. K. Sahoo, Phys. Rev. A 86, 033416 (2012). [18] B. K. Sahoo and B. Arora, Phys. Rev. A 87, 023402 (2013). [19] A. Miffre et al., Eur. Phys. J. D 38, 353 (2006). [20] C. R. Ekstrom et al., Phys. Rev. A 51, 3883 (1995). [21] W. F. Holmgren et al., Phys. Rev. A 81, 053607 (2010). [22] E. D. Palik, Handbook of Optical Constants of Solids , Academic Press, San Diego (1985). [23] A. Derevianko, J. F. Babb and A. Dalgarno, Phys. Rev. A83, 052704 (2001). [24] M. Marinescu, H. R. Sadeghpour and A. Dalgarno, Phys. Rev. A49, 982 (1994). [25] J. Mitroy and M. W. J. Bromley, Phys. Rev. A 68, 052714 (2003). [26] L. W. Wansbeek et al., Phys. Rev. A 78, 012515 (2008); Erratum: Phys. Rev. A 82, 029901 (2010) [27] I. Russier-Antoine et al., J. Phys. B 33, 2753 (2000). [28] A. Pashov et al., Eur. Phys. J. D 46, 241 (2008). [29] C. Chin et al., Phys. Rev. A 70, 032701 (2004).

1404.1488v2.Gilbert_damping_in_noncollinear_ferromagnets.pdf

arXiv:1404.1488v2 [cond-mat.mtrl-sci] 27 Nov 2014Gilbert damping in noncollinear ferromagnets Zhe Yuan,1,∗Kjetil M. D. Hals,2,3Yi Liu,1Anton A. Starikov,1Arne Brataas,2and Paul J. Kelly1 1Faculty of Science and Technology and MESA+Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands 2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 3Niels Bohr International Academy and the Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark The precession and damping of a collinear magnetization dis placed from its equilibrium are well described by the Landau-Lifshitz-Gilbert equation. The th eoretical and experimental complexity of noncollinear magnetizations is such that it is not known h ow the damping is modified by the noncollinearity. We use first-principles scattering theor y to investigate transverse domain walls (DWs) of the important ferromagnetic alloy Ni 80Fe20and show that the damping depends not only on the magnetization texture but also on the specific dynamic modes of Bloch and N´ eel DWs in ways that were not theoretically predicted. Even in the highly di sordered Ni 80Fe20alloy, the damping is found to be remarkably nonlocal. PACS numbers: 72.25.Rb, 75.60.Ch, 75.78.-n, 75.60.Jk Introduction. —The key common ingredient in various proposed nanoscale spintronics devices involving mag- netic droplet solitons [ 1], skyrmions [ 2,3], or magnetic domain walls (DWs) [ 4,5], is a noncollinear magneti- zation that can be manipulated using current-induced torques (CITs) [ 6]. Different microscopic mechanisms have been proposed for the CIT including spin trans- fer [7,8], spin-orbit interaction with broken inversion symmetry in the bulk or at interfaces [ 9–11], the spin- Halleffect[ 12]orproximity-inducedanisotropicmagnetic properties in adjacent normal metals [ 13]. Their contri- butions are hotly debated but can only be disentangled if the Gilbert damping torque is accurately known. This is not the case [ 14]. Theoretical work [ 15–19] suggest- ing that noncollinearity can modify the Gilbert damping due to the absorption of the pumped spin current by the adjacent precessing magnetization has stimulated exper- imental efforts to confirm this quantitatively [ 14,20]. In this Letter, we use first-principles scattering calculations to show that the Gilbert damping in a noncollinear alloy can be significantly enhanced depending on the partic- ular precession modes and surprisingly, that even in a highly disordered alloy like Ni 80Fe20, the nonlocal char- acterofthe dampingis verysubstantial. Ourfindingsare important for understanding field- and/or current-driven noncollinear magnetization dynamics and for designing new spintronics devices. Gilbert damping in Ni 80Fe20DWs.—Gilbert damping is in general described by a symmetric 3 ×3 tensor. For a substitutional, cubic binary alloy like Permalloy, Ni80Fe20, this tensor is essentially diagonal and isotropic and reduces to scalar form when the magnetization is collinear. A value of this dimensionless scalar calculated from first-principles, αcoll= 0.0046, is in good agree- ment with values extracted from room temperature ex- periments that range between 0.004 and 0.009 [ 21]. In a one-dimensional (1D) transverse DW, the Gilbert damp-ing tensor is still diagonal but, as a consequence of the lowered symmetry [ 22], it contains two unequal compo- nents. The magnetization in static N´ eel or Bloch DWs (a) (b) (c) φ θ x y z φ θ 0 0.1 0.2 0.3 0.4 1/( /h w) (nm -1 )00.01 0.02 0.03 _eff Néel Bloch jSO =0 50 20 10 5 3 /h w (nm) _oeff _ieff FIG. 1. (color online). Sketch of N´ eel (a) and Bloch (b) DWs. (c) Calculated effective Gilbert damping parameters for Permalloy DWs (N´ eel, black lines; Bloch, red lines) as a function of the inverse of the DW width λw. Without spin- orbit coupling, calculations for the two DW types yield the same results (blue lines). The green dot represents the valu e of Gilbert damping calculated for collinear Permalloy. For each value of λw, we typically consider 8 different disorder configurations and the error bars are a measure of the spread of the results.2 liesinsidewelldefinedplanesthatareillustratedinFig. 1. An angle θrepresents the in-plane rotation with respect to the magnetizationin the left domainand it variesfrom 0 toπthrough a 180◦DW. If the plane changes in time, as it does when the magnetization precesses, an angle φ can be used to describe its rotation. We define an out- of-plane damping component αocorresponding to varia- tion inφ, and an in-plane component αicorresponding to time-dependent θ. Rigid translation of the DW, i.e. making the DW center rwvary in time, is a specific ex- ample of the latter. For Walker-profile DWs [ 23], an effective (dimension- less) in-plane ( αeff i) and out-of-plane damping ( αeff o) can be calculated in terms of the scattering matrix Sof the system using the scattering theory of magnetization dis- sipation [ 24,25]. Both calculated values are plotted in Fig.1(c) as a function of the inverse DW width 1 /λwfor N´ eel and Bloch DWs. Results with the spin-orbit cou- pling (SOC) artificially switched off are shown for com- parison; because spin space is then decoupled from real space, the results for the two DW profiles are identical and both αeff iandαeff ovanish in the large λwlimit con- firming that SOC is the origin of intrinsic Gilbert damp- ing for collinear magnetization. With SOC switched on, N´ eel and Bloch DWs have identical values within the numerical accuracy, reflecting the negligibly small mag- netocrystalline anisotropy in Permalloy. Both αeff iand αeff oapproach the collinear value αcoll[21], shown as a green dot in the figure, in the wide DW limit. For finite widths, theyexhibit aquadraticandapredominantlylin- ear dependence on 1 /(πλw), respectively, both with and withoutSOC;forlargevaluesof λw, thereisahintofnon- linearity in αeff o(λw). However, phenomenological theo- ries [15–17] predict that αeff ishould be independent of λw and equal to αcollwhileαeff oshould be a quadratic func- tion of the magnetization gradient. Neither of these pre- dicted behaviours is observed in Fig. 1(c) indicating that existing theoretical models of texture-enhanced Gilbert damping need to be reexamined. Theαeffshown in Fig. 1(c) is an effective damping constant because the magnetization gradient dθ/dzof a Walker profile DW is inhomogeneous. Our aim in the following is to understand the physical mechanisms of texture-enhanced Gilbert damping with a view to deter- mining how the local damping depends on the magneti- zation gradient, as well as the corresponding parameters for Permalloy, and finally expressing these in a form suit- able for use in micromagnetic simulations. In-plane damping αi.—To get a clearer picture of how the in-plane damping depends on the gradient, we calcu- late the energy pumping Er≡Tr/parenleftBig ∂S ∂rs∂S† ∂rs/parenrightBig for a finite lengthLof a Bloch-DW-type spin spiral (SS) centered atrs. In this SS segment (SSS), dθ/dzis constant ex- cept at the ends. Figure 2(b) showsthe resultscalculated without SOC for a single PermalloySSS with dθ/dz= 6◦0 10 20 30 40 L (nm) 020 40 Er (nm -2 )Without smearing With smearing 0 4 2 6Winding angle ( /) 0 1 2 3 4 Number of SSSs z0n//L de/dz L L (c) (a) (b) FIG. 2. (color online). (a) Sketch of the magnetization gra- dient for two SSSs separated by collinear magnetization wit h (green, dashed) and without (red, solid) a broadening of the magnetization gradient at the ends of the SSSs. The length of each segment is L. (b) Calculated energy pumping Eras a function of Lfor asingle Permalloy Bloch-DW-typeSSSwith- out SOC. The upper horizontal axis shows the total winding angle of the SSS. (c) Calculated energy pumping Erwithout SOC as a function of the number of SSSs that are separated by a stretch of collinear magnetization. per atomic layer; Fig. 1(c) shows that SOC does not in- fluence the quadratic behaviour essentially. Eris seen to be independent of Lindicating there is no dissipation whendθ/dzis constant in the absence of SOC. In this case, the only contribution arises from the ends of the SSS where dθ/dzchanges abruptly; see Fig. 2(a). If we replace the step function of dθ/dzby a Fermi-like func- tion with a smearing width equal to one atomic layer, Er decreasessignificantly(greensquares). Formultiple SSSs separated by collinear magnetization, we find that Eris proportional to the number of segments; see Fig. 2(c). What remains is to understand the physical origin of the damping at the ends of the SSSs. Rigid translation of a SSS or of a DW allows for a dissipative spin cur- rentj′′ s∼ −m×∂z∂tmthat breaks time-reversal sym- metry [19]. The divergence of j′′ sgives rise to a local dissipative torque, whose transverse component is the enhancement of the in-plane Gilbert damping from the magnetizationtexture. After straightforwardalgebra, we obtain the texture-enhanced in-plane damping torque α′′/bracketleftbig (m·∂z∂tm)m×∂zm−m×∂2 z∂tm/bracketrightbig ,(1) whereα′′is a material parameter with dimensions of length squared. In 1D SSs or DWs, Eq. ( 1) leads to the local energy dissipation rate ˙E(r) = (α′′Ms/γ)∂tθ∂t(d2θ/dz2) [25], where Msis the satura- tion magnetization and γ=gµB//planckover2pi1is the gyromagnetic ratio expressed in terms of the Land´ e g-factor and the Bohr magneton µB. This results shows explicitly that the in-plane damping enhancement is related to finite d2θ/dz2. Using the calculated data in Fig. 1(c), we ex-3 tract a value for the coefficient α′′= 0.016 nm2that is independent of specific textures m(r) [25]. Out-of-plane damping αo.—We begin our analysis of the out-of-plane damping with a simple two-band free- electron DW model [ 25]. Because the linearity of the damping enhancement does not depend on SOC, we ex- amine the SOC free case for which there is no differ- ence between N´ eel and Bloch DW profiles and we use N´ eel DWs in the following. Without disorder, we can use the known φ-dependence of the scattering matrix for this model [ 31] to obtain αeff oanalytically, αeff o=gµB 4πAMsλw/summationdisplay k/bardbl/parenleftbigg/vextendsingle/vextendsingle/vextendsinglerk/bardbl ↑↓/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsinglerk/bardbl ↓↑/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingletk/bardbl ↑↓/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingletk/bardbl ↓↑/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg ≈gµB 4πAMsλwh e2GSh,. (2) whereAis the cross sectional area and the convention used for the reflection ( r) and transmission ( t) probabil- ity amplitudes is shown in Fig. 3(a). Note that |tk/bardbl ↑↓|2and |tk/bardbl ↓↑|2are of the order of unity and much larger than the othertwotermsbetweenthebracketsunlesstheexchange splitting is very large and the DW width very small. It is then a good approximation to replace the quantities in bracketsbythenumberofpropagatingmodesat k/bardbltoob- tain the second line of Eq. ( 2), where GShis the Sharvin conductance that only depends on the free-electron den- sity. Equation ( 2) shows analytically that αeff ois pro- portional to 1 /λwin the ballistic regime. This is repro- duced by the results of numerical calculations for ideal free-electron DWs shown as black circles in Fig. 3(b). Introducing site disorder [ 32] into the free-electron model results in a finite resistivity. The out-of-plane damping calculated for disordered free-electron DWs ex- hibits a transition as a function of its width. For narrow DWs (ballistic limit), αeff ois inversely proportional to λw and the green, red and blue circles in Fig. 3(b) tend to becomeparalleltothevioletlineforsmallvaluesof λw. If λwis sufficiently large, αeff obecomes proportional to λ−2 w in agreement with phenomenological predictions [ 15–17] where the diffusive limit is assumed. This demonstrates the different behaviour of αeff oin these two regimes. We can construct an expression that describes both the ballistic and diffusive regimes by introducing an ex- plicit spatial correlation in the nonlocal form of the out- of-plane Gilbert damping tensor that was derived using the fluctuation-dissipation theorem [ 15] [αo]ij(r,r′) =αcollδijδ(r−r′)+α′D(r,r′;l0) ×[m(r)×∂zm(r)]i[m(r′)×∂z′m(r′)]j.(3) Hereα′isamaterialparameterwithdimensionsoflength squared and Dis a correlation function with an effective correlation length l0. In practice, we use D(r,r′;l0) = 1√πAl0e−(z−z′)2/l2 0, which reduces to δ(r−r′) in the dif- fusive limit ( l0≪λw) and reproduces earlier results [ 15– 17]. In the ballistic limit, both α′andl0are infinite,0.01 0.05 0.1 0.5 1/( /h w) (nm -1 )10 -4 10 -3 10 -2 10 -1 _oeff Ballistic l=2.7 !1 cm l=25 !1 cm l=94 !1 cm 100 50 30 20 10 5 2/h w (nm) ~1/ hw ~1/ hw2(a) (b) and . By definition, for weak splitting 1, but for all commonplace s the Fermi wavelength 2 is orders of magnitude smaller than . This implies a wall resistance that is vanishingly small, because of the exponential depen- dence. For the example of iron, 2 is only 1 or 2 A , depending on which band is in question, whilst the wall thickness is some thousands of A . This leads to a 10 . The physical reason for this is that waves are only scattered very much by potential steps that are abrupt on the scale of the wavelength of that wave, as sketched in figure 13. For strong splitting ( it was found to be necessary to restrict the culation to a very narrow wall, viz. me 1. In practice this means mic abruptness. In this case a variable ¼ ð ÞÞ , trivially connected to the definitions of in equations (2) and (3), determines the DW ce. The obvious relationship with the Stearns definition of polarisation, equation (3), emphasises that the theory is essentially one of tunnelling between one domain and the next. The DW resistance vanishes as 1, as might be d. As !1 uivalent to unity), the material becomes half-metallic and the wall resistance also !1 . A multi-domain half-metal, with no opportunity for spin relaxation, is an insulator, no matter how high is. Cabrera and Falicov satisfied themselves that, once the diamagnetic Lorentz force e that give rise to additional resistance at the wall were properly treated [178], their theory could account for the results found in the Fe whiskers. However, it does not describe most cases encountered experimentally because the condition Abrupt Figure 13. Spin-resolved potential profiles and resulting wavefunctions at abrupt and wide (adiabatic) domain walls. The wavefunctions are travelling from left to right. In the adiabatic case, the wavelengths of the two wavefunctions are exchanged, but the change in potential energy is slow enough that there is no change in the amplitude of the transmitted wave. When the wall is abrupt the wavelength change is accompanied by substantial reflection, lting in a much lower transmitted amplitude (the reflected part of the wavefunction is not shown). This gives rise to domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010 V↑ V↓ ↓ ↑ e±ik↑z e±ik↓z e±ik↓z e±ik↑z . By definition, for weak splitting 1, but for all commonplace mi wavelength 2 is orders of magnitude smaller than . This a wall resistance that is vanishingly small, because of the exponential depen- e of iron, 2 is only 1 or 2 A , depending on which band is in question, whilst the wall thickness is some thousands of A . This leads to a 10 . The physical reason for this is that waves are only scattered very much by potential steps that are abrupt on the scale of the wavelength of that wave, as d in figure 13. it was found to be necessary to restrict the to a very narrow wall, 1. In practice this means abruptness. In this case a variable ¼ ð ÞÞ , trivially to the definitions of in equations (2) and (3), determines the DW . The obvious relationship with the Stearns definition of polarisation, on (3), emphasises that the theory is essentially one of tunnelling between DW resistance vanishes as 1, as might be d. As !1 to , the material becomes half-metallic !1 . A multi-domain half-metal, with no opportunity is an insulator, no matter how high to additional resistance at the wall were properly treated ld account for the results found in the Fe whiskers. However, it does not describe most cases encountered experimentally because the condition at abrupt to right. In the of the two wavefunctions are exchanged, but the change in is slow enough that there is no change in the amplitude of the transmitted is abrupt the wavelength change is accompanied by substantial reflection, in a much lower transmitted amplitude (the reflected part of the wavefunction is not to domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010 t↑↑ t↓↓ ↓↓ t↑↓ t↓↑ FIG. 3. (color online). (a) Cartoon of electronic transport in a two-band, free-electron DW. The global quantization axis of the system is defined by the majority and minority spin states in the left domain. (b) Calculated αeff ofor two- band free-electron DWs as a function of 1 /(πλw) on a log-log scale. The black circles show the calculated results for the clean DWs, whichare in perfect agreement with theanalytica l model Eq. ( 2), shown as a dashed violet line. When disorder (characterized by the resistivity ρcalculated for the corre- sponding collinear magnetization) is introduced, αeff oshows a transition from a linear dependence on 1 /λwfor narrow DWs toaquadraticbehaviourfor wideDWs. The solid lines arefits using Eq. ( S24). The dashed orange lines illustrate quadratic behaviour. but the product α′D(r,r′;l0) =α′/(√πAl0) is finite and related to the Sharvin conductance of the system [ 33], consistent with Eq. ( 2). We then fit the calculated val- ues ofαeff oshown in Fig. 3(b) using Eq. ( S24) [25]. With the parameters α′andl0listed in Table I, the fit is seen to be excellent over the whole range of λw. The out- of-plane damping enhancement arises from the pumped spin current j′ s∼∂tm×∂zmin a magnetization tex- ture [15,17], where the magnitude of j′ sis related to the TABLEI. Fitparameters usedtodescribe thedampingshown in Fig.1for Permalloy DWs and in Fig. 3for free-electron DWs with Eq. ( S24). The resistivity is determined for the corresponding collinear magnetization. System ρ(µΩ cm) α′(nm2)l0(nm) Free electron 2 .69 45 .0 13 .8 Free electron 24 .8 1 .96 4 .50 Free electron 94 .3 0 .324 2 .78 Py (ξSO= 0) 0 .504 23 .1 28 .3 Py (ξSO/negationslash= 0) 3 .45 5 .91 13 .14 conductivity [ 15]. This is the reason why α′is larger in a system with a lower resistivity in Table I.l0is a mea- sure of how far the pumped transverse spin current can propagate before being absorbed by the local magnetiza- tion. It is worth distinguishing the relevant characteris- tic lengths in microscopic spin transport that define the diffusive regimes for different transport processes. The mean free path lmis the length scale for diffusive charge transport. The spin-flip diffusion length lsfcharacterizes the length scale for diffusive transport of a longitudinal spin current, and l0is the corresponding length scale for transverse spin currents. Only when the system size is larger than the corresponding characteristic length can transport be described in a local approximation. We can use Eq. ( S24) to fit the calculated αeff oshown in Fig.1for Permalloy DWs. The results are shown in Fig.S4. Since the values of αeff owe calculate for N´ eel and Bloch DWs are nearly identical, we take their aver- age for the SOC case. Intuitively, we would expect the out-of-plane damping for a highly disordered alloy like Permalloy to be in the diffusive regime corresponding to a shortl0. But the fitted values of l0are remarkably large, as long as 28.3 nm without SOC. With SOC, l0 is reduced to 13.1 nm implying that nonlocal damping can play an important role in nanoscale magnetization textures in Permalloy, whose length scale in experiment is usually about 100 nm and can be reduced to be even smaller than l0by manipulating the shape anisotropy of experimental samples [ 34,35]. As shown in Table I,l0is positively correlated with the conductivity. The large value of l0and the low re- sistivity of Permalloy can be qualitatively understood in terms of its electronic structure and spin-dependent scat- tering. The Ni and Fe potentials seen by majority-spin electrons around the Fermi level in Permalloy are almost identical [ 25] so that they are only very weakly scattered. The Ni and Fe potentials seen by minority-spin electrons are howeverquite different leading to strongscattering in transport. The strong asymmetric spin-dependent scat- tering can also be seen in the resistivity of Permalloy calculated without SOC, where ρ↓/ρ↑>200 [21,36]. As a result, conduction in Permalloy is dominated by the weakly scattered majority-spin electrons resulting in a low total resistivity and a large value of l0. This short- circuit effect is only slightly reduced by SOC-induced spin-flipscatteringbecausetheSOCin3 dtransitionmet- als is in energy terms small compared to the bandwidth and exchange splitting. Indeed, αeff o−αcollcalculated with SOC (the red curve in Fig. S4) shows a greater cur- vature at large widths than without SOC, but is still quite different from the quadratic function characteristic ofdiffusive behaviourforthe widest DWs wecould study. Bothαeff iandαeff ooriginate from locally pumped spin currents proportional to m×∂tm. Because of the spa- tially varying magnetization, the spin currents pumped totheleftandrightdonotcancelexactlyandthenetspin0.02 0.05 0.1 0.2 0.5 1/(πλw) (nm-1)0.0010.010.05αoeff-αcoll ξSO≠0 ξSO=040 30 20 15 10 5 3 2πλw (nm) ~1/λw ~1/λw2 FIG. 4. (color online). Calculated out-of-plane damping αeff o−αcollfrom Fig. 1plotted as a function of 1 /(πλw) on a log-log scale. The solid lines are fitted using Eq. ( S24). The dashed violet and orange lines illustrate linear and quadra tic behaviour, respectively. current contains two components, j′′ s∼ −m×∂z∂tm[19] andj′ s∼∂tm×∂zm[15,17]. For out-of-plane damping, ∂zmis perpendicular to ∂tmso there is large enhance- ment due to the lowest order derivative. For the rigid motion of a 1D DW, ∂zmis parallel to ∂tmso thatj′ s vanishes. The enhancement of in-plane damping arising fromj′′ sdue to the higher-orderspatial derivative of mag- netization is then smaller. Conclusions.— We have discovered an anisotropic texture-enhanced Gilbert damping in Permalloy DWs using first-principles calculations. The findings are ex- pressed in a form [Eqs. ( 1) and (S24)] suitable for ap- plication to micromagnetic simulations of the dynamics of magnetization textures. The nonlocal character of the magnetization dissipation suggests that field and/or cur- rentdrivenDW motion, whichis alwaysassumedto be in the diffusive limit, needs to be reexamined. The more ac- curate form of the damping that we propose can be used to deduce the CITs in magnetization textures where the usual way to study them quantitatively is by comparing experimental observations with simulations. Current-drivenDWs movewith velocities that arepro- portional to β/αwhereβis the nonadiabatic spin trans- fer torque parameter. The order of magnitude spread in values of βdeduced for Permalloy from measurements of the velocities of vortex DWs [ 37–40] may be a result of assumingthat αis a scalarconstant. Ourpredictions can be tested by reexamining these studies using the expres- sions for αgiven in this paper as input to micromagnetic calculations. We would like to thank Geert Brocks and Taher Am- laki for useful discussions. This work was financially supported by the “Nederlandse Organisatie voor Weten- schappelijk Onderzoek” (NWO) through the research programme of “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) and the supercomputer facilities5 of NWO “Exacte Wetenschappen (Physical Sciences)”. It was also partly supported by the Royal Netherlands Academy of Arts and Sciences (KNAW). A.B. acknowl- edges the Research Council of Norway, grant no. 216700. ∗Present address: Institut f¨ ur Physik, Johannes Gutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128 Mainz, Germany; zyuan@uni-mainz.de [1] S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, P. K. Muduli, E. Iacocca, A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. Hoefer, and J. ˚Akerman, Science339, 1295 (2013) . [2] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature465, 901 (2010) . [3] A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnology 8, 152 (2013) . 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B 89, 020403(R) (2014) .1 Supplementary Material for “Gilbert damping in noncolline ar ferromagnets” Zhe Yuan,1,∗Kjetil M. D. Hals,2,3Yi Liu,1Anton A. Starikov,1Arne Brataas,2and Paul J. Kelly1 1Faculty of Science and Technology and MESA+Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands 2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 3Niels Bohr International Academy and the Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark I. COMPUTATIONAL DETAILS. Taking the concrete example of Walker profile domain walls (DWs), the effective (dimensionless) in-plane and out-of-plane damping parameters can be expressed in terms of the scattering matrix Sof the system as, re- spectively, αeff i=gµBλw 8πAMsTr/parenleftbigg∂S ∂rw∂S† ∂rw/parenrightbigg , (S1) αeff o=gµB 8πAMsλwTr/parenleftbigg∂S ∂φ∂S† ∂φ/parenrightbigg ,(S2) using the scattering theory of magnetization dissipation [S1,S2]. Heregis the Land´ e g-factor,µBis the Bohr magneton, λwdenotes the DW width, Ais the cross sec- tional area, and Msis the saturation magnetization. It is interesting to compare the scheme for calculat- ing the Gilbert damping of DWs using Eqs. ( S1) and (S2) [S1,S2] with that used for collinear magnetiza- tion [S3,S4]. Both of them are based upon the energy pumping theory [ S2,S3]. To calculate the damping αcoll for the collinear case, the magnetization is made to pre- cessuniformlyandthelocalenergydissipationishomoge- neous throughout the ferromagnet. The total energy loss due to Gilbert damping is then proportional to the vol- ume of the ferromagnetic material and the homogenous local damping αcollcan be determined from the damp- ing per unit volume. When the magnetization of a DW is made to change either by moving its center rwor varying its orientation φ, this results in a relatively large preces- sion at the center of the DW; the further from the center, the less the magnetization changes. The local contribu- tion to the total energydissipationofthe DWis weighted by the magnitude of the magnetization precession when rworφvaries. For a fixed DW width, the total damping is not proportional to the volume of the scattering region but converges to a constant once the scattering region is large compared to the DW. In practice, αeff iandαeff ocal- culated using Eqs. ( S1) and (S2) are well converged for a scattering region 10 times longer than λw. Effectively, αeffcan be regardedas a weighted averageof the (dimen- sionless) damping constant in the region of a DW. In the wide DW limit, αeff iandαeff oboth approach αcollwith spin-orbit coupling (SOC) and vanish in its absence. To evaluate the effective Gilbert damping of a DWusing Eqs. ( S1) and (S2), we attached semiinfinite (cop- per) leads to a finite length of Ni 80Fe20alloy (Permal- loy, Py) and rotated the local magnetization to make a 180◦DW using the Walker profile. Specifically, we usedm= (sechz−rw λw,0,tanhz−rw λw) for N´ eel DWs and m= (−tanhz−rw λw,−sechz−rw λw,0) for Bloch DWs. The scatteringpropertiesofthedisorderedregionwereprobed by studying how Bloch waves in the Cu leads incident from the left or right sides weretransmitted and reflected [S4,S5]. Thescatteringmatrixwasobtainedusingafirst- principles “wave-function matching” scheme [ S6] imple- mented with tight-binding linearized muffin-tin orbitals (TB-LMTOs) [ S7]. SOC was included using a Pauli Hamiltonian. The calculations were rendered tractable by imposing periodic boundary conditions transverse to the transport direction. It turned out that good results could be achieved even when these so-called “lateral su- percells” were quite modest in size. In practice, we used 5×5 lateral supercells and the longest DW we consid- ered was more than 500 atomic monolayers thick. After embedding the DW between collinear Py and Cu leads, the largest scattering region contained 13300 atoms. For every DW width, we averaged over about 8 random dis- order configurations. A potential profile for the scattering region was con- structed within the framework of the local spin den- sity approximation of density functional theory as fol- lows. For a slab of collinear Py binary alloy sandwiched between Cu leads, atomic-sphere-approximation (ASA) potentials [ S7] were calculated self-consistently without SOC using a surface Green’s function (SGF) method im- plemented [ S8] with TB-LMTOs. Chargeand spin densi- ties for binary alloy AandBsites were calculated using the coherent potential approximation [ S9] generalized to layer structures [ S8]. For the scattering matrix calcu- lation, the resulting ASA potentials were assigned ran- domly to sites in the lateral supercells subject to mainte- nance of the appropriate concentration of the alloy [ S6] and SOC was included. The exchange potentials are ro- tated in spin space [ S10] so that the local quantization axis for each atomic sphere follows the DW profile. The DW width is determined in reality by a competition be- tween interatomic exchange interactions and magnetic anisotropy. For a nanowire composed of a soft mag- netic material like Py, the latter is dominated by the2 shape anisotropy that arises from long range magnetic dipole-dipole interactions and depends on the nanowire profile. Experimentallyitcanbetailoredbychangingthe nanowire dimensions leading to the considerable spread of reported DW widths [ S11]. In electronic structure cal- culations, that do not contain magnetic dipole-dipole in- teractions, we simulate a change of demagnetization en- ergy by varying the DW width. In this way we can study the dependence of Gilbert damping on the magnetization gradient by performing a series of calculations for DWs with different widths. 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 120 ×120 grid. The transport calculations includ- ing SOC were performed with a 32 ×32 2D BZ grid for a 5×5 lateral supercell, which is equivalent to a 160 ×160 grid in the 1 ×1 2D BZ. II. EXTRACTING α′′ We first briefly derive the form of the in-plane damp- ing. It has been argued phenomenologically [ S12] that for a noncollinear magnetization texture varying slowly in time the lowest order term in an expansion of the transverse component of the spin current in spatial and time derivatives that breaks time-reversal symmetry and is therefore dissipative is j′′ s=−ηm×∂z∂tm, (S3) whereηis a coefficient depending on the material and mis a unit vector in the direction of the magnetization. The divergence of the spin current, ∂zj′′ s=−η/parenleftbig ∂zm×∂z∂tm+m×∂2 z∂tm/parenrightbig ,(S4) gives the corresponding dissipative torque exerted on the local magnetization. While the second term in brackets in Eq. (S4) is perpendicular to m, the first term contains both perpendicular and parallel components. Since we are only interested in the transverse component of the torque, we subtract the parallel component to find the damping torque τ′′=−η/braceleftbig (1−mm)·(∂zm×∂z∂tm)+m×∂2 z∂tm/bracerightbig =−η/braceleftbig [m×(∂zm×∂z∂tm)]×m+m×∂2 z∂tm/bracerightbig =η/bracketleftbig (m·∂z∂tm)m×∂zm−m×∂2 z∂tm/bracketrightbig .(S5) The Landau-Lifshitz-Gilbert equation including the damping torque τ′′reads ∂tm=−γm×Heff+αcollm×∂tm+γτ′′ Ms =−γm×Heff+αcollm×∂tm +α′′/bracketleftbig (m·∂z∂tm)m×∂zm−m×∂2 z∂tm/bracketrightbig ,(S6)where the in-plane damping parameter α′′≡γη/Mshas the dimension of length squared. In the following, we explain how α′′can be extracted from calculations on Walker DWs and show that it is ap- plicable to other profiles. The formulation is essentially independent of the DW type (Bloch or N´ eel) and we use a Bloch DW in the following derivation for which m(z) = [cosθ(z),sinθ(z),0], (S7) whereθ(z) represents the in-plane rotation (see Fig. 1 in the paper). The local energy dissipation associated with a time-dependent θis given by [ S2] γ Ms˙E(z) =αcoll∂tm·∂tm +α′′/bracketleftbig (m·∂z∂tm)∂tm·∂zm−∂tm·∂2 z∂tm/bracketrightbig .(S8) For the one-dimensional profile Eq. ( S7), this can be sim- plified as γ Ms˙E(z) =αcoll/parenleftbiggdθ dt/parenrightbigg2 −α′′dθ dtd dt/parenleftbiggd2θ dz2/parenrightbigg .(S9) Substituting into Eq. ( S9) the Walker profile θ(z) =−π 2−arcsin/parenleftbigg tanhz−rw λw/parenrightbigg ,(S10) that we used in the calculations, we obtain for the total energy dissipation associated with the motion of a rigid DW for which ˙θ= ˙rwdθ/drw, ˙E=/integraldisplay d3r˙E(z) =2MsA γλw/parenleftbigg αcoll+α′′ 3λ2w/parenrightbigg ˙r2 w.(S11) Comparing this to the energy dissipation expressed in terms of the effective in-plane damping αeff i[S2] ˙E=2MsA γλwαeff i˙r2 w, (S12) we arrive at αeff i(λw) =αcoll+α′′ 3λ2w. (S13) Using Eq. ( S13), we perform a least squares linear fitting ofαeff ias a function of λ−2 wto obtain αcollandα′′. The fitting is shown in Fig. S1and the parameters are listed in Table SI. Note that αcollis in perfect agreement with independent calculations for collinear Py [ S4]. To confirm that α′′is independent of texture, we con- sider another analytical DW profile in which the in-plane rotation is described by a Fermi-like function, θ(z) =−π+π 1+ez−rF λF. (S14) HererFandλFdenote the DW center and width, re- spectively. Substituting Eq. ( S14) into Eq. ( S9), we find the energy dissipation for “Fermi” DWs to be ˙E=π2MsA 6γλF/parenleftbigg αcoll+α′′ 5λ2 F/parenrightbigg ˙r2 F,(S15)3 0 0.5 1.0 1.5 1/λw2 (nm-2)00.0050.0100.015αieff Bloch Néel ξSO=0Walker FIG. S1. Calculated αeff ifor Walker-profile Permalloy DWs. N´ eel DWs: black circles, Bloch DWs: red circles. Without SOC, calculations for the twoDWtypesyield thesame results (blue circles). The dashed lines are linear fits using Eq. ( S13). which suggests the effective in-plane damping αeff i(λF) =αcoll+α′′ 5λ2 F. (S16) Eq. (S16) is plotted as solid lines in Fig. S2with the values of αcollandα′′taken from Table SI. Since the energy pumping can be expressed in terms of the scattering matrix Sas ˙E=/planckover2pi1 4πTr/parenleftbigg∂S ∂t∂S† ∂t/parenrightbigg =/planckover2pi1 4πTr/parenleftbigg∂S ∂rF∂S† ∂rF/parenrightbigg ˙r2 F,(S17) we can calculate the effective in-plane damping for a Fermi DW from the Smatrix to be αeff i=3/planckover2pi1γλF 2π3MsATr/parenleftbigg∂S ∂rF∂S† ∂rF/parenrightbigg .(S18) We plot the values of αeff icalculated using the derivative of the scattering matrix Eq. ( S18) as circles in Fig. S2. The good agreement between the circles and the solid lines demonstratesthe validity ofthe form ofthe in-plane damping torque in Eq. ( S6) and that the parameter α′′ does not depend on a specific magnetization texture. TABLE SI. Fit parameters to describe the in-plane Gilbert damping in Permalloy DWs. DW type αcoll α′′(nm2) Bloch (4.6 ±0.1)×10−30.016±0.001 N´ eel (4.5 ±0.1)×10−30.016±0.001 ξSO=0 (2.0 ±1.0)×10−60.017±0.0010 0.5 1.0 1.5 2.0 2.5 3.0 1/λF2 (nm-2)00.0050.0100.015αieff Bloch ξSO=0Fermi FIG. S2. Calculated αeff ifor Permalloy Bloch DWs (red cir- cles) with the Fermi profile Eq. ( S14). The blue circles are results calculated without SOC. The solid lines are the an- alytical expression Eq. ( S16) using the parameters listed in TableSI. III. THE FREE-ELECTRON MODEL USING MUFFIN-TIN ORBITALS We take constant potentials, V↑=−0.2 Ry,V↓= −0.1Ry inside atomic sphereswith an exchangesplitting ∆V= 0.1 Ry between majority and minority spins and a Fermi level EF= 0. The atomic spheres are placed on a face-centered cubic (fcc) lattice with the lattice constant of nickel, 3.52 ˚A. The magnetic moment on each atom is then 0.072µB. The transport direction is along the fcc [111]. In the scattering calculation, we use a 300 ×300 01020 30 4050 60 L (nm)306090102030AR (fΩ m2)456(a) (b) (c)ρ=2.69±0.06 µΩ cm ρ=24.8±0.5 µΩ cm ρ=94.3±4.4 µΩ cm FIG.S3. Resistancecalculatedforthedisorderedfree-ele ctron model as a function of the length of the scattering region for three values of V0, the disorder strength: 0.05 Ry (a), 0.15 Ry (b) and 0.25 Ry (c). The lines are the linear fitting used to determine the resistivity.4 k-point mesh in the 2D BZ. The calculated Sharvin con- ductances for majority and minority channels are 0.306 and 0.153 e2/hper unit cell, respectively, compared with analytical values of 0.305 and 0.153. To mimic disordered free-electron systems, we intro- duce a 5 ×5 lateral supercell and distribute constant potentials uniformly in the energy range [ −V0/2,V0/2] whereV0is some given strength [ S13] and spatially at random on every atomic sphere in the scattering re-gion. The calculated total resistance as a function of the lengthLof the (disordered) scattering region is shown in Fig.S3withV0= 0.05 Ry (a), 0.15 Ry (b) and 0.25 Ry (c). The resistivity increases with the impurity strength as expected and can be extracted with a linear fitting AR(L) =AR0+ρL. For each system, we calculate about 10randomconfigurationsand takethe averageofthe cal- culatedresults. Wellconvergedresultsareobtainedusing a 32×32k-point mesh for the 5 ×5 supercell. IV. FITTING α′ANDl0 With a nonlocal Gilbert damping, α(r,r′), the energy dissipation rate is given by [ S2] ˙E=Ms γ/integraldisplay d3r˙m(r)·/integraldisplay d3r′α(r,r′)·˙m(r′). (S19) If we consider the out-of-plane damping of a N´ eel DW, i.e. for which the angle φvaries in time (see Fig. 1 in the paper), we have ˙m(r) =˙φsechz−rw λwˆy. (S20) Considering again a Walker profile, we find the explicit form of the out- of-plane damping matrix element αo(z,z′) =αcollδ(z−z′)+α′ λ2wsechz−rw λwsechz′−rw λw1√πAl0e−(z−z′ l0)2. (S21) Substituting Eq. ( S21) and Eq. ( S20) into Eq. ( S19), we obtain explicitly the energy dissipation rate ˙E=2MsAλw γαcoll˙φ2+MsAα′˙φ2 √πγl0λ2w/integraldisplay dzsech2z−rw λw/integraldisplay dz′sech2z′−rw λwe−(z−z′ l0)2 . (S22) The calculated effective out-of-plane Gilbert damping for a DW with th e Walker profile is related to the energy dissipation rate as [ S2] ˙E=2MsAλw γαeff o˙φ2. (S23) Comparing Eqs. ( S22) and (S23), we arrive at αeff o=αcoll+α′ 2√πλ3wl0/integraldisplay dzsech2z−rw λw/integraldisplay dz′sech2z′−rw λwe−(z−z′ l0)2 . (S24) The last equation is used to fit α′andl0toαeff ocalculated for different λw. For Bloch DWs, it is straightforward to repeat the above derivation and find the same result, Eq. ( S24). V. BAND STRUCTURES OF NI AND FE IN PERMALLOY In the coherent potential approximation (CPA) [ S8, S9], the single-site approximation involves calculating auxiliary (spin-dependent) potentials for Ni and Fe self- consistently. In our transport calculations, these auxil- iary potentials are distributed randomly in the scattering region. It is instructive to place the Ni potentials (for majority- and minority-spin electrons) on an fcc latticeand to calculate the band structure non-self-consistently. Then we do the same using the Fe potentials. The cor- responding band structures are plotted in Fig. S4. At the Fermi level, where electron transport takes place, the majority-spin bands for Ni and Fe are almost identi- cal, including their angular momentum character. This means that majority-spin electrons in a disordered al- loy see essentially the same potentials on all lattice sites and are only very weakly scattered in transport by the randomly distributed Ni and Fe potentials. In contrast,5 -9-6-303E-EF (eV)Majority Spin Minority Spin X Γ L-9-6-303E-EF (eV) X Γ LNi Ni Fe Fe FIG. S4. Band structures calculated with the auxiliary Ni and Fe atomic sphere potentials and Fermi energy that were calculated self-consistently forNi 80Fe20usingthecoherentpo- tential approximation. The red bars indicates the amount of scharacter in each band. the minority-spin bands are quite different for Ni and Fe. Thiscanbeunderstoodintermsofthe differentexchange splitting between majority- and minority-spin bands; the calculated magnetic moments of Ni and Fe in Permalloy in the CPA are 0.63 and 2.61 µB, respectively. The ran- dom distribution of Ni and Fe potentials in Permalloy then leads to strong scattering of minority-spin electrons in transport.∗Present address: Institut f¨ ur Physik, Johannes Gutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128 Mainz, Germany; zyuan@uni-mainz.de [S1] K. M. D. Hals, A. K. Nguyen, and A. Brataas, Phys. Rev. Lett. 102, 256601 (2009) . [S2] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. B 84, 054416 (2011) . [S3] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 101, 037207 (2008) . [S4] A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010) . [S5] Z. Yuan, Y. Liu, A. A. Starikov, P. J. Kelly, and A. Brataas, Phys. Rev. Lett. 109, 267201 (2012) . [S6] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W. Bauer, Phys. Rev. B 73, 064420 (2006) . [S7] O. K. Andersen, Z. Pawlowska, and O. Jepsen, Phys. Rev. B 34, 5253 (1986) . [S8] I. Turek, V. Drchal, J. Kudrnovsk´ y, M. ˇSob, and P. Weinberger, Electronic Structure of Disordered Al- loys, Surfaces and Interfaces (Kluwer, Boston-London- Dordrecht, 1997). [S9] P. Soven, Phys. Rev. 156, 809 (1967) . [S10] S. Wang, Y. Xu, and K. Xia, Phys. Rev. B 77, 184430 (2008) . [S11] O. Boulle, G. Malinowski, and M. Kl¨ aui, Mat. Science and Eng. R 72, 159 (2011) . [S12] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale, Phys. Rev. B 79, 094415 (2009) . [S13] A. K. Nguyen and A. Brataas, Phys. Rev. Lett. 101, 016801 (2008) .

2403.17732v1.On_a_class_of_nonautonomous_quasilinear_systems_with_general_time_gradually_degenerate_damping.pdf

arXiv:2403.17732v1 [math.AP] 26 Mar 2024On a class of nonautonomous quasilinear systems with general time-gradually-degenerate damping Richard De la cruz∗and Wladimir Neves† March 28, 2024 Abstract Inthispaper, westudytwosystemswithatime-variable coeffi cientandgeneral time-gradually-degenerate damping. More explicitly, we construct the Riemann solutio ns to the time-variable coefficient Zeldovich approximation and time-variable coefficient pressureless g as systems both with general time-gradually- degenerate damping. Applying the method of similar variabl es and nonlinear viscosity, we obtain classical Riemann solutions and delta shock wave solutions. Keywords: Pressurelessgasdynamicssystem, Zeldovichtypeapproximatesy stem, time-gradually-degenerate damping, Riemann problem, delta shock solution. 1 Introduction One can find many problems from Continuum Physics that are mathem atically modeled by balance laws, that is to say, systems of partial differential equations in the following div ergence form ∂u ∂t+d/summationdisplay j=1∂Fj(u) ∂xj=G(u), (1) where (t,x)∈Rd+1 +≡(0,∞)×Rdis the set of independent variables, u∈Rndenotes the unknown vector field, Fj∈Rnis called the flux function and G∈Rnis the vector production, absorption, or damping term. The first component t >0 is the time variable and x∈Rdis the space variable. Moreover, when G≡0 equation (1) is called a system of conservation laws. In fact, denoting Aj(·) =DFj(·), that is the Jacobian matrices of the fluxes, the system ( 1) falls in the general class of nonhomogeneous quasilinear first-ord er systems of partial differential equations ∂u ∂t+d/summationdisplay j=1Aj(u)∂u ∂xj=G(u). (2) Albeit, there are important applications that require to consider sy stems where the coefficients Ajand Gin (2) may depend also on the independent variables ( t,x), for instance to take into account material inhomogeneities, or some special geometries, also external action s, etc., see Francesco Oliveri [ 22] and references therein. Therefore, one has to study the general nonautonomo us quasilinear system of partial differential equations ∂ui ∂t+d/summationdisplay j=1Aj i(t,x,u)∂u ∂xj=Gi(t,x,u),(i= 1,...,n). ∗School of Mathematics and Statistics, Universidad Pedag´ o gica y Tecnol´ ogica de Colombia, 150003, Tunja, Colombia. E -mail: richard.delacruz@uptc.edu.co †Instituto de Matem´ atica, Universidade Federal do Rio de Ja neiro, Cidade Universit´ aria 21945-970, Rio de Janeiro, Br azil. E-mail: wladimir@im.ufrj.br 1We are interested in studying these types of systems, more precis ely, a particular class of such systems which is the 2×2 systems, ( n= 2,d= 1), when Ai≡Ai(t,u), and thus the companion function Gi=Gi(t,u), (i= 1,2). Moreover, in this case, we recover in a simple way the divergence form. Indeed, taking especially, Ai(t,u) =αi(t)Ai(u) andGi(t,u) =σi(t)Gi(u), we may write the above system as ∂u1 ∂t+α1(t)∂F1(u1,u2) ∂x=σ1(t)G1(u1,u2), ∂u2 ∂t+α2(t)∂F2(u1,u2) ∂x=σ2(t)G2(u1.u2).(3) Related to system ( 3), let us start our study by considering the following class of nonaut onomous quasilin- ear systems with time-variable coefficients and time-dependent (line ar) damping represented by the following systems: ρt+α(t)(ρu)x= 0, ut+α(t)(u2 2)x=−σ(t)u,(4) and also /braceleftBigg ρt+α(t)(ρu)x= 0, (ρu)t+α(t)(ρu2)x=−σ(t)ρu,(5) where 0 ≤α∈L1([0,∞)), 0≤σ∈L1 loc([0,∞)), the unknown ρcan be interpreted as some density, and uis the velocity vector field which carries the density ρ. Companion to ( 4) and (5) the initial data is given by (ρ(x,0),u(x,0)) = (ρ0(x),u0(x)) =/braceleftBigg (ρ−,u−),ifx <0, (ρ+,u+),ifx >0,(6) for arbitrary constant states u±andρ±>0. Therefore, we are considering in fact the Riemann problem, which is the building block of the Cauchy problem. At this point, we would like to address the reader to [ 21], where it is studied the following generalized Boussinesq system with variable-coefficients, (compare it with the s ystem (4)), ut+α1(t)(u2 2)x+β1(t)ux+γ1(t)ρx= 0, ρt+α2(t)(ρu)x+β2(t)ρx+γ2(t)uxxx= 0, whereαi,βi,γi, (i= 1,2), are time-dependent coefficients relevant to density, dispersio n and viscosity of the fluid. The above system can model the propagation of weakly disper sive and long weakly nonlinear surface waves in shallow water. The authors, under a selection of the spect ral parameters, showed the existence of soliton solutions applying the Darboux transformation and symbolic c omputation. Oneobservesthatthesecondequationofthesystem( 4)istheBurgersequationwithtimevariablecoefficients [8]. In particular, the time variable coefficients can provide more usefu l models in many complicated physical situations [ 8,12,28]. The homogeneous case of the system ( 4), that is to say σ(t) = 0 for all t≥0, is the following time variable coefficient system ρt+α(t)(ρu)t= 0, ut+α(t)(u2 2)x= 0,(7) which can be interpreted as an extension of Zeldovich approximation system [26,30]. In particular, the system (7) withα(·)≡1 is used to model the evolution of density inhomogeneities of matter in the universe [ 24, B. Late nonlinear stage, 3. Sticky dust]. Further, let us recall that, the system ( 4) belongs to the class of triangular systems of conservation laws, that arises in a wide variety of models in physics and engineering, see for example 2[15,23] and references therein. For this reason, the triangular system s have been studied by many authors and several rigorous results have been obtained for them. In [ 4,6], the Riemann problem was solved to the system ( 4) withα(·)≡1 andσ(·) equals to a positive constant, where Delta shocks have to be cons idered. Recently, based on the method of similar variables proposed in [ 6], Li [19] studied the Riemann problem to the system ( 4) withα(·)≡1 andσ(t) =µ 1+twith physical parameter µ >0. In the literature the external term σ(t) =µ (1+t)θuwith physical parameters µ >0 andθ≥0 is called a time-gradually-degenerate damping [11,20], and it represents the time-gradually-vanishing friction effect. On the other hand, the homogeneous case of the system ( 5) is the following time variable coefficient system /braceleftBigg ρt+α(t)(ρu)t= 0, (ρu)t+α(t)(ρu2)x= 0,(8) which can be seen as an extension of pressureless gas dynamics system [26,30]. We recall that gas dynamics with zero pressure is a simplified scenario where the pressure of the gas is assumed to be negligible, accounting for high-speed flows or rarefied gases. The first study for the us ual pressureless gas dynamics system, that is (8) withα(·)≡1, is due to Bouchut [ 1] in 1994. In that paper it was studied the existence of solutions to t he Riemann problem for the pressureless gas dynamics system, introd ucing a notion of measure solution and delta shock waves were obtained. However, uniqueness was not studied . Moreover, the existence of a weak solution to the Cauchy problem w as first obtained independently by E, Rikov, Sinai [ 9] in 1996, and Brenier, Grenier [ 3] in 1998. In particular, the authors in [ 9] show that, the standard entropy condition ( ρΦ(ρ))t+(ρuΦ(ρ))x≤0 in the sense of distributions, where Φ is a convex function, is not enough to express a uniqueness criterion for weak solutions t o the Cauchy problem. Conversely, Wang and Ding [ 27] proved that the pressureless gas dynamics system has a unique w eak solution using the Oleinik entropy condition when the initial data ρ0,u0are both bounded measurable functions. However, the solution for the Cauchy problem for the pressureless gas dynamics system is in general a Radon measure [ 9]. In 2001, Huang and Wang [ 14] studied the Cauchy problem for the system ( 8), when initial data ρ0,u0are respectively a Radon measure and a bounded measurable function. Then, they showed the uniqueness of weak solutions under the Oleinik entropy condition together with an energ y condition in the sense that, ρu2weakly converges to ρ0u2 0ast→0. We recall that, a particular case of Radon measure solution is the delta shock wave solution. A delta shock wave solution is a type of nonclassical wa ve solution in which at least one state variable may develop a Dirac measure. Actually, on physical grounds , delta shock solutions typically display concentration occurrence in a complex system [ 2,18]. On the other hand, it is well known that the solution for the Riemann problem to the pressureless gas dynamics system involv es vacuum and delta shock wave solution and the classical Riemann solutions satisfy the Lax entropy conditio n while delta shock wave solution is unique under an over-compressive entropy condition [ 25,29]. In a similar way, Keita and Bourgault [ 17] solved the Riemann problem for the pressureless system with linear damping, th at is, the system ( 5) withα(·)≡1 and σ(·)≡const., showing vacuum states and delta shock solution and uniqueness un der the Lax entropy condition and over-compressive entropy condition, respectively. Finally, De la cruz and Juajibioy [ 5] obtained delta shock solutions for a generalized pressureless system with linear damping. 1.1 Equivalent ×non-equivaqlent systems Since we considered α1=α2=αin both systems ( 4), (5), one may ask when these systems are equivalent or not. Indeed, we observe first that for smooth solutions an elemen tary manipulation of the second equation of (5) reads ρ(ut+α(t)(1 2u2)x)+u(ρt+α(t)(ρu)x) =−σ(t)ρu. Therefore, due to the first equation of ( 5) and for ρ/ne}ationslash= 0, the above equation reduces to the equation ( 4)2, and thus for smooth solutions the system ( 5) is equivalent to the system ( 4). Albeit, the question remains open when the solutions are non-regular. 3Once placed the above question, we observe that Keita and Bourga ult [17], recently in 2019, studied the Riemann problem for the Zeldovich approximation and pressureless g as dynamics systems with linear damping withσ=const. > 0. Moreprecisely, they analyzedin that paper the Riemann problemt o the followingsystems: ρt+(ρu)x= 0, ut+(u2 2)x=−σu,(9) /braceleftBigg ρt+(ρu)x= 0, (ρu)t+(ρu2)x=−σρu,(10) with initial data given by ( 6), and it was proved that 1.u−< u+. The solution of the Riemann problem ( 9)-(6) and (10)-(6) is given by (ρ,u)(x,t) = (ρ−,u−e−σt), x < u −1−e−σt σ, (0,σx eσt−1), u −1−e−σt σ≤x≤u+1−e−σt σ, (ρ+,u+e−σt), x > u +1−e−σt σ. 2.u−> u+. The solution of the Riemann problem ( 9)-(6) is given by (ρ,u)(x,t) = (ρ−,u−e−σt), x <u−+u+ 2σ(1−e−σt), (w(t)δ(x−u−+u+ 2σ(1−e−σt)),uδ(t)), x=u−+u+ 2σ(1−e−σt), (ρ+,u+e−σt), x >u−+u+ 2σ(1−e−σt),(11) where w(t) =(ρ++ρ−)(u−−u+) 2σ(1−e−σt) and uδ(t) =u−+u+ 2e−σt. However, the solution of the Riemann problem ( 10)-(6) is given by (ρ,u)(x,t) = (ρ−,u−e−σt), x <√ρ+u++√ρ−u−√ρ++√ρ−(1−eσt), (w(t)δ(x−/integraldisplayt 0uδ(s)ds),uδ(t)), x=√ρ+u++√ρ−u−√ρ++√ρ−(1−eσt), (ρ+,u+e−σt), x >√ρ+u++√ρ−u−√ρ++√ρ−(1−eσt), where w(t) =√ρ−ρ+(u−−u+) σ(1−e−σt) and uδ(t) =√ρ+u++√ρ−u−√ρ++√ρ−e−σt. Consequently, Keita, Bourgault showed that the systems ( 9) and (10) are equivalent for smooth and also for two contact-discontinuity solutions, but they differ for delta shoc k solutions. Therefore, it should be expected that a similar scenario of delta shocks are presented here as well, an d the systems ( 4) and (5) are not equivalent for these types of solutions. We remark that the problems here be come much more complicated since σ(·) besides non-constant is just a locally summable function. 42 The Zeldovich Type Approximate System In this section, we study the Riemann problem to the time-variable co efficient Zeldovich’s approximate system and time-variable linear damping, that is to say ( 4)-(6). We extended some ideas from [ 4] to construct the viscous solutions to the system ( 4), see (12) below. After we show that the family of viscous solutions {(ρε,uε)} converges to a solution of the Riemann problem ( 4)-(6). Foru−< u+, classical Riemann solutions are obtained. Whenu−> u+, we show that a delta shock solution is a solution to the Riemann proble m (4)-(6). 2.1 Parabolic regularization Givenε >0, we consider the following parabolic regularization for the system ( 4), ρε t+α(t)(ρεuε)x=εβ(t)ρε xx, uε t+1 2α(t)((uε)2)x+σ(t)uε=εβ(t)uε xx,(12) where conveniently we define β(t) :=α(t)exp(−/integraltextt 0σ(s)ds). We search for ( ρε,uε) be an approximate solution of problem ( 4)-(6), which is defined by the parabolic approximation ( 12) with initial data given by (ρε(x,0),uε(x,0)) = (ρ0(x),u0(x)), (13) where (ρ0,u0) is given by ( 6). Then, the main issue of this section is to solve problem ( 12) with initial data ( 13). To this end, we use the auxiliary function u(x,t) =/hatwideux(x,t)e−/integraltextt 0σ(τ)dτand a version of Hopf-Cole transformation which enable us to obtain an explicit solution of the viscous system ( 12)-(13). The function /hatwideuwill be explained during the proof of the following Proposition 2.1. Under the assumptions on the functions α,β,σ, the explicit solution of the problem (12)-(13) is given by ρε(x,t) =∂xWε(x,t)anduε(x,t) =u+bε +(x,t)+u−bε −(x,t) bε +(x,t)+bε −(x,t)exp(−/integraldisplayt 0σ(s)ds), where Wε(x,t) =ρ−/parenleftBig x−u−/integraltextt 0β(s)ds/parenrightBig bε −(x,t)+ρ+/parenleftBig x−u+/integraltextt 0β(s)ds/parenrightBig bε +(x,t) bε −(x,t)+bε +(x,t) +(ρ+−ρ−)(ε/integraltextt 0β(s)ds)1/2exp/parenleftBig −x2 4ε/integraltextt 0β(s)ds/parenrightBig π1/2(bε −(x,t)+bε +(x,t)) and bε ±(x,t) :=±1 (4πε/integraltextt 0β(s)ds)1/2/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy. Proof.1. Firstly we observe that, if ( /hatwideρ,/hatwideu) solves /braceleftBigg /hatwideρt+α(t)e−/integraltextt 0σ(τ)dτ/hatwideρx/hatwideux=εβ(t)/hatwideρxx, /hatwideut+1 2α(t)e−/integraltextt 0σ(τ)dτ(/hatwideux)2=εβ(t)/hatwideuxx,(14) with the initial condition given by (ρ(x,0),/hatwideu(x,0)) =/braceleftBigg (ρ−x,u−x),ifx <0, (ρ+x,u+x),ifx >0, 5then (ρε,uε) defined by ( /hatwideρε x,/hatwideuxe−/integraltextt 0σ(τ)dτ) solves the problem ( 12)-(13). Indeed, let us recall the generalized Hopf-Cole transformation, see [ 13,4,16], that is /braceleftBigg /hatwideρε=Cεe/hatwideu 2ε, /hatwideuε=−2εln(Sε).(15) Then, from system ( 14) and the generalized Hopf-Cole transformation ( 15), we have /braceleftBigg Cε t=εβ(t)Cε xx, Sε t=εβ(t)Sε xx,(16) with initial data given by (Cε(x,0),Sε(x,0)) =/braceleftBigg (ρ−xe−u−x 2ε,e−u−x 2ε),ifx <0, (ρ+xe−u+x 2ε,e−u+x 2ε),ifx >0.(17) 2. Now, the solution to the problem ( 16)-(17) in terms of the heat kernel is /braceleftBigg Cε(x,t) =aε −(x,t)+aε +(x,t), Sε(x,t) =bε −(x,t)+bε +(x,t),(18) where aε ±(x,t) :=±ρ± (4πε/integraltextt 0β(s)ds)1/2/integraldisplay±∞ 0yexp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy and bε ±(x,t) :=±1 (4πε/integraltextt 0β(s)ds)1/2/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy. Moreover, we have /integraldisplay±∞ 0∂y/parenleftBigg exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds/parenrightBigg/parenrightBigg exp/parenleftBig −u±y 2ε/parenrightBig dy=−exp/parenleftBigg −x2 4ε/integraltextt 0β(s)ds/parenrightBigg +u± 2ε/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy.(19) On the other hand, it follows that /integraldisplay±∞ 0∂y/parenleftBigg exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds/parenrightBigg/parenrightBigg exp/parenleftBig −u±y 2ε/parenrightBig dy=/integraldisplay±∞ 0(x−y) 2ε/integraltextt 0β(s)dsexp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy =x 2ε/integraltextt 0β(s)ds/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy −/integraldisplay±∞ 0y 2ε/integraltextt 0β(s)dsexp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy. (20) Therefore, from ( 19) and (20) we obtain /integraldisplay±∞ 0yexp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy=2ε/integraldisplayt 0β(s)ds·exp/parenleftBigg −x2 4ε/integraltextt 0β(s)ds/parenrightBigg +/parenleftbigg x−u±/integraldisplayt 0β(s)ds/parenrightbigg/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy.(21) 63. Finally, we observe that ∂x/parenleftBigg/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy/parenrightBigg =−/integraldisplay±∞ 0∂y/parenleftBigg exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds/parenrightBigg/parenrightBigg exp/parenleftBig −u±y 2ε/parenrightBig dy and from ( 19) we have ∂x/parenleftBigg/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy/parenrightBigg = exp/parenleftBigg −x2 4ε/integraltextt 0β(s)ds/parenrightBigg −u± 2ε/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy.(22) Therefore, we may write from ( 18) and (21) that Cε(x,t) =ρ−/bracketleftBigg −(ε/integraltextt 0β(s)ds)1/2 π1/2exp/parenleftBigg −x2 4ε/integraltextt 0β(s)ds/parenrightBigg +/parenleftbigg x−u−/integraldisplayt 0β(s)ds/parenrightbigg bε −(x,t;1)/bracketrightBigg +ρ+/bracketleftBigg (ε/integraltextt 0β(s)ds)1/2 π1/2exp/parenleftBigg −x2 4ε/integraltextt 0β(s)ds/parenrightBigg +/parenleftbigg x−u+/integraldisplayt 0β(s)ds/parenrightbigg bε +(x,t;1)/bracketrightBigg . Moreover, from ( 18) and (22) we have Sε x(x,t) =−1 2ε(u−bε −(x,t)+u+bε +(x,t)). Applying the generalized Hopf-Cole transformation ( 15), it follows that ρε(x,t) =/hatwideρε x(x,t) = (Cε(x,t)/Sε(x,t))x, uε(x,t) =−2εSε x Sεexp(−/integraldisplayt 0σ(s)ds), and hence the proof is complete. Remark 1. One observes that, the solution ( ρε,uε) of the problem ( 12)-(13) is absolutely continuous with respect to time t >0, and smooth in x∈R. 2.2 The Riemann problem In this section, we study the Riemann problem to the system ( 4) withσ(t)≥0 for allt≥0, which means that the damping can degenerate in some open interval contained in (0 ,∞). To obtain the Riemann solution to the problem ( 4) with initial data ( 6) we use the viscosity system with time-dependent damping ( 12) with initial data ( 13) and analyze the limit behavior as ε→0+of the solutions (ρε,uε) obtained in the previous section. To follow, we write bε ±(x,t) as bε ±(x,t) =±1 (4πε/integraltextt 0β(s)ds)1/2/integraldisplay±∞ 0exp/parenleftBigg −(x−y)2 4ε/integraltextt 0β(s)ds−u±y 2ε/parenrightBigg dy =±1 (πBε(t))1/2exp/parenleftbigg−x2+(x−x±(t))2 Bε(t)/parenrightbigg/integraldisplay±∞ 0exp/parenleftbigg −(y+x±(t)−x)2 Bε(t)/parenrightbigg dy =1 π1/2exp/parenleftbigg−x2+(x−x±(t))2 Bε(t)/parenrightbigg/integraldisplay∞ ±(Bε(t))1/2(x±(t)−x)exp(−y2)dy =1 π1/2exp/parenleftbigg−x2+(x−x±(t))2 Bε(t)/parenrightbigg Iε,t ±, 7wherex±(t) =u±/integraltextt 0β(s)ds,Bε(t) = 4ε/integraltextt 0β(s)ds, and Iε,t ±=/integraldisplay∞ ±(Bε(t))1/2(x±(t)−x)exp(−y2)dy. Asε→0+, due to the asymptotic expansion of the (complementary) error f unction (see [ 10]), we have Iε,t ±= ∞/summationdisplay n=0(−1)n(2n)! n!/parenleftbigg(Bε(t))1/2 ±2(x±(t)−x)/parenrightbigg2n+1 exp/parenleftbigg −(x±(t)−x)2 Bε(t)/parenrightbigg ,if±(x±(t)−x)>(Bε(t))1/2, 1 2π1/2,ifx±(t) =x, π1/2−∞/summationdisplay n=0(−1)n(2n)! n!/parenleftbigg(Bε(t))1/2 ∓2(x±(t)−x)/parenrightbigg2n+1 exp/parenleftbigg −(x±(t)−x)2 Bε(t)/parenrightbigg ,if±(x±(t)−x)<−(Bε(t))1/2, and therefore we obtain bε ±(x,t) = ±Q± π1/2exp/parenleftBigx2 Bε(t)/parenrightBig ,if±(x±(t)−x)>(Bε(t))1/2, 1 2exp/parenleftBig −x2 Bε(t)/parenrightBig ,ifx±(t) =x, exp/parenleftBig−x2+(x±(t)−x)2 Bε(t)/parenrightBig ±Q± π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig ,if±(x±(t)−x)<−(Bε(t))1/2,(23) where Q±=∞/summationdisplay n=0(−1)n(2n)! n!/parenleftBig(Bε(t))1/2 2(x±(t)−x)/parenrightBig2n+1 =ε1/2/parenleftBigg/parenleftbig/integraltextt 0β(s)ds/parenrightbig1/2 x±(t)−x−2ε/parenleftBig/parenleftbig/integraltextt 0β(s)ds/parenrightbig1/2 x±(t)−x/parenrightBig3 +12ε2/parenleftBig/parenleftbig/integraltextt 0β(s)ds/parenrightbig1/2 x±(t)−x/parenrightBig5 −···/parenrightBigg . 2.2.1 Classical Riemann solutions: u−≤u+. In this case, we have the following Theorem 2.1. Suppose that u−≤u+. Let(ρε,uε)be the solution of the viscosity problem (12)-(13). Then, the limit lim ε→0+(ρε(x,t),uε(x,t)) = (ρ(x,t),u(x,t)) exists in the sense of distributions, and the pair (ρ(x,t),u(x,t))solves the time-variable coefficient Zeldovich approximate system and time-dependent damping (4)with initial data (6). In addition, if u−< u+, then (ρ(x,t),u(x,t)) = (ρ−,u−exp(−/integraltextt 0σ(s)ds)),ifx < x−(t), (0,x/integraltextt 0β(s)dsexp(−/integraltextt 0σ(s)ds)),ifx−(t)< x < x +(t), (ρ+,u+exp(−/integraltextt 0σ(s)ds)),ifx > x+(t), and when u−=u+, then (ρ(x,t),u(x,t)) =/braceleftBigg (ρ−,u−exp(−/integraltextt 0σ(s)ds)),ifx < x−(t), (ρ+,u−exp(−/integraltextt 0σ(s)ds)),ifx > x−(t). 8Proof.1. First, let us consider the case x−x−(t)<−(Bε(t))1/2. Forε >0 sufficiently small, due to approxi- mations given by ( 23), we may write Wε(x,t)≈ρ−(x−x−(t))cε −−ρ+(Bε(t))1/2 2π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig +(ρ+−ρ−)(Bε(t))1/2 2π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig cε −+(Bε(t))1/2 2π1/2(x+(t)−x), wherecε −= exp/parenleftBig −x2+(x−(t)−x)2 Bε(t)/parenrightBig −(Bε(t))1/2 2π1/2(x−(t)−x)exp/parenleftBig −x2 Bε(t)/parenrightBig . Therefore, we obtain Wε(x,t)≈ρ−(x−x−(t))exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBig (24) and uε(x,t)≈(Bε(t))1/2 2π1/2/parenleftBig u+ x+(t)−x−u− x−(t)−x/parenrightBig +u−exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBigexp(−/integraldisplayt 0σ(s)ds). (25) 2. Similarly, if x−(t)+(Bε(t))1/2< x < x +(t)−(Bε(t))1/2, then we approximate Wεas Wε(x,t)≈ρ−(x−x−(t))/hatwidecε −+ρ+(x−x+(t))/hatwidecε ++(ρ+−ρ−)(Bε(t))1/2 2π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig /hatwidecε −+/hatwidecε + where/hatwidecε ±=±1 π1/2/parenleftBig (Bε(t))1/2 2(x±(t)−x)−(Bε(t))3/2 4(x±(t)−x)3/parenrightBig exp/parenleftBig −x2 Bε(t)/parenrightBig . Therefore, Wε(x,t)≈Bε(t)/parenleftBig ρ+ (x+(t)−x)2−ρ− (x−(t)−x)2/parenrightBig 2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBig +Bε(t)/parenleftBig 1 (x−(t)−x)3−1 (x+(t)−x)3/parenrightBig (26) and uε(x,t) =u+ x+(t)−x+u− x−x−(t)+∞/summationtext n=1(−1)n(2n)!(Bε(t))n n!4n/parenleftBig u+ (x+(t)−x)2n+1+u− (x−x−(t))2n+1/parenrightBig 1 x+(t)−x+1 x−x−(t)+∞/summationtext n=1(−1)n(2n)!(Bε(t))n n!4n/parenleftBig 1 (x+(t)−x)2n+1+1 (x−x−(t))2n+1/parenrightBigexp(−/integraldisplayt 0σ(s)ds).(27) Moreover, if x+(t)−x <−(Bε(t))1/2, then we have Wε(x,t)≈ρ−(Bε(t))1/2 2π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig +ρ+(x−x+(t))cε ++(ρ+−ρ−)(Bε(t))1/2 2π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig (Bε(t))1/2 2π1/2(x−x−(t))exp/parenleftBig −x2 Bε(t)/parenrightBig +cε + wherecε += exp/parenleftBig −x2+(x+(t)−x)2 Bε(t)/parenrightBig −(Bε(t))2 2π1/2(x−x+(t))exp/parenleftBig −x2 Bε(t)/parenrightBig , and therefore we get Wε(x,t)≈ρ+(x−x+(t))exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBig (28) and uε(x,t)≈u+exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig u+ x+(t)−x−u− x−(t)−x/parenrightBig exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBigexp(−/integraldisplayt 0σ(s)ds). (29) 93. Now, for the case u−< u+, from (24), (26), and (28) we have lim ε→0+Wε(x,t) =W(x,t) = ρ−(x−x−(t)),ifx < x−(t), 0, ifx−(t)< x < x +(t), ρ+(x−x+(t)),ifx > x+(t) and from ( 25), (27), and (29) we have lim ε→0+uε(x,t) =u(x,t) = u−exp(−/integraltextt 0σ(s)ds),ifx < x−(t), x/integraltextt 0β(s)dsexp(−/integraltextt 0σ(s)ds),ifx−(t)< x < x +(t), u+exp(−/integraltextt 0σ(s)ds),ifx > x+(t). Sinceuε(x,t) is bounded on compact subsets of R2 +={(x,t) :x∈R,t >0}anduε(x,t)→u(x,t) pointwise as ε→0+, then uε(x,t)→u(x,t) in the sense of distribution. Also, Wε(x,t) is bounded on compact subsets of R2 +andWε(x,t)→W(x,t) pointwise as ε→0+, then Wε(x,t)→W(x,t) in the sense of distributions and so Wε x(x,t) converges in the distributional sense to Wx(x,t). From Proposition 2.1, we have that lim ε→0+ρε(x,t) = ρ(x,t) exists in the sense of distribution and ρ(x,t) =Wx(x,t) = ρ−,ifx < x−(t), 0,ifx−(t)< x < x +(t), ρ+,ifx > x+(t). For the case u−=u+, we have lim ε→0+(ρε(x,t),uε(x,t)) = (ρ(x,t),u(x,t)) =/braceleftBigg (ρ−,u−exp(−/integraltextt 0σ(s)ds)),ifx < x−(t), (ρ+,u−exp(−/integraltextt 0σ(s)ds)),ifx > x−(t). Finally, it is not difficult to show that ( ρ(x,t),u(x,t)) solves ( 4), and thus we omit the details. 2.2.2 Delta shock wave solutions: u−> u+. In this section, we study the Riemann problem to the system ( 4) with initial data ( 6) whenu−> u+. Let us recall that, in particular when α(·)≡1 andσ(·)≡σ=const. > 0, the solution is not bounded and contains a weighted delta measure supported on a smooth curve (see [ 17]), which is a delta shock solution given by ( 11). Here, we have a more general context with similar results. Therefo re, we first define the meaning of a two-dimensional weighted delta function. Definition 2.1. Givenw∈L1((a,b)), with−∞< a < b < ∞, and a smooth curve L≡ {(x(s),t(s)) :a < s < b }, we say that w(·)δLis a two-dimensional weighted delta function supported on L, when for each test function ϕ∈C∞ 0(R×[0,∞)), /an}bracketle{tw(·)δL,ϕ(·,·)/an}bracketri}ht=/integraldisplayb aw(s)ϕ(x(s),t(s))ds. Now, the following definition tells us when a pair ( ρ,u) is a delta shock wave solution to the Riemann problem ( 4)-(6). Definition 2.2. A distribution pair ( ρ,u) is called a delta shock wave solution of the problem ( 4) and (6) in the sense of distributions, when there exists a smooth curve Land a function w(·), such that ρanduare represented respectively by ρ=/hatwideρ(x,t)+wδL, u=u(x,t) 10with/hatwideρ,u∈L∞(R×(0,∞)), and satisfy for each the test function ϕ∈C∞ 0(R×(0,∞)), < ρ,ϕ t>+< αρu,ϕ x>= 0, /integraldisplay/integraldisplay R2 +/parenleftBig uϕt+α(t) 2u2ϕx−σ(t)uϕ/parenrightBig dxdt= 0, where < ρ,ϕ >=/integraldisplay/integraldisplay R2 +/hatwideρϕdxdt+/an}bracketle{twδL,ϕ/an}bracketri}ht, and < αρu,ϕ > =/integraldisplay/integraldisplay R2 +α(t)/hatwideρuϕdxdt +/an}bracketle{tα(·)wuδδL,ϕ/an}bracketri}ht. Moreover, u|L=uδ(·). Placed the previous definitions, we are going to show a solution with a d iscontinuity on x=x(t) for the system (4) of the form (ρ(x,t),u(x,t)) = (ρ−(x,t),u−(x,t)),ifx < γ(t), (w(t)δL,uδ(t)),ifx=γ(t), (ρ+(x,t),u+(x,t)),ifx > γ(t), whereρ±(x,t),u±(x,t) are piecewise smooth solutions of system ( 4),δLis the Dirac measure supported on the curveγ∈C1, andγ,w, anduδare to be determined. Then, we have the following Theorem 2.2. Suppose u−> u+. Let(ρε,uε)be the solution of the problem (12)-(13). Then the limit lim ε→0+(ρε(x,t),uε(x,t)) = (ρ(x,t),u(x,t)) exists in the sense of distributions and (ρ(x,t),u(x,t))solves the problem (4)-(6). In addition, (ρ(x,t),u(x,t)) = (ρ−,u−exp(−/integraldisplayt 0σ(s)ds)),ifx < x(t), (w(t)δ(x−x(t)),u−+u+ 2exp(−/integraldisplayt 0σ(s)ds)),ifx=x(t), (ρ+,u+exp(−/integraldisplayt 0σ(s)ds)),ifx > x(t), where w(t) =1 2(ρ−+ρ+)(u−−u+)/integraldisplayt 0α(s)exp(−/integraldisplays 0σ(τ)dτ)ds, x(t) =u−+u+ 2/integraldisplayt 0α(s)exp(−/integraldisplays 0σ(τ)dτ)ds. Proof.1. First, since u−> u+, it follows that x−(t)> x+(t). Forε >0 sufficiently small, if x−x−(t)> (Bε(t))1/2, then we may write from ( 23), Wε(x,t)≈ρ+(x−x+(t))exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBig and uε(x,t)≈u+exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig u+ x+(t)−x−u− x−(t)−x/parenrightBig exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBigexp(−/integraldisplayt 0σ(s)ds). 11Ifx+(t)−x <−(Bε(t))1/2andx=x−(t), then Wε(x,t)≈−ρ−(Bε(t))1/2 2π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig +ρ+(x−x+(t))exp/parenleftBig −x2+(x+(t)−x)2 Bε(t)/parenrightBig 1 2exp/parenleftBig −x2 Bε(t)/parenrightBig +exp/parenleftBig −x2+(x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2(x+(t)−x)exp/parenleftBig −x2 Bε(t)/parenrightBig and uε(x,t)≈u− 2exp/parenleftBig −x2 Bε(t)/parenrightBig +u+exp/parenleftBig −x2+(x+(t)−x)2 Bε(t)/parenrightBig +u+(Bε(t))1/2 2π1/2(x+(t)−x)exp/parenleftBig −x2 Bε(t)/parenrightBig 1 2exp/parenleftBig −x2 Bε(t)/parenrightBig +exp/parenleftBig −x2+(x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2(x+(t)−x)exp/parenleftBig −x2 Bε(t)/parenrightBigexp(−/integraldisplayt 0σ(s)ds). Ifx+(t)+(Bε(t))1/2≤x≤x−(t)−(Bε(t))1/2, then Wε(x,t)≈ρ−(x−x−(t))exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +ρ+(x−x+(t))exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBig and uε(x,t)≈u−exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +u+exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig u+ x+(t)−x−u− x−(t)−x/parenrightBig exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +exp/parenleftBig (x+(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBigexp(−/integraldisplayt 0σ(s)ds). Ifx+(t)−x >(Bε(t))1/2, then Wε(x,t)≈ρ−(x−x−(t))exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBig and uε(x,t)≈u−exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig u+ x+(t)−x−u− x−(t)−x/parenrightBig exp/parenleftBig (x−(t)−x)2 Bε(t)/parenrightBig +(Bε(t))1/2 2π1/2/parenleftBig 1 x+(t)−x−1 x−(t)−x/parenrightBigexp(−/integraldisplayt 0σ(s)ds). Ifx−x−(t)<−(Bε(t))1/2andx=x+(t), then Wε(x,t)≈ρ−(x−x−(t))exp/parenleftBig −x2+(x−(t)−x)2 Bε(t)/parenrightBig +ρ+(Bε(t))1/2 2π1/2exp/parenleftBig −x2 Bε(t)/parenrightBig exp/parenleftBig −x2+(x−(t)−x)2 Bε(t)/parenrightBig −(Bε(t))1/2 2π1/2(x−(t)−x)exp/parenleftBig −x2 Bε(t)/parenrightBig +1 2exp/parenleftBig −x2 Bε(t)/parenrightBig and uε(x,t)≈u−exp/parenleftBig −x2+(x−(t)−x)2 Bε(t)/parenrightBig −u−(Bε(t))1/2 2π1/2(x−(t)−x)exp/parenleftBig −x2 Bε(t)/parenrightBig +u+ 2exp/parenleftBig −x2 Bε(t)/parenrightBig exp/parenleftBig −x2+(x−(t)−x)2 Bε(t)/parenrightBig −(Bε(t))1/2 2π1/2(x−(t)−x)exp/parenleftBig −x2 Bε(t)/parenrightBig +1 2exp/parenleftBig −x2 Bε(t)/parenrightBigexp(−/integraldisplayt 0σ(s)ds). Therefore, we have that lim ε→0+Wε(x,t) =/braceleftBigg ρ−(x−x−(t)),if (x−x+(t))2−(x−x−(t))2<0, ρ+(x−x+(t)),if (x−x+(t))2−(x−x−(t))2>0. Observe that ( x−x+(t))2−(x−x−(t))2= 2(x−(t)−x+(t))(x−x−(t)+x+(t) 2), and defining x−(t)+x+(t) 2=:x(t), 12we get lim ε→0+Wε(x,t) =/braceleftBigg ρ−(x−x−(t)),ifx < x(t), ρ+(x−x+(t)),ifx > x(t). SinceWε(x,t) is bounded on compact subsets of R2 +andWε(x,t)→W(x,t) pointwise as ε→0+, then Wε(x,t)→W(x,t) in the sense of distribution and so Wε x(x,t) convergesin the distributional sense to Wx(x,t). From Proposition 2.1we have that lim ε→0+ρε(x,t) =ρ(x,t) exists in the sense of distribution and ρ(x,t) =Wx(x,t) = ρ−,ifx < x(t), (x−(t)−x+(t))ρ−+ρ+ 2δ(x−x(t)),ifx=x(t) ρ+,ifx > x(t).(30) Analogously, we obtain u(x,t) = u−exp(−/integraldisplayt 0σ(s)ds),ifx < x(t), u−+u+ 2exp(−/integraldisplayt 0σ(s)ds),ifx=x(t), u+exp(−/integraldisplayt 0σ(s)ds),ifx > x(t).(31) 2. Now, we show that ρandu, defined respectively by ( 30), (31) solve the Riemann problem ( 4)-(6) in the sense of Definition 2.2. Indeed, for any test function ϕ∈C∞ 0(R×R+) we have < ρ,ϕ t>+< αρu,ϕ x>=/integraldisplay∞ 0/integraldisplay R(ρϕt+α(t)ρuϕx)dxdt +/integraldisplay∞ 0ρ−+ρ+ 2(x−(t)−x+(t))/parenleftbigg ϕt+α(t)u−+u+ 2exp(−/integraldisplayt 0σ(s)ds)ϕx/parenrightbigg dt =/integraldisplay∞ 0/integraldisplayx(t) −∞(ρ−ϕt+α(t)ρ−u−exp(−/integraldisplayt 0σ(s)ds)ϕx)dxdt+/integraldisplay∞ 0/integraldisplay∞ x(t)(ρ+ϕt+α(t)ρ+u+exp(−/integraldisplayt 0σ(s)ds)ϕx)dxdt +/integraldisplay∞ 0ρ−+ρ+ 2(x−(t)−x+(t))/parenleftbigg ϕt+α(t)u−+u+ 2exp(−/integraldisplayt 0σ(s)ds)ϕx/parenrightbigg dt =−/contintegraldisplay −(α(t)ρ−u−exp(−/integraldisplayt 0σ(s)ds)ϕ)dt+(ρ−ϕ)dx+/contintegraldisplay −(α(t)ρ+u+exp(−/integraldisplayt 0σ(s)ds)ϕ)dt+(ρ+ϕ)dx +/integraldisplay∞ 0ρ−+ρ+ 2(x−(t)−x+(t))/parenleftbigg ϕt+α(t)u−+u+ 2exp(−/integraldisplayt 0σ(s)ds)ϕx/parenrightbigg dt =/integraldisplayt 0/parenleftbigg α(t)(ρ−u−−ρ+u+)exp(−/integraldisplayt 0σ(s)ds)−(ρ−−ρ+)dx(t) dt/parenrightbigg ϕdt +/integraldisplay∞ 0ρ−+ρ+ 2(x−(t)−x+(t))dϕ dtdt =/integraldisplayt 0/parenleftbigg α(t)(ρ−u−−ρ+u+)exp(−/integraldisplayt 0σ(s)ds)−(ρ−−ρ+)dx(t) dt/parenrightbigg ϕdt −/integraldisplay∞ 0d dt/parenleftbiggρ−+ρ+ 2(x−(t)−x+(t))/parenrightbigg ϕdt= 0, 13and /integraldisplay∞ 0/integraldisplay R/parenleftbigg uϕt+α(t) 2u2ϕx−σ(t)uϕ/parenrightbigg dxdt=/integraldisplay∞ 0/integraldisplay R/parenleftbigg uϕt+α(t) 2u2ϕx/parenrightbigg dxdt−/integraldisplay∞ 0/integraldisplay Rσ(t)uϕdxdt =/integraldisplay∞ 0/integraldisplayx(t) −∞u−exp(−/integraldisplayt 0σ(s)ds)/parenleftbigg ϕt+α(t) 2u−exp(−/integraldisplayt 0σ(s)ds)ϕx/parenrightbigg dxdt +/integraldisplay∞ 0/integraldisplay∞ x(t)u+exp(−/integraldisplayt 0σ(s)ds)/parenleftbigg ϕt+α(t) 2u+exp(−/integraldisplayt 0σ(s)ds)ϕx/parenrightbigg dxdt−/integraldisplay∞ 0/integraldisplay Rσ(t)uϕdxdt =−/contintegraldisplay −/parenleftbiggα(t) 2u2 −exp(−2/integraldisplayt 0σ(s)ds)ϕ/parenrightbigg dt+/parenleftbigg u−exp(−/integraldisplayt 0σ(s)ds)ϕ/parenrightbigg dx +/integraldisplay∞ 0/integraldisplayx(t) −∞σ(t)u−exp(−/integraldisplayt 0σ(s)ds)ϕdxdt +/contintegraldisplay −/parenleftbiggα(t) 2u2 +exp(−2/integraldisplayt 0σ(s)ds)ϕ/parenrightbigg dt+/parenleftbigg u+exp(−/integraldisplayt 0σ(s)ds)ϕ/parenrightbigg dx +/integraldisplay∞ 0/integraldisplay∞ x(t)σ(t)u+exp(−/integraldisplayt 0σ(s)ds)ϕdxdt−/integraldisplay∞ 0/integraldisplay Rσ(t)uϕdxdt =/integraldisplay∞ 0/parenleftbiggα(t) 2(u2 −−u2 +)exp(−/integraldisplayt 0σ(s)ds)−(u−−u+)dx(t) dt/parenrightbigg ϕexp(−/integraldisplayt 0σ(s)ds)dt= 0. 3. Finally, we observe that, for each t≥0, u+α(t)exp(−/integraldisplayt 0σ(τ)dτ)<dx(t) dt< u+α(t)exp(−/integraldisplayt 0σ(τ)dτ), which is an entropy condition to the system ( 4). 3 Pressureless Type Gas Dynamics System The main issue of this section is to study the Riemann problem of the pr essureless gas system with variable coefficient and time-variable linear damping ( 5). We introduce a similar variable to reduce the system ( 5) to hyperbolic conservation laws with variable coefficient to solve the Riem ann problem with u−< u+. To the caseu−> u+, similar to [ 5], we use a nonlinear viscous system and using a similar variable we obtain viscous solutions that converge to a delta shock solution of the Riemann pro blem (5)-(6). 3.1 Classical Riemann solutions. We observe that under transformation /hatwideu(x,t) =u(x,t)e/integraltextt 0σ(r)drthe system ( 5) is equivalent to /braceleftBigg ρt+α(t)e−/integraltextt 0σ(r)dr(ρ/hatwideu)x= 0, (ρ/hatwideu)t+α(t)e−/integraltextt 0σ(r)dr(ρ/hatwideu2)x= 0,(32) with the initial data ( 6). Using the similar variable ξ=x/integraltextt 0α(s)e−/integraltexts 0σ(r)drds, (33) the system ( 32) can be written as/braceleftBigg −ξρξ+(ρ/hatwideu)ξ= 0, −ξ(ρ/hatwideu)ξ+(ρ/hatwideu2)ξ= 0,(34) 14and the initial condition ( 6) changes to the boundary condition (ρ(±∞),/hatwideu(±∞)) = (ρ±,u±). Now, we note that any smooth solution of the system ( 34) satisfies /parenleftbigg/hatwideu−ξ ρ /hatwideu(/hatwideu−ξ)ρ(2/hatwideu−ξ)/parenrightbigg/parenleftbiggρξ /hatwideuξ/parenrightbigg =/parenleftbigg0 0/parenrightbigg and it provides either the general solution (constant state) ρ(ξ) =constant and /hatwideu(ξ) =constant, ρ/ne}ationslash= 0, or the singular solution ρ(ξ) = 0 for all ξand/hatwideu(ξ) =ξ, called the vacuum state . Thus the smooth solutions of system (34) only contain constants and vacuum solutions. For a bounded disco ntinuity at ξ=η, the Rankine-Hugoniot condition holds, that is to say, /braceleftBigg −η(ρ−−ρ+)+(ρ−/hatwideu−−ρ+/hatwideu+) = 0, −η(ρ−/hatwideu−−ρ+/hatwideu+)+(ρ−/hatwideu2 −−ρ+/hatwideu2 +) = 0, which holds when η=u−=u+. Therefore, two states ( ρ−,u−) and (ρ+,u+) can be connected by a contact discontinuity if and only if u−=u+. Thus, the contact discontinuity is characterized by ξ=u−=u+. Summarizing, we obtainthe solutionwhich consistsoftwocontactdis continuitiesand avacuum statebesides two constant states. Therefore, the solution can be expressed as (ρ(ξ),/hatwideu(ξ)) = (ρ−,u−),ifξ < u−, (0,ξ),ifu−≤ξ≤u+, (ρ+,u+),ifξ > u+. Sinceu(x,t) =/hatwideu(x,t)e−/integraltextt 0σ(r)drandξ=x/integraltextt 0α(s)e−/integraltexts 0σ(r)drds, then for u−< u+the Riemann solution to the system (5) is (ρ(x,t),u(x,t)) = (ρ−,u−e−/integraltextt 0σ(r)dr),ifx < u−/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds, (0,xe−/integraltextt 0σ(r)dr /integraltextt 0α(s)e−/integraltexts 0σ(r)drds),ifu−/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds≤x≤u+/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds, (ρ+,u+e−/integraltextt 0σ(r)dr),ifx > u+/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds. 3.2 Delta shock wave solutions. Givenε >0, we consider the following parabolic regularization to the system ( 5), /braceleftBiggρε t+α(t)(ρεuε)x= 0, (ρεuε)t+α(t)(ρε(uε)2)x=εβ∗(t)uε xx−σ(t)ρεuε,(35) whereβ∗(t) =α(t)exp(−/integraltextt 0σ(s)ds)/integraltextt 0α(s)exp(−/integraltexts 0σ(r)dr)ds, with initial condition (ρε(x,0),uε(x,0)) = (ρ0(x),u0(x)), (36) where (ρ0,u0) is given by ( 6). Under the transformation /hatwideuε(x,t) =uε(x,t)e/integraltextt 0σ(r)drthe system ( 35) becomes ρε t+α(t)e−/integraltextt 0σ(r)dr(ρε/hatwideuε)x= 0, (ρε/hatwideuε)t+α(t)e−/integraltextt 0σ(r)dr(ρε(/hatwideuε)2)x=εβ∗(t)/hatwideuε xx,(37) 15and the initial condition ( 36) becomes (ρε(x,0),/hatwideuε(x,0)) = (ρε 0(x),/hatwideuε 0(x)) =/braceleftBigg (ρ−,u−),ifx <0, (ρ+,u+),ifx >0(38) for arbitrary constant states u±andρ±>0 as well. By using the similar variable ( 33) the system ( 37) can be written as −ξρε ξ+(ρε/hatwideuε)ξ= 0, −ξ(ρε/hatwideuε)ξ+(ρε(/hatwideuε)2)ξ=ε/hatwideuε ξξ(39) and the initial data ( 38) changes to the boundary condition (ρ(±∞),/hatwideu(±∞)) = (ρ±,u±) (40) for arbitrary constant states u−> u+andρ±>0. The existence of solutions to the system ( 39) with boundary condition ( 40) was shown in Theorem 3 of [ 5]. More explicitly, in [ 5], the following result was obtained: Proposition 3.1. There exists a weak solution (ρε,/hatwideuε)∈L1 loc((−∞,+∞))×C2((−∞,+∞))to the boundary problem (39)-(40). From Theorem 2 in [ 5], we have that for each ε >0, the function /hatwideuεsatisfies /braceleftBigg ε(/hatwideuε)′′(ξ) = (ρε(ξ)(/hatwideu−ξ))(/hatwideuε)′(ξ), /hatwideuε(±∞) =u±, with′=d dξand ρε(ξ) =/braceleftBigg ρε 1(ξ),if−∞< ξ < ξε ς, ρε 2(ξ),ifξε ς< ξ <+∞, whereξε ςsatisfies/hatwideuε(ξε ς) =ξε ς, ρ1(ξ) =ρ−exp/parenleftBigg −/integraldisplayξ −∞(/hatwideuε(s))′ /hatwideuε(s)−sds/parenrightBigg andρ2(ξ) =ρ+exp/parenleftbigg/integraldisplay∞ ξ(/hatwideuε(s))′ /hatwideuε(s)−sds/parenrightbigg . Definition 3.1. A distribution pair ( ρ,u) is called a delta shock wave solution of the problem ( 5) and (6) in the sense of distributions, when there exist a smooth curve Land a function w(·), such that ρanduare represented respectively by ρ=/hatwideρ(x,t)+wδLandu=u(x,t), with/hatwideρ,u∈L∞(R×(0,∞)), and satisfy for each the test function ϕ∈C∞ 0(R×(0,∞)), /braceleftBigg < ρ,ϕ t>+< αρu,ϕ x>= 0, < ρu,ϕ t>+< αρu2,ϕx>=< σρu,ϕ >,(41) where < ρ,ϕ >:=/integraldisplay/integraldisplay R2 +/hatwideρϕdxdt+/an}bracketle{twδL,ϕ/an}bracketri}ht and for some smooth function G, < αρG(·),ϕ >:=/integraldisplay/integraldisplay R2 +α(t)/hatwideρG(u)ϕdxdt+/an}bracketle{tα(·)wG(uδ)δL,ϕ/an}bracketri}ht. Moreover, u|L=uδ(t). 16Now, we denote ς= lim ε→0+ξε ς= lim ε→0+/hatwideuε(ξε ς) =/hatwideu(ς). Then, according to Theorem 4 in [ 5], we have lim ε→0+(ρε(ξ),/hatwideuε(ξ)) = (ρ−,u−), ifξ < ς, (w0δ(ξ−ς),uδ),ifξ=ς, (ρ+,u+), ifξ > ς, whereρεconverges in the sense of distributions to the sum of a step functio n and a Dirac measure δwith weight w0=−ς(ρ−−ρ+)+(ρ−u−−ρ+u+) anduδ=/hatwideu(ς). Moreover, ( ς,w0,uδ) satisfies ς=uδ, w0=−ς(ρ−−ρ+)+(ρ−u−−ρ+u+), w0uδ=−ς(ρ−u−−ρ+u+)+(ρ−u2 −−ρ+u2 +),(42) and the over-compressive entropy condition u+< uδ< u−. (43) Observe that from the system ( 42) we have (ρ−−ρ+)u2 δ−2(ρ−u−−ρ+u+)uδ+(ρ−u2 −−ρ+u2 +) = 0, which implies uδ=√ρ−u−−√ρ+u+√ρ−−√ρ+oruδ=√ρ−u−+√ρ+u+√ρ−+√ρ+. One remarks that, when uδ=√ρ−u−+√ρ+u+√ρ−+√ρ+the entropy condition is valid while uδ=√ρ−u−−√ρ+u+√ρ−−√ρ+ does not satisfy the entropy condition. Moreover, using the seco nd equation of the system ( 42) anduδ=√ρ−u−+√ρ+u+√ρ−+√ρ+, we obtain w0=√ρ−ρ+(u−−u+). Therefore, when ρ−=ρ+, from (42) we obtain 2(u−−u+)uδ−(u2 −−u2 +) = 0 and hence we have uδ=1 2(u−+u+) andw0=ρ−(u−−u+). Finally, using the similar variable ( 33), we have obtained the following result Proposition 3.2. Suppose u−> u+. Let(ρε(x,t),/hatwideuε(x,t))be the solution of the problem (37)-(38). Then the limit lim ε→0+(ρε(x,t),/hatwideuε(x,t)) = (ρ(x,t),/hatwideu(x,t))exists in the distribution sense. Moreover, (ρ(x,t),/hatwideu(x,t))is given by (ρ−,u−),ifx < uδ/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds, (w0/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds·δ(x−uδ/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds),uδ),ifx=uδ/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds, (ρ+,u+),ifx > uδ/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds, wherew0=√ρ−ρ+(u−−u+)anduδ=√ρ−u−+√ρ+u+√ρ−+√ρ+, whenρ−/ne}ationslash=ρ+. For the case ρ−=ρ+, it follows that, w0=ρ−(u−−u+)anduδ=1 2(u−+u+). In addition, the solution is unique under the over-compres sive entropy condition (43). Remark 2. The condition ( 42) is necessary and sufficient to guarantee the existence of delta sh ock solutions to the problem ( 37)-(38) withε= 0. In fact, there are two delta shock solutions. Now, the over-c ompressive entropy condition ( 43), (see the above proposition), was sufficient to obtain the uniquen ess of the delta shock solution. 17From the above proposition and since u(x,t) =/hatwideu(x,t)e−/integraltextt 0σ(r)dr, we can establish a solution to the system (5)withinitialdata( 6). Moreover,multiplyingtheentropycondition( 43)byα(t)wegetα(t)u+< uδα(t)< α(t) for allt≥0 and again using that u(x,t) =/hatwideu(x,t)e−/integraltextt 0σ(r)dr, we have extended the entropy condition ( 43) to the following entropy condition to the system ( 5), λ(ρ+,u+)e−/integraltextt 0σ(r)dr<dx(t) dt< λ(ρ−,u−)e−/integraltextt 0σ(r)dr,for allt≥0, (44) whereλ(ρ,u) =αuis the eigenvalue associated to system ( 5). Then, we have the following Theorem 3.1. Suppose u−> u+. Then the Riemann problem (5)-(6)admits under the entropy condition (44) a unique delta shock solution of the form (ρ(x,t),u(x,t)) = (ρ−,u−e−/integraltextt 0σ(r)dr),ifx < x(t), (w(t)δ(x−x(t)),uδ(t)),ifx=x(t), (ρ+,u+e−/integraltextt 0σ(r)dr),ifx > x(t),(45) where for ρ−/ne}ationslash=ρ+, w(t) =√ρ−ρ+(u−−u+)/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds, u δ(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt 0σ(r)dr,and x(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds. For the case ρ−=ρ+, it follws that w(t) =ρ−(u−−u+)/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds, u δ(t) =1 2(u−+u+)e−/integraltexts 0σ(r)dr,and x(t) =1 2(u−+u+)/integraldisplayt 0α(s)e−/integraltexts 0σ(r)drds. Proof.Suppose that ρ−/ne}ationslash=ρ+. Therefore, in orderto show that ( ρ,u), given by ( 45), is a solution to the problem (5)-(6), we consider any test function ϕ∈C∞ 0(R×(0,∞)) and compute < ρu,ϕ t>+< ρu2,ϕx>=/integraldisplay∞ 0/integraldisplay R(ρuϕt+α(t)ρu2ϕx)dxdt+/integraldisplay∞ 0w(t)(uδ(t)ϕt+α(t)u2 δ(t)ϕx)dt =/integraldisplay∞ 0/integraldisplayx(t) −∞(ρ−u−e−/integraltextt 0σ(r)drϕt+α(t)ρ−u2 −e−2/integraltextt 0σ(r)drϕx)dxdt +/integraldisplay∞ 0/integraldisplay∞ x(t)(ρ+u+e−/integraltextt 0σ(r)drϕt+α(t)ρ+u2 +e−2/integraltextt 0σ(r)drϕx)dxdt +/integraldisplay∞ 0w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt 0σ(r)dr/parenleftbigg ϕt+α(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt 0σ(r)drϕx/parenrightbigg dt =−/contintegraldisplay −/parenleftBig α(t)ρ−u2 −e−2/integraltextt 0σ(r)drϕ/parenrightBig dt+/parenleftBig ρ−u−e−/integraltextt 0σ(r)drϕ/parenrightBig dx +/contintegraldisplay −/parenleftBig α(t)ρ+u2 +e−2/integraltextt 0σ(r)drϕ/parenrightBig dt+/parenleftBig ρ+u+e−/integraltextt 0σ(r)drϕ/parenrightBig dx +/integraldisplay∞ 0/integraldisplay Rσ(t)ρuϕdxdt +/integraldisplay∞ 0w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt 0σ(r)dr/parenleftbigg ϕt+dx(t) dtϕx/parenrightbigg dt 18=/integraldisplay∞ 0α(t)(ρ−u2 −−ρ+u2 +)e−2/integraltextt 0σ(r)drϕdt−/integraldisplay∞ 0dx(t) dt(ρ−u−−ρ+u+)e−/integraltextt 0σ(r)drϕdt +/integraldisplay∞ 0/integraldisplay Rσ(t)ρuϕdxdt +/integraldisplay∞ 0w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt 0σ(r)drdϕ(t) dtdt =/integraldisplay∞ 0α(t)(ρ−u2 −−ρ+u2 +)e−2/integraltextt 0σ(r)drϕdt−/integraldisplay∞ 0dx(t) dt(ρ−u−−ρ+u+)e−/integraltextt 0σ(r)drϕdt +/integraldisplay∞ 0/integraldisplay Rσ(t)ρuϕdxdt −/integraldisplay∞ 0d dt/parenleftbigg w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt 0σ(r)dr/parenrightbigg ϕdt =/integraldisplay∞ 0/integraldisplay Rσ(t)ρuϕdxdt +/integraldisplay∞ 0σ(t)w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt 0σ(r)drdt=< σρu,ϕ >, which implies the second equation of ( 41). With a similar argument, it is possible to obtain the first equation of (41) and the case when ρ−=ρ+. The uniqueness of the solution will be obtained under the entropy c ondition (44). 4 Riemann problem to the systems (4) and (5) with σ(·)≡0 In this section, we consider σ(t) =µν(t) whereµ >0 is a parameter, ν(t)≥0 for allt≥0, andν∈L1 loc([0,∞)). According to the Sections 2.2.1and3.1, ifu−< u+, the systems ( 4) and (5) with initial data ( 6) have the solution (ρ(x,t),u(x,t)) = (ρ−,u−exp(−µ/integraltextt 0ν(s)ds)), ifx < x−(t), (0,x/integraltextt 0α(s)exp(−µ/integraltexts 0ν(r)dr)dsexp(−µ/integraltextt 0ν(s)ds)),ifx−(t)< x < x +(t), (ρ+,u+exp(−µ/integraltextt 0ν(s)ds)), ifx > x+(t), wherex±(t) =u±/integraltextt 0α(s)exp(−µ/integraltexts 0ν(r)dr)ds. Ifu−> u+, the the solution for the problem ( 4)-(6) is (ρ(x,t),u(x,t)) = (ρ−,u−exp(−µ/integraltextt 0ν(s)ds)), ifx < x(t), (w(t)δ(x−x(t)),u−+u+ 2exp(−µ/integraltextt 0ν(s)ds)),ifx=x(t), (ρ+,u+exp(−µ/integraltextt 0ν(s)ds)), ifx > x(t), wherew(t) =1 2(ρ−+ρ+)(u−−u+)/integraltextt 0α(s)exp(−µ/integraltexts 0ν(τ)dτ)dsandx(t) =u−+u+ 2/integraltextt 0α(s)exp(−µ/integraltexts 0ν(τ)dτ)ds while the solution to the problem ( 5)-(6) is (ρ(x,t),u(x,t)) = (ρ−,u−e−µ/integraltextt 0ν(r)dr,ifx < x(t), (w(t)δ(x−x(t)),uδ(t)),ifx=x(t), (ρ+,u+e−µ/integraltextt 0ν(r)dr,ifx > x(t), wherew(t) =√ρ+ρ−(u−−u+)/integraltextt 0α(s)e−µ/integraltexts 0ν(r)drds,uδ(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+e−µ/integraltextt 0ν(r)dr, andx(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+/integraltextt 0α(s)e−µ/integraltexts 0ν(r)drdsifρ−/ne}ationslash=ρ+andw(t) =ρ−(u−−u+)/integraltextt 0α(s)e−µ/integraltexts 0ν(r)drds, uδ(t) =1 2(u−+u+)e−µ/integraltextt 0ν(r)dr, andx(t) =1 2(u−−u+)/integraltextt 0α(s)e−µ/integraltexts 0ν(r)drdsifρ−=ρ+. One observes that the solutions given above are explicit with respec t to the parameter µ >0, and also we have lim µ→0+exp(−µ/integraldisplayt 0ν(s)ds) = 1 and lim µ→0+/integraldisplayt 0α(s)exp(−µ/integraldisplays 0ν(r)dr)ds=/integraldisplayt 0α(s)ds. Therefore, the Riemann solution to the problems ( 4) and (5) withσ(t) = 0 for all t≥0 and initial data ( 6) is given by (ρ(x,t),u(x,t)) = (ρ−,u−),ifx < u−/integraltextt 0α(s)ds, (0,x/integraltextt 0α(s)ds),ifu−/integraltextt 0α(s)ds < x < u +/integraltextt 0α(s)ds, (ρ+,u+),ifx > u+/integraltextt 0α(s)ds. 19ifu−< u+. Ifu−> u+, then the Riemann solution to the problem ( 4) withσ(t) = 0 for all t≥0 and initial data (6) is (ρ(x,t),u(x,t)) = (ρ−,u−), ifx < x(t), (w(t)δ(x−x(t)),u−+u+ 2),ifx=x(t), (ρ+,u+), ifx > x(t), wherew(t) =1 2(ρ−+ρ+)(u−−u+)/integraltextt 0α(s)dsandx(t) =u−+u+ 2/integraltextt 0α(s)dsand the Riemann solution to the problem ( 5)-(6) withσ(t) = 0 for all t≥0 is given by (ρ(x,t),u(x,t)) = (ρ−,u−), ifx < x(t), (w(t)δ(x−x(t)),uδ(t)),ifx=x(t), (ρ+,u+), ifx > x(t), wherew(t) =√ρ+ρ−(u−−u+)/integraltextt 0α(s)ds,uδ(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+,andx(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+/integraltextt 0α(s)dsifρ−/ne}ationslash=ρ+ andw(t) =ρ−(u−−u+)/integraltextt 0α(s)ds,uδ(t) =1 2(u−+u+), andx(t) =1 2(u−−u+)/integraltextt 0α(s)dsifρ−=ρ+. 5 Comments and Extensions The main goal of this section is to present comments and extensions of ongoing work on the topic developed in this paper. We studied in this paper, the Riemann problems to the time-variable co efficient Zeldovich approximate system (4) and time-variablecoefficientpressurelessgassystem ( 5) both with generaltime-gradually-degenerate damping. Similar to the results obtained by Keita and Bourgault in [ 17] to the Riemann problems ( 4)-(6) and (5)-(6) both with α(·)≡1 andσ(·)≡σ=const. > 0, we have that the systems ( 4) and (5), where αandσ are non-negative functions that dependents of time t, are equivalent for smooth and two-contact-discontinuity solutions but they differ for delta shock solutions. Moreover, we sh ow that the uniqueness is obtained under an over-compressive entropy condition. Itisinterestingtoremarkthat, whywehavetofixthesignof α(·) solvingthe Riemannproblem. Indeed, they only need to have one sign (positive or negative) to maintain the Lax e ntropy (in shocks) and over-compressive entropy condition in delta shocks (as we need the characteristics n ot to be inverted). Clearly, the sign of σ(·) justifies the physical meaning of damping. Now, we would like to mention another direction of the work developed here, see [ 7]. Also related to system (3), we consider the following nonautonomous quasilinear systems: ρt+α1(t)(ρu)x= 0, ut+α2(t)(u2 2)x=−σ(t)u, and also /braceleftBigg ρt+α1(t)(ρu)x= 0, (ρu)t+α2(t)(ρu2)x=−σ(t)ρu, whereαi∈L1([0,∞)), (i= 1,2), and 0 ≤σ∈L1 loc([0,∞)). It is not absolutely clear that, all the strategies appliedinthispaperworkwiththesesystems,infact, thisisnotthec ase. Indeed, when α1/ne}ationslash=α2theconstruction of shocks, rarefactions, contact discontinuities, and delta shoc k solutions is not easy due to the behavior of the under- or over-compressibility of the eigenvalues and left or right s tates. This stands as the focal point of our ongoing research efforts. Data availability statement Data sharing does not apply to this article as no data sets were gene rated or analyzed during the current study. 20Conflict of Interest The authorRichardDe lacruzacknowledgesthe supportreceivedf rom UniversidadPedag´ ogicay Tecnol´ ogicade Colombia. TheauthorWladimirNeveshasreceivedresearchgrantsf romCNPqthroughthegrants313005/2023- 0, 406460/2023-0, and also by FAPERJ (Cientista do Nosso Estado ) through the grant E-26/201.139/2021. References [1] F. Bouchut, On zero pressure gas dynamics. In: Advances in Kinetic Theory and Computing . Series on Advances in Mathematics for Applied Sciences, vol. 22, pp. 171–190 . World Scientific, Singapore (1994). [2] Y. Brenier, Solutions with concentration to the Riemann problem f or one-dimensional Chaplygin gas dy- namics, J. Math. Fluid Mech. 7, S326–S331 (2005). [3] Y. Brenier and E. Grenier, Sticky particles and scalar conservat ion laws, SIAM Journal on Numerical Analysis, 35(6), 2317-2328 (1998). [4] R. De la cruz, Riemann problem for a 2 ×2 hyperbolic system with linear damping, Acta Appl. Math. 170, 631-647 (2020). [5] R. De la cruz and J.C. Juajibioy, Delta shock solution for generalize d zero-pressure gas dynamics system with linear damping, Acta Appl. Math. 177(1), 1-25 (2021). [6] R. De la cruz and J.C. Juajibioy, Vanishing viscosity limit for Riemann s olutions to a 2 ×2 hyperbolic system with linear damping, Asymptotic Analysis 127(3), 275-296 (2022). [7] R. De la cruz and W. Neves, On an interpolated system between Ke yfitz-Kranzer and Pressureless type systems. In preparation. [8] R. De lacruz, Y-G. Lu, and X-t. Wang, Riemann problemfora gene ralvariablecoefficientBurgersequation with time-dependent damping, International Journal of Non-Linear Mechanics 162, 104703 (2024). [9] W.E,Y-G.Rykov,andY-G.Sinai, GeneralizedVariationalPrinciple s,GlobalWeakSolutionsandBehavior with Random Initial Data for Systems of Conservation Laws Arising in Adhesion Particle Dynamics, Commun.Math.Phys.177, 349-380 (1996). [10] R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretati on, Academic Press, London and New York (1973). [11] S. Geng, Y. Lin, and M. Mei, Asymptotic behavior of solutions to E uler equations with time-dependent damping in critical case. SIAM J Math Anal 52, 1463–1488 (2020). [12] W.P. Hong, On B¨ acklund transformation for a generalized Burg ers equation and solitonic solutions, Phys. Lett. A268, 81–84 (2000). [13] E. Hopf, The partial differential equation ut+uux=εuxx. Commun Pure Appl Math 3(3), 201–230(1950). [14] F. Huang and Z. Wang, Well posedness for pressureless flow, Communications in Mathematical Physics , 222, 117-146, (2001). [15] E.L. Isaacson and B. Temple, Analysis of a singular hyperbolic sys tem of conservation laws, J. Differential Equations 65, 250-268 (1986). [16] K.T. Joseph, A Riemann problem whose viscosity solution contain δ-measures.Asymptot. Anal. 7, 105–120 (1993). [17] S. Keita and Y. Bourgault, Eulerian droplet model: Delta-shock w aves and solution of the Riemann problem, J. Math. Anal. Appl. 472, 1001-1027 (2019). 21[18] D.J. Korchinski, Solution of a Riemann problem for a 2 ×2 system of conservation laws possessing no classical weak solution, thesis, Adelphi University (1977). [19] S. Li, Riemann problem for a hyperbolic system with time-gradually -degenerate damping, Boundary Value Problems , volume 2023, Article number 109 (2023). [20] H. Li, J. Li, M. Mei, and K. Zhang, Convergence to nonlinear diffus ion waves for solutions of p-system with time-dependent damping. Journal of Mathematical Analysis and Ap plications 456(2), 849–871 (2017). [21] D-X. Meng, Y-T. Gao, L. Wang, X-L. Gai, and G-D. Lin, Interac tion of solitons in a variable-coefficient generalized Boussinesq system in shallow water, Phys. Scr. 82, 045012 (2010). [22] F. Oliveri, General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form. International Journal of Non-Linear Mechanics 47, 53–60 (2012). [23] C. Sackand and H. Schamel, Nonlinear dynamics in expanding plasm as,Phys. Lett. A 110, 206-212 (1985). [24] S.F. Shandarin and Y.B. Zeldovich, Large-scalestructure of th e universe: Turbulence, intermittency, struc- tures in a self-gravitating medium, Rev. Modern Phys. 61, 185-220 (1989). [25] W.C. Sheng and T. Zhang, The Riemann problem for the transpor tation equations in gas dynamics. In: Memoirs of the American Mathematical Society, vol. 137. Am. Math. Soc., Providence (1999). [26] M. Vergassola, B. Dubrulle, U. Frisch, and A. Noullez, Burgers’ equation, devil’s staircases and the mass distribution function for large-scale structures. 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0912.5521v1.Spin_torque_and_critical_currents_for_magnetic_vortex_nano_oscillator_in_nanopillars.pdf

1 Spin torque and critical currents for magnet ic vortex nano-oscillator in nanopillars Konstantin Y. Guslienko1,2*, Gloria R. Aranda1, and Julian M. Gonzalez1 1Dpto. Fisica de Materiales, Universidad de l Pais Vasco, 20018 Donostia-San Sebastian, Spain 2IKERBASQUE, the Basque Foundation for Science, 48011 Bilbao, Spain We calculated the main dynamic parameters of the spin polarized current induced magnetic vortex oscillations in nanopillars, such as the range of current density, wh ere a vortex steady oscillation state exists, the oscillation frequency and orbit radius. We accounted for both the non-linear vortex frequency and non-linear vortex damping. To describe the vortex excitations by the spin polarized current we used a generalized Thiele approach to motion of the vortex core as a collective coordinate. All the results are represented via the free layer sizes, saturation magnetiza tion, Gilbert damping and the degree of the spin polarization of the fixed layer. Pr edictions of the developed model can be checked experimentally. Key words: spin polarized current, magnetic nanopillar, nano-oscilla tors, magnetic vortex *Corresponding author. Electronic mail: sckguslk@ehu.es 2 Now excitations of the microwave oscillatio ns in magnetic nanopilla rs, nanocontacts and tunnel junctions by spin polarized curren t as well as the current induced domain wall motions in nanowires are perspective applications of spintronics.1 A general theoretical approach to microwave generation in nanopillars/nanocontacts driven by spin-polarized current based on the universal model of an auto- oscillator with negative damping and nonlinear frequency shift was de veloped recently by Slavin and Tiberkevich [see Ref. 2 and references therein]. Th e model was applied to the case of a spin-torque oscillator (STO) excited in a uniformly magnetized free layer of nanopillar, and explains the main experimentally observed effects such as the power and frequency of the gene rated microwave signal. However, the low generated power ~ 1 nW of such STO prevents their practic al applications. Recently extremely narrow linewidth of 0.3 MHz and relatively high generated power was detected for the magnetic vortex (strongly non-uniform st ate) nano-oscillat ors in nanopillars.3 The considerable microwave power emission from a vortex STO in magnetic tunnel junctions was observed.4 It was established that the permanent perpendicular to the plane (CPP) spin polarized current I can excite vortex motion in free layer of th e nanopillar if the current intens ity exceeds some critical value, Ic1.5 Then, in the interval Ic1 <I <Ic2 (Ic2 is a second critical current) th ere is microwave generation at a frequency which smoothly increase with the current increasing.6 This frequency corresponds to the vortex oscillations with a stationary orbit determ ined by the current value. If the current exceeds a critical value Ic27,8 the vortex steady state is not stable anym ore, presumably because the vortex reaches a critical velocity9 and reverses its core. The vortex with op posite core cannot be excited for the given current sign I and the oscillations stop. Such excitation scheme is different from the current-in-plane (CIP) case, where one needs to apply a.c. CIP of about the resonance frequency to excite the vortex motion.10 Low value of Ic1 and high value of Ic2 make the vortex CPP nano-os cillators attractive for applications as microwave devices. However, the calcu lations of the spin torque term (ST force) gave contradictory results for the ST force and Ic1. The standard STO approach of Ref. 2 is not applicable to 3 the vortex dynamics due to specific damping term. The problem was reduced to the problem of vortex core reversal in the perpe ndicular to the layer plane magnetic field in Ref. 7 but vortex steady state dynamics was not accounted. Using the Thiele approach the ST force was calculated in Refs. 5, 8 which differs in 2 times from one calculate d form the energy dissipation balance.6 Non-linearity of the main governing parameters and the Oersted field of curre nt were not accounted or accounted incorrectly. The critical current Ic2 has not yet been calculated. In this Letter we present a simple and effectiv e approach to calculate the ST force and the critical currents of the vortex STO in nanopillar. The appro ach is based on the Thiele formulation of the non- uniform magnetization dynamics11 and conception of the linear spin excitations.12 The nanopillar device consists of two ferromagnetic layers , typically FeNi or Co and a nonma gnetic metallic spacer, typically Cu, arranged in a vertical stack (Fig . 1). Magnetization of one layer is fixed (this layer is the so called polarizer), whereas the magnetization of the second layer of the nanopillar ()t,rM is free to rotate. The current of spin polarized elec trons transfers some torque sτ from the polarizer, which excites magnetization dynamics of the free layer. We st art from the Landau-Lifshi tz equation of motion s LLG eff τ mm Hm m γ α γ +× + × −=   , where sM/Mm= , Mss is the saturation magnetization, γ>0 is the gyromagnetic ratio, Heff is the effective field, and LLGα is the Gilbert damping. We use the ST term in the form suggested by Slonczewski,13 () Pm mτ × × =Jsσ , where ()sLMe2/η σ== , η is the current spin polarization ( η=0.2 for FeNi), e is the electron charge, L is the free layer (dot) thickness, J is the current density, and z PP= is the unit vector of the polarizer magnetization ( P=+1/-1). We assume the positive vortex core polarization p=+1, P=+1 and define the current (flow of the positive charges) as positive I>0 when it flows from the polarizer to free layer. The spin polarized curr ent can excite a vortex motion in the free layer if only IpP > 0 (only the electrons bringing a magnetic moment from the polarizer to free layer opposite to the core polarization can excite a vortex motion). Except p, the vortex 4 is described by its core position in the free layer, X=(X,Y), and chirality C=±1 .14 Let denote the Slonczewski´s energy density which correspond s to the spin polarized current as sw. Then, using the Thiele approach and the ST field Pm m × =∂ ∂ a ws/ , the ST force acting on th e vortex in the free layer can be written as ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ∂∂× ⋅ =∂∂−= ∫ ∫ α αα Xd aL dVwXFs STmmρ P2, (1) where J Masσ = , α=x, y, ()ϕρ, =ρ is the in-plane radius vector, the derivative is taken with respect to the vortex core position X assuming an ansatz () ( ) [] t t Xρmρm , ,= (m dependence on thickness coordinate z is neglected). X has sense of the amplitude of the vortex gyrotropic eigenmode. We use representation of m-components by the spherical angles ΦΘ, (Fig. 1) as ) cos, sin sin, cos (sin Θ Φ Θ Φ Θ =m and find the expression for the ST force Xρ F∂Φ∂Θ =∫2 2sindaLST . (2) In the main approximation we use the decompositions ( ) () () ( ) ρX Xρ ˆ cos ,0⋅ + =Θ = ρ ρ g m mz z , () ( ) [] ϕ ϕ ρ cos sin ,0 0 Y X m − +Φ= Φ Xρ , where ()ρ0 zm , 0Φ are the static vortex core profile and phase, () () ( )22 2 2 2/ 1 4 ρ ρ ρ ρ + + = c pc g is the excitation amplitude of the z-component of the vortex magnetization ( RRcc/ = , cR is the vortex core radius, ρ, X are normalized to the dot radius R) and ()() ρ ρ ρ / 12 0 −= m is the gyrotropic mode profile.12 One can conclude from Eq. (2) that only moving vortex core contribute to the ST force because the contribu tion of the main dot area where 2/π=Θ is equal to zero due to vanishing integrals on azimuthal angle φ from the gradient of the vortex phase Φ∂X 5 (it was checked accounting in ()Xρ,Φ the terms up to cubic terms in X α-components). This is a reason why the ST contribution is relatively small bei ng comparable with the damping contribution. The integration in Eq. (2) yields the ST force ()Xz F × = ˆaLSTπ . This force contributes to the Thiele’s equation of motion ST D W FX XGX + + −∂=×   ˆ , where γ π / 2ˆs pLMzG= is the gyrovector, Dˆ is the damping tensor. The vortex energy ()XW and restoring force WR X F −∂= can be calculated from an appropriate model14 (the force balance is shown in Fig. 2). For circular steady st ate vortex core motion the XωX ×= relation holds, which allows calculating Jc1. To calculate the vortex steady orbit radius X=sR we need, however, to account non-linear on X α terms in the vortex damping and frequency (the account only non-linear frequency as in Ref. 5 is not sufficient). The gyr ovector also depends on X, but this dependence is essential only for the vortex core p reversal, where G changes its sign. As we show below, the most important non-linearity co mes from the damping tensor defined as () β ααβγαX XdVMDs LLG∂∂⋅∂∂−=∫m mX , (3) or () [] Φ∂Φ∂Θ +Θ∂Θ∂ −= ∫ β α β α αβ γ α2 2sin /ρd LM Ds LLG in ΦΘ, -representation Accounting αβ αβ δD D= and introducing dimensionless damping parameter 0 /> −= GD d15 we can write the equation for a steady state vortex motion with the orbit radius X=sR : () ()ϕωST G F sRs Gsd = from which RRss/ = and the critical currents Jc1, Jc2 can be found. In the second order non-linear approximation ()2 1 0 sd dsd + = , ()2 1 0 s sG ω ω ω + = and aLRs FSTπϕ= , whereGω is the vortex precession frequency, 01>ω is a function of the dot aspect ratio β=L/R calculated from the vortex energy decomposition ()sW i n s e r i e s o f R s /X= . It can be shown that () ( ) [] 3/41 9/200 β βγ βω − =sM and () () βω βω0 1 4≈ for quite wide range of β= 0.01-0.2 of practi cal interest, whereas considerably larger 6 non-linearity () () 8.42 /0 1 =βωβω was calculated in Ref. 5 due to in correct account of the magnetostatic energy. We use the pole free model of the shifted vortex ()[] tXρm, , where the dynamic magnetization satisfies the strong pinning boundary condition at the dot circumference16 R=ρ . The damping parameters are () () 2/ / ln8/50 c LLG RR d + =α , () 4/3/8 /2 2 1 − =c LLG RR d α . We need also to account for the Oersted field of the current, wh ich leads to contribution to the vo rtex frequency proportional to the current density () ( ) J Jeω βω βω ω + = =0 0 0 , , where () ( ) CcRe ξ γ π ω / 15/8= , () R Rc8/ 2/12ln151 − +=ξ is the correction for the finite core radius cR<<R. In the linear approximation we get the equality of the damping force and the ST force (negative damp ing) as the condition to find the value of Jc1, with the contribution of the Oersted field of the current accounted. The values of FST, Jc1 coincide with calculation of Ref. 6 conducted by the method of the damping energy balance and differ by the multiplayer 2 from Refs. 5, 8. The first critical current is ()e c d d J ω γσω0 00 1 2/ / − = and the steady state vortex orbit radius is () 1 /1− =cJJ Js λ , 1cJJ> , ()[]10 1 011 2 21 ω ωγσλd J dJ cc += (4) In this approximation the vortex trajectory radius ()Js increases as square root of the current overcriticality ()1 1/c cJ JJ− (for the typical parameters and R=80-120 nm we get λ=0.25-0.30) and the vortex frequency () ( )1 12 0 1 / ω λ ω βω ω − + + =c e G JJ J increases linearly with J increasing. The vortex steady orbit can exist until the m oving vortex crosses the dot border s=1 or its velocity X reaches the critical velocity cυ defined in Ref. 9. The later allows to write equation for the s econd critical current Jc2 as () ()c G RJsJ υ ω =. Substituting to this expression the equations for ()JGω and ()Js derived above 7 we get a cubic equation for Jc2 in the form () [ ] R xx J d Jc ce c λυ λω ω γσ / 2/2/1 2 1 1 0 1 = + + , () 1 /1 2 − =c cJ J x . This equation has one positive root xc and the value of Jc2 can be easily calculated (Fig. 3). The former condition ( s=1) gives the second critical current ()12 2 /11c c J J λ+=′ . More detailed analysis shows that both the mechanisms of the hi gh current instability of the vortex motion are possible depending on the dot sizes L, R, and the critical current is th e lower value of the currents Jc2, J’c2. The vortex core reversal inside the dot occurs for large enough R (> 100 nm) and L. For the typical sizes L=10 nm, R=120 nm and C=1, the critical currents are Jc1=6.3 106 A/cm2 (Ic1=2.9 mA), Jc2=1.13 108 A/cm2 (Ic2=51 mA), and for L= 5 nm, R=100 nm we get Jc1=1.8 106 A/cm2 (Ic1=0.56 mA), J’c2=2.7 107 A/cm2 (I’c2=8.4 mA). In summary, we calculated the main physical para meters of the spin polar ized CPP current induced vortex oscillations in na nopillars, such as the cr itical current densities Jc1, Jc2, the vortex steady state oscillations frequency and orbit radius. All the results are represented via the free layer sizes ( L, R), saturation magnetization, Gilbert damp ing and the degree of the spin polarization of the fixed layer. These parameters can be obtained from independent e xperiments. We demonstrated that the generalized Thiele approach is applicable to the problem of the vortex STO excitations by the CPP spin polarized current. The spin transfer torque force is related to the vortex core only. The authors thank J. Grollier and A.K. Khvalkovsk iy for fruitful discussions. K.G. and G.R.A. acknowledge support by IKERBASQUE (the Basque Science Foundation) and by the Program JAE-doc of the CSIC (Spain), respectively. The author s thank UPV/EHU (SGIker Arina) and DIPC for computation tools. The work was part ially supported by the SAIOTEK grant S-PC09UN03. 8 References 1 G. Tatara, H. Kohno, and J. Shibata, Phys. Rep . 468, 213 (2008). 2 A. Slavin and V. Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009). 3 V.S. Pribiag, I.N. Krivorotov, G.D. Fuchs et al., Nature Phys . 3, 498 (2007). 4 A. Dussaux, B. Georges, J. Grollier et al. , submitted to Nature Phys. (2009). 5 B. A. Ivanov an d C. E. Zaspel, Phys. Rev. Lett. 99, 247208 (2007). 6 A.V. Khvalkovskiy, J. Grollier, A. Dussaux, K.A. Zvezdin, and V. Cros, Phys. Rev . B 80, 140401 (2009). 7 J.-G. Caputo, Y. Gaididei, F.G. Mertens and D.D. Sheka, Phys. Rev. Lett. 98, 056604 (2007); D.D. Sheka, Y. Gaididei, and F.G. Mertens, Appl. Phys. Lett. 91, 082509 (2007). 8 Y. Liu, H. He, and Z. Zhang, Appl. Phys. Lett. 91, 242501 (2007). 9 K.Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev. Lett . 100, 027203 (2008); K.-S. Lee et al., Phys. Rev. Lett. 101, 267206 (2008). 10 S. Kasai, Y. Nakatani, K. Koba yashi, H. Kohno, and T. Ono, Phys. Rev. Lett. 97, 107204 (2006); K. Yamada, S. Kasai, Y. Naka tani, K. Kobayashi, and T. Ono, Appl. Phys. Lett. 93, 152502 (2008). 11 A. A. Thiele, Phys. Rev. Lett . 30, 230 (1973). 12 K.Y. Guslienko, A.N. Slavin, V. Tiberkevich, S. Kim, Phys. Rev. Lett. 101, 247203 (2008). 13 J. Slonczewski, J. Magn. Magn. Mat . 159, L1 (1996); J. Magn. Magn. Mat . 247, 324 (2002). 14 K.Y. Guslienko, J. Nanosci. Nanotechn. 8, 2745 (2008). 15 K.Y. Guslienko, Appl. Phys. Lett. 89, 022510 (2006). 16 K. Y. Guslienko et al., J. Appl. Phys . 91, 8037 (2002); V. Novosad et al. , Phys. Rev . B 72, 024455 (2005). 9 Captions to the Figures Fig. 1. Sketch of the magnetic nan opillar with the coordinate system used. The upper (free) layer is in the vortex state with non- uniform magnetization distribution. The polarizer layer (red color) is in uniform magnetization state w ith the magnetization along Oz axis. The positive current I (vertical arrow) flows from the polarizer to free layer. Fig. 2. Top view of the free laye r with the moving vortex. The arrows denote the force balance for the vortex core. The spin torque ( FST), damping ( FD), restoring (RF) and gyro- ( FG) forces are defined in the text. The vortex core steady trajectory Rs is marked by orange color. The vortex chirality is C=+1. Fig. 3. Dependence of the critical currents Jc1 (solid red line), Jc2 (dashed green line) and J’c2 of the vortex motion instability on the radius R of the free layer. L= 10 nm, Ms =800 G, η =0.2, 01 .0=LLGα , γ/2 =2.95 MHz/Oe, Rc=12 nm. The vortex STO motion is stable at Jc1 < J < min( Jc2, J’c2). 10 Fig. 1. 11 Fig. 2. 12 Fig. 3. 60 80 100 120 14056789101112 Jc2 J'c2Current density, J (107 A/cm2) Dot radius, R (nm)FeNi Ms=800 G L=10 nm Jc1x10

2206.03218v2.Decay_property_of_solutions_to_the_wave_equation_with_space_dependent_damping__absorbing_nonlinearity__and_polynomially_decaying_data.pdf

arXiv:2206.03218v2 [math.AP] 11 Aug 2022DECAY PROPERTY OF SOLUTIONS TO THE WAVE EQUATION WITH SPACE-DEPENDENT DAMPING, ABSORBING NONLINEARITY, AND POLYNOMIALLY DECAYING DATA YUTA WAKASUGI Abstract. We study the large time behavior of solutions to the semiline ar wave equation with space-dependent damping and absorbing n onlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and th e decay rate of the initial data at the spatial infinity determine the decay r ates of the energy and the L2-norm of the solution. In Appendix, we also give a survey of ba sic results on the local and global existence of solutions and th e properties of weight functions used in the energy method. 1.Introduction We study the initial-boundary value problem of the wave equation with space- dependent damping and absorbing nonlinearity ∂2 tu−∆u+a(x)∂tu+|u|p−1u= 0, t >0,x∈Ω, u(t,x) = 0, t > 0,x∈∂Ω, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(1.1) Here, Ω = Rnwithn≥1, or Ω⊂Rnwithn≥2 is an exterior domain, that is,Rn\Ω is compact. We also assume that the boundary ∂Ω of Ω is of class C2. When Ω = Rn, the boundary condition is omitted and we consider the initial value problem. The unknown function u=u(t,x) is assumed to be real-valued. The function a(x) denotes the coefficient of the damping term. Throughout this paper, we assume that a∈C(Rn) is nonnegative and bounded. The semilinear term|u|p−1u, wherep >1, is the so-called absorbing nonlinearity, which assists the decay of the solution. The aim of this paper is to obtain the decay estimates of the energy E[u](t) :=1 2/integraldisplay Ω(|∂tu(t,x)|2+|∇u(t,x)|2)dx+1 p+1/integraldisplay Ω|u(t,x)|p+1dx(1.2) and the weighted L2-norm /integraldisplay Ωa(x)|u(t,x)|2dx of the solution. Date: August 12, 2022. 2020Mathematics Subject Classification. 35L71, 35L20, 35B40. Key words and phrases. wave equation, space-dependent damping, absorbing nonlin earity. 12 Y. WAKASUGI First, for the energy E[u](t), we observe from the equation (1.1) that d dtE[u](t) =−/integraldisplay Ωa(x)|∂tu(t,x)|2dx, which gives the energy identity E[u](t)+/integraldisplayt 0/integraldisplay Ωa(x)|∂tu(s,x)|2dxds=E[u](0). Sincea(x) is nonnegative, the energy is monotone decreasing in time. Theref ore, a naturalquestionarisesastowhethertheenergytendstozeroa stimegoestoinfinity and, if that is true, what the actual decay rate is. Moreover, we c an expect that the amplitude of the damping coefficient a(x), the power pof the nonlinearity, and the spatial decay of the initial data ( u0,u1) will play crucial roles for this problem. Our goal is to clarify how these three factors determine the decay property of the solution. Before going to the main result, we shall review previous studies on t he asymp- totic behavior of solutions to linear and nonlinear damped wave equat ions. The study of the asymptotic behavior of solutions to the damped wa ve equation goes back to the pioneering work by Matsumura [52]. He studied the initial value problem of the linear wave equation with the classical damping /braceleftbigg∂2 tu−∆u+∂tu= 0, t > 0,x∈Rn, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Rn.(1.3) In this case the energy of the solution uis defined by EL(t) :=1 2/integraldisplay Rn(|∂tu(t,x)|2+|∇u(t,x)|2)dx. (1.4) By using the Fourier transform, he proved the so-called Matsumur a estimates /ba∇dbl∂k t∂γ xu(t)/ba∇dblL∞≤C(1+t)−n 2m−k−|γ| 2/parenleftbig /ba∇dblu0/ba∇dblLm+/ba∇dblu1/ba∇dblLm+/ba∇dblu0/ba∇dblH[n 2]+k+|γ|+1+/ba∇dblu1/ba∇dblH[n 2]+k+|γ|/parenrightbig , /ba∇dbl∂k t∂γ xu(t)/ba∇dblL2≤C(1+t)−n 2(1 m−1 2)−k−|γ| 2(/ba∇dblu0/ba∇dblLm+/ba∇dblu1/ba∇dblLm+/ba∇dblu0/ba∇dblHk+|γ|+/ba∇dblu1/ba∇dblHk+|γ|−1) (1.5) for 1≤m≤2,k∈Z≥0, andγ∈Zn ≥0, and applied them to semilinear problems. In particular, the above estimate implies (1+t)EL(t)+/ba∇dblu(t)/ba∇dbl2 L2 ≤C(1+t)−n(1 m−1 2)(/ba∇dblu0/ba∇dblLm+/ba∇dblu1/ba∇dblLm+/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2)2.(1.6) This indicates that the spatial decay of the initial data improves the time decay of the solution. Moreover, the decay rate in the estimates (1.5) suggeststhat th e solution of (1.3) is approximated by a solution of the corresponding heat equation ∂tv−∆v= 0, t >0,x∈Rn. This is the so-called diffusion phenomenon and firstly proved by Hsiao a nd Liu [18] for the hyperbolic conservation law with damping. There are many improvements and generalizations of the Matsumur a estimates and the diffusion phenomenon for (1.3). We refer the reader to [7, 1 7, 20, 21, 28, 33, 41, 44, 51, 55, 59, 61, 76, 78, 86, 99] and the references th erein.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 3 Next, we consider the initial boundary value problem of the linear wav e equation with space-dependent damping ∂2 tu−∆u+a(x)∂tu= 0, t > 0,x∈Ω, u(t,x) = 0, t > 0,x∈∂Ω, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(1.7) Mochizuki [56] firstly studied the case Ω = Rn(n/\e}atio\slash= 2) and showed that if a(x)≤ C/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αwithα >1, then the wave operator exists and is not identically vanishing. Namely, the energy EL(t) defined by (1.4) of the solution does not decay to zero in general, and the solution behaves like a solution of the wave equatio n without damping. This means that if the damping is sufficiently small at the spat ial infinity, then the energy ofthe solution does not decay to zero in general. H is result actually includesthetimeandspacedependentdamping, andgeneralizations inthedamping coefficientsand domainscan be found in Mochizuki and Nakazawa[57], Matsuyama [54], and Ueda [90]. On the other hand, for (1.7) with Ω = Rn, from the result by Matsumura [53], we see that if u0,u1∈C∞ 0(Rn) anda(x)≥C/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1, thenEL(t) decays to zero as t→ ∞(seealsoUesaka[91]). Theseresultsindicatethatforthedampingc oefficient a(x) =/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α, the value α= 1 is critical for the energy decay or non-decay. Regardingthe precise decayrate ofthe solution to (1.7), Todorov aand Yordanov [89] proved that if Ω = Rn,a(x) is positive, radial and satisfies a(x) =a0|x|−α+ o(|x|−α) (|x| → ∞) with some α∈[0,1), and the initial data has compact support, then the solution satisfies (1+t)EL(t)+/integraldisplay Rna(x)|u(t,x)|2dx≤C(1+t)−n−α 2−α+δ(/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2)2, whereδ >0 is arbitrary constant and Cdepends on δand the support of the data. We note that if we formally take α= 0 and δ= 0, then the decay rate coincides with that of (1.6). The proof of [89] is based on the weighted energy method with the weight function t−n−α 2−α+2δexp/parenleftbigg −/parenleftbiggn−α 2−α−δ/parenrightbiggA(x) t/parenrightbigg , whereA(x) is a solution of the Poisson equation ∆ A(x) =a(x). Such weight functions were firstly introduced by Ikehata and Tanizawa [36] and Ikehata [32] for damped wave equations. Some generalizations of the principal p art to variable coefficients were made by Radu, Todorova, and Yordanov [71, 72]. The assumption of the radial symmetry of a(x) was relaxed by Sobajima and the author [81]. More- over, in [83, 84], the compactness assumption on the support of th e initial data was removed and polynomially decaying data were treated. The point is th e use of a suitable supersolution of the corresponding heat equation a(x)∂tv−∆v= 0 having polynomial order in the far field. This approach is also a main too l in this paper. For the diffusion phenomenon, we refer the reader to [40, 6 8, 73, 74, 80, 82, 92]. When the damping coefficient is critical for the energy decay, the sit uation be- comes more delicate. Ikehata, Todorova, and Yordanov [38] stud ied (1.7) in the case where Ω = Rn(n≥3),a(x) satisfies a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1≤a(x)≤a1/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1with some4 Y. WAKASUGI a0,a1>0, and the initial data has compact support. They obtained the dec ay estimates EL(t) =/braceleftBigg O(t−a0) (1< a0< n), O(t−n+δ) (a0≥n) ast→ ∞with arbitrarysmall δ >0. This indicates that the decay rate depends on the constant a0. Similar results in the lower dimensional cases and the optimality of the above estimates under additional assumptions were also obt ained in [38]. We also mention that a(x) is not necessarily positive everywhere. It is known that the so-called geometric control condition (GCC) introduced b y Rauch and Taylor [75] and Bardos, Lebeau, and Rauch [2] is sufficient for the en ergy decay of solutions with initial data in the energy space. For the problem (1.7) w ith Ω =Rn, (GCC) is read as follows: There exist constants T >0 andc >0 such that for any (x0,ξ0)∈Rn×Sn−1, we have 1 T/integraldisplayT 0a(x0+sξ0)ds≥c. For this and related topics, we refer the reader to [1, 5, 9, 29, 45 , 58, 67, 68, 101]. We note that for a(x) =/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αwithα >0, (GCC) is not fulfilled. We note that for the linear wave equation with time-dependent damp ing ∂2 tu−∆u+b(t)∂tu= 0, the asymptotic behavior of the solution can be classified depending o n the behavior ofb(t). See [93, 94, 95, 96, 97, 98]. Thirdly, we consider the semilinear problem ∂2 tu−∆u+∂tu=f(u), t > 0,x∈Ω, u(t,x) = 0, t > 0,x∈∂Ω, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(1.8) Whenf(u) =|u|p−1uor±|u|pwithp >1, the nonlinearity works as a sourcing term and it may cause the singularity of the solution in a finite time. In t his case, it is known that there exists the critical exponent pF(n) = 1+2 n, that is, if p > pF(n), then (1.8) admits the global solution for small initial data; if p < pF(n), then the solution may blow up in finite time even for the small initial data. The num ber pF(n) is the so-called Fujita critical exponent named after the pioneerin g work by Fujita [10] for the semilinear heat equation. When Ω = Rnandf(u) =±|u|p, Todorova and Yordanov [87] determined the critical exponent for compactly supported initial data. Later on, Zhang [100] and Kirane and Qafsaoui [46] proved that the critical case p=pF(n) belongs to the blow-up case. There are many improvements and related studies to the results ab ove. The compactness assumption of the support of the initial data were re moved by [13, 20, 21, 36, 60]. The diffusion phenomenon for the global solution was proved by [11, 13, 42, 43]. The case where Ω is the half space or the exterior do main was studied by [24, 26, 30, 31, 69, 70, 77] Also, estimates of lifespan fo r blowing-up solutions were obtained by [48, 49, 62, 27, 22, 24, 23]. Whenf(u) =|u|p−1u, the global existence part can be proved completely the same way as in the case f(u) =±|u|p. However, regardingthe blow-up of solutions, the same proof as before works only for n≤3, since the fundamental solution ofSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 5 the linear damped wave equation is not positive for n≥4, which follows from the explicit formula of the linear wave equation (see e.g., [76, p.1011]). Ike hata and Ohta [35] obtained the blow-up of solutions for the subcritical case p < pF(n). The critical case p=pF(n) withn≥4 seems to remain open. Whenf(u) =−|u|p−1uwithp >1, the nonlinearity works as an absorbing term. In this case with Ω = Rn, Kawashima, Nakao, and Ono [44] proved the large data global existence. Moreover, decay estimates of solutions we re obtained for p >1 +4 n. Later on, Nishihara and Zhao [65] and Ikahata, Nishihara, and Zha o [34] studied the case 1 < p≤1+4 n. From their results, we have the energy estimate (1+t)E[u](t)+/ba∇dblu(t)/ba∇dbl2 L2≤C(I0)(1+t)−2(1 p−1−n 4), (1.9) where I0:=/integraldisplay Rn/parenleftbig |u1(x)|2+|∇u0(x)|2+|u0(x)|p+1+|u0(x)|2/parenrightbig /a\}b∇acketle{tx/a\}b∇acket∇i}ht2mdx, m > 2/parenleftbigg1 p−1−n 4/parenrightbigg and we recall that E[u](t) is defined by (1.2). Also, the asymptotic behavior was discussed by [41, 12, 15, 16, 34, 63]. There seems no result for ext erior domain cases. Finally, weconsiderthesemilinearproblemwithspace-dependentdam pingwhich is slightly more general than (1.1): ∂2 tu−∆u+a(x)∂tu=f(u), t > 0,x∈Ω, u(t,x) = 0, t > 0,x∈∂Ω, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω. When the nonlinearity works as a sourcing term, we expect that the re is the critical exponent as in the case a(x)≡1. Indeed, in the case where Ω = Rn,f(u) =±|u|p, the initial data has compact support, and a(x) is positive, radial, and satisfies a(x) =a0|x|−α+o(|x|−α) (|x| → ∞) withα∈[0,1), Ikehata, Todorova, and Yordanov [37] determined the critical exponent as pF(n−α) = 1 +2 n−α. The estimate of lifespan for blowing-up solutions was obtained in [24, 27]. T he blow-up of solutions for the case f(u) =|u|p−1useems to be an open problem. Recently, Sobajima [79] studied the critical damping case a(x) =a0|x|−1in an exterior domain Ω with n≥3, and proved the small data global existence of solutions under the conditions a0> n−2 andp >1+4 n−2+min{n,a0}. The blow-up part was investigated by [25, 50, 79]. In particular, when Ω is the ou tside a ball withn≥3,a0≥n, andf(u) =±|u|p, the critical exponent is determined as p=pF(n−1). Moreover, in Ikeda and Sobajima [25], the blow-up of solutions wa s obtained for Ω = Rn(n≥3), 0≤a0<(n−1)2 n+1,f(u) =±|u|pwithn n−1< p≤ pS(n+a0), where pS(n) is the positive root of the quadratic equation 2+(n+1)p−(n−1)p2= 0 and is the so-called Strauss exponent. We remark that pS(n+a0)> pF(n−1) holds ifa0<(n−1)2 n+1. From this, we can expect that the critical exponent changes depending on the value a0. For the absorbing nonlinear term f(u) =−|u|p−1uin the whole space case Ω =Rnwas studied by Todorova and Yordanov [88] and Nishihara [64]. In [64], for compactly supported initial data, the following two results were proved:6 Y. WAKASUGI (i) Ifa(x) =a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αwith some a0>0 andα∈[0,1), then we have (1+t)E[u](t)+/integraldisplay Rna(x)|u(t,x)|2dx≤C(1+t)−n−α 2−α+δ with arbitrary small δ >0; (ii) Ifa0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α≤a(x)≤a1/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αwith some a0,a1>0 andα∈[0,1), then we have (1+t)E[u](t)+/integraldisplay Rna(x)|u(t,x)|2dx≤C (1+t)−4 2−α(1 p−1−n−α 4)(p > psubc(n,α)), (1+t)−2 p−1log(2+t) (p=psubc(n,α)), (1+t)−2 p−1 (p < psubc(n,α)), where psubc(n,α) := 1+2α n−α. (1.10) We note that the decay rate in (i) is the same as that of the linear pro blem (1.7) and it is better than that of (ii) if p > pF(n−α). This means pF(n−α) is critical in the sense of the effect of the nonlinearity to the decay rate of the energy. Moreover, (ii) shows that the second critical exponent psubc(n,α) appears and it divides the decay rate of the energy. We also note that the estimate for the c asep > psubc(n,α) corresponds to the estimate (1.9). Thus, we may interpret the sit uation in the following way: When the damping is weak in the sense of a(x)∼ /a\}b∇acketle{tx/a\}b∇acket∇i}ht−αwith α∈(0,1), we cannot obtain the same type energy estimate as in (1.9) for a ll p >1, and the decay rate becomes worse under or on the second critic al exponent psubc(n,α). Our main goal in this paper is to give a generalization of the results ( i) and (ii) above. In recent years, semilinear wave equations with time-dependent da mping have been intensively studied. For the progress of this problem, we refe r the reader to Sections 1 and 2 in Lai, Schiavone, and Takamura [47]. We also refer to [66] and the references therein for a recent study of semilinear wave equa tions with time and space dependent damping. To state our results, we define the solution. Definition 1.1 (Mild and strong solutions) .LetAbe the operator A=/parenleftbigg0 1 ∆−a(x)/parenrightbigg defined on H:=H1 0(Ω)×L2(Ω)with the domain D(A) = (H2(Ω)∩H1 0(Ω))×H1 0(Ω). LetU(t)denote the C0-semigroup generated by A. Let(u0,u1)∈ HandT∈(0,∞]. A function u∈C([0,T);H1 0(Ω))∩C1([0,T);L2(Ω)) is called a mild solution of (1.1)on[0,T)ifU=t(u,∂tu)satisfies the integral equation U(t) =U(t)/parenleftbiggu0 u1/parenrightbigg +/integraldisplayt 0U(t−s)/parenleftbigg0 −|u|p−1u/parenrightbigg dsSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 7 inC([0,T);H). Moreover, when (u0,u1)∈D(A), a function u∈C([0,T);H2(Ω))∩C1([0,T);H1 0(Ω))∩C2([0,T);L2(Ω)) is said to be a strong solution of (1.1)on[0,T)ifusatisfies the equation of (1.1) inC([0,T);L2(Ω)). IfT=∞, we call ua global (mild or strong) solution. First, we prepare the existence and regularity of the global solutio n. Proposition 1.2. LetΩ =Rnwithn≥1, orΩ⊂Rnwithn≥2be an exterior domain with C2-boundary. Let a(x)∈C(Rn)be nonnegative and bounded. Let 1< p <∞(n= 1,2),1< p≤n n−2(n≥3), (1.11) and let(u0,u1)∈H1 0(Ω)×L2(Ω). Then, there exists a unique global mild solution u to(1.1). If we further assume (u0,u1)∈(H2(Ω)∩H1 0(Ω))×H1 0(Ω), thenubecomes a strong solution to (1.1). Remark 1.3. The assumption ∂Ω∈C2is used to ensure D(A) = (H2(Ω)∩ H1 0(Ω))×H1 0(Ω)(see Cazenave and Haraux [6, Remark 2.6.3] and Brezis [4, Theo- rem9.25] ). The restriction of the range of pin(1.11)is due to the use of Gagliardo– Nirenberg inequality (see Section A.2). The proof of Proposition 1.2 is standard. However, for reader’s co nvenience, we will give an outline of the proof in the appendix. To state our result, we recall that E[u](t) andpsubc(n,α) are defined by (1.2) and (1.10), respectively. The main result of this paper reads as follo ws. Theorem 1.4. LetΩ =Rnwithn≥1orΩ⊂Rnwithn≥2be an exterior domain with C2-boundary. Let psatisfy(1.11)and(u0,u1)∈H1 0(Ω)×L2(Ω), and letube the corresponding global mild solution of (1.1). Then, the followings hold. (i)Assume that a∈C(Rn)is positive and satisfies lim |x|→∞|x|αa(x) =a0 (1.12) with some constants α∈[0,1)anda0>0. Moreover, we assume that the initial data satisfy I0[u0,u1] :=/integraldisplay Ω/bracketleftbig (|u1(x)|2+|∇u0(x)|2+|u0(x)|p+1)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+|u0(x)|2/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α/bracketrightbig /a\}b∇acketle{tx/a\}b∇acket∇i}htλ(2−α)dx <∞ (1.13) with some λ∈[0,n−α 2−α). Then, we have (1+t)E[u](t)+/integraldisplay Ωa(x)|u(t,x)|2dx≤CI0[u0,u1](1+t)−λ fort≥0with some constant C=C(n,a,p,λ)>0. (ii)Assume that a∈C(Rn)is positive and satisfies a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α≤a(x)≤a1/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α8 Y. WAKASUGI with some constants α∈[0,1),a0,a1>0. Moreover, we assume that the initial data satisfy the condition I0[u0,u1]<∞with some λ∈[0,∞), where I0[u0,u1]is defined by (1.13). Then, we have (1+t)E[u](t)+/integraldisplay Ωa(x)|u(t,x)|2dx ≤C(I0[u0,u1]+1) × (1+t)−λ(λ <min{4 2−α(1 p−1−n−α 4),2 p−1}), (1+t)−λlog(2+t) (λ= min{4 2−α(1 p−1−n−α 4),2 p−1}, p/\e}atio\slash=psubc(n,α)), (1+t)−λ(log(2+ t))2(λ=4 2−α(1 p−1−n−α 4) =2 p−1,i.e., p=psubc(n,α)), (1+t)−4 2−α(1 p−1−n−α 4)(λ >4 2−α(1 p−1−n−α 4), p > p subc(n,α)), (1+t)−2 p−1log(2+t) (λ >2 p−1, p=psubc(n,α)), (1+t)−2 p−1 (λ >2 p−1, p < p subc(n,α)) fort≥0with some constant C=C(n,a,p,λ)>0. Remark 1.5. Under the assumptions of (i), the both conclusions of (i) and (ii) are true. In Figure 1, the decay rates of/integraldisplay Ωa(x)|u(t,x)|2dxis classified in the case (n,α) = (3,0.5)(for ease of viewing, the figure is multiplied by 7and0.75in the horizontal and vertical axis, respectively). Remark 1.6. From the proof of the above theorem, we also have the followin g estimates for the L2-norm of uwithout the weight a(x): Under the assumptions on (i) withλ∈[α 2−α,n−α 2−α), we have /integraldisplay Ω|u(t,x)|2dx≤C(1+t)−λ+α 2−α fort >0; Under the assumptions on (ii) with λ∈[α 2−α,∞), we have /integraldisplay Ω|u(t,x)|2dx ≤C (1+t)−λ+α 2−α (λ <min{4 2−α(1 p−1−n−α 4),2 p−1}), (1+t)−λ+α 2−αlog(2+t) (λ= min{4 2−α(1 p−1−n−α 4),2 p−1}, p/\e}atio\slash=psubc(n,α)), (1+t)−λ+α 2−α(log(2+ t))2(λ=4 2−α(1 p−1−n−α 4) =2 p−1,i.e., p=psubc(n,α)), (1+t)−4 2−α(1 p−1−n−α 4)+α 2−α(λ >4 2−α(1 p−1−n−α 4), p > p subc(n,α)), (1+t)−2 p−1+α 2−αlog(2+t) (λ >2 p−1, p=psubc(n,α)), (1+t)−2 p−1+α 2−α (λ >2 p−1, p < p subc(n,α)) fort >0. Remark 1.7. (i) Theorem 1.4 generalizes the result of Nishihara [64]to the exte- rior domain, general damping coefficient a(x)satisfying (1.12), and polynomially decaying initial data satisfying (1.13). (ii) For the simplest case Ω =Rnanda(x)≡1, the result of Theorem 1.4 (ii) extends that of Ikehata, Nishihara, and Zhao [34], in the sense that our estimate in the region λ >2/parenleftBig 1 p−1−n 4/parenrightBig coincides with their estimate (1.9). Moreover, theSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 9 n−α 2−αn−α α 1 psubc(n,α) pF(n−α)pλ (1+t)−λ(1+t)−4 2−α(1 p−1−n−α 4)(1+t)−2 p−1 (1+t)−λlog(2+t) (1+t)−λlog(2+t)(1+t)−2 p−1log(2+t) (1+t)−λ(log(2+ t))2 (1+t)−n−α 2−α+δλ=4 2−α(1 p−1−n−α 4) λ=2 p−1 Figure 1. Classification of decay rates in p-λplane when ( n,α) = (3,1 2) result of Theorem 1.4 (i) in the case p > pF(n)is better than the estimate obtained in[34]. Hence, our result still has a novelty. Remark 1.8. The optimality of the decay rates in Theorem 1.4 is an open pro blem. We expect that the estimate in the case (i) is optimal if p > pF(n−α) = 1+2 n−α, since the decay rate is the same as that of the linear problem (1.7)obtained by [84]. On the other hand, in the critical case p=pF(n−α), the estimates in Theorem 1.4 will be improved in view of the known results [15, 16]for the classical damping (1.8) in the whole space. Moreover, the optimality in the subcriti cal casep < pF(n−α) is a difficult problem even when a(x)≡1andΩ =Rn, and we have no idea so far. The strategy of the proof of Theorem 1.4 is as follows. For the both parts (i) and (ii), we apply the weighted energy method. The difficulty is how to e stimate the weighted L2-norm of the solution. To overcome it, we take different approache s for (i) and (ii). First, for the part (i), we apply the weighted energy method developed by [83, 84]. We shall use a suitable supersolution of the cor responding heat equation a(x)∂tv−∆v= 0 as the weight function. Next, for the part (ii), we shall use the same type weight function as in Ikehata, Nishihara, an d Zhao [34] with a modification to fit the space-dependent damping case. In this cas e the absorbing semilinear term helps to estimate the weighted L2-norm of the solution.10 Y. WAKASUGI The rest of the paper is organized in the following way. In the next se ction, we prepare the definitions and properties of the weight functions use d in the proof. Sections 3 and 4 are devoted to the proof of Theorem 1.4 (i) and (ii), respectively. In Appendix A, we give a proof of Proposition 1.2. Finally, in Appendix B, we prove the properties of weight functions stated in Section 2. We end up this section with introducing notations used throughout t his paper. The letter Cindicates a generic positive constant, which may change from line to line. In particular, C(∗,···,∗) denotes a constant depending only on the quantities in the parentheses. For x= (x1,...,x n)∈Rn, we define /a\}b∇acketle{tx/a\}b∇acket∇i}ht=/radicalbig 1+|x|2. We sometimes use BR(x0) ={x∈Rn;|x−x0|< R}forR >0 andx0∈Rn. LetLp(Ω) be the usual Lebesgue space equipped with the norm /ba∇dblf/ba∇dblLp= /parenleftbigg/integraldisplay Ω|f(x)|pdx/parenrightbigg1/p (1< p <∞), esssup x∈Ω|f(x)| (p=∞). In particular, L2(Ω) is a Hilbert space with the innerproduct (f,g)L2:=/integraldisplay Ωf(x)g(x)dx. LetHk(Ω) with a nonnegative integer kbe the Sobolev space equipped with the innerproduct and the norm (f,g)Hk=/summationdisplay |α|≤k(∂αf,∂αg)L2,/ba∇dblf/ba∇dblHk=/radicalbig (f,f)Hk, respectively. C∞ 0(Ω) denotes the space of smooth functions on Ω with compact support. Hk 0(Ω) is the completion of C∞ 0(Ω) with respect to the norm /ba∇dbl·/ba∇dblHk. For an interval I⊂R, a Banach space X, and a nonnegative integer k,Ck(I;X) stands for the space of k-times continuously differentiable functions from ItoX. 2.Preliminaries In this section, we prepare weight functions for the weighted ener gy method used in the proof of Theorem 1.4. These lemmas were shown in [77, 81, 83, 84], however, for the conve nience, we give a proof of them in the appendix. Following [81], we first take a suitable approximate solution of the Poiss on equa- tion ∆A(x) =a(x), which will be used for the construction of the weight function. Lemma 2.1 ([81, 84]) .Assume that a(x)∈C(Rn)is positive and satisfies the condition lim|x|→∞|x|αa(x) =a0with some constants α∈(−∞,min{2,n})and a0>0. Letε∈(0,1). Then, there exist a function Aε∈C2(Rn)and positive constants c=c(n,a,ε)andC=C(n,a,ε)such that for x∈Rn, we have (1−ε)a(x)≤∆Aε(x)≤(1+ε)a(x), (2.1) c/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α≤Aε(x)≤C/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α, (2.2) |∇Aε(x)|2 a(x)Aε(x)≤2−α n−α+ε. (2.3) Forthe constructionofourweightfunction, wealsoneed the follow ingKummer’s confluent hypergeometric function.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 11 Definition 2.2 (Kummer’s confluent hypergeometric functions) .Forb,c∈Rwith −c /∈N∪{0}, Kummer’s confluent hypergeometric function of first kind is defined by M(b,c;s) =∞/summationdisplay n=0(b)n (c)nsn n!, s∈[0,∞), where(d)nis the Pochhammer symbol defined by (d)0= 1and(d)n=/producttextn k=1(d+ k−1)forn∈N; note that when b=c,M(b,b;s)coincides with es. Forε∈(0,1/2), we define /tildewideγε=/parenleftbigg2−α n−α+2ε/parenrightbigg−1 , γε= (1−2ε)/tildewideγε. (2.4) Definition 2.3. Forβ∈R, define ϕβ,ε(s) =e−sM(γε−β,γε;s), s≥0. SinceM(γε,γε,s) =es, we remark that ϕ0,ε(s)≡1. Roughly speaking, if we formally take ε= 0, then {ϕβ,0}β∈Rgives a family of self-similar profiles of the equation |x|−α∂tv= ∆vwith the parameter β. See [83] for more detailed explanation. The next lemma states basic properties of ϕβ,ε. Lemma 2.4. The function ϕβ,εdefined in Definition 2.3 satisfies the following properties. (i)ϕβ,ε(s)satisfies the equation sϕ′′(s)+(γε+s)ϕ′(s)+βϕ(s) = 0. (2.5) (ii)If0≤β < γε, thenϕβ,ε(s)satisfies the estimates kβ,ε(1+s)−β≤ϕβ,ε(s)≤Kβ,ε(1+s)−β with some constants kβ,ε,Kβ,ε>0. (iii)For every β≥0,ϕβ,ε(s)satisfies |ϕβ,ε(s)| ≤Kβ,ε(1+s)−β with some constant Kβ,ε>0. (iv)For every β∈R,ϕβ,ε(s)andϕβ+1,ε(s)satisfy the recurrence relation βϕβ,ε(s)+sϕ′ β,ε(s) =βϕβ+1,ε(s). (v)For every β∈R, we have ϕ′ β,ε(s) =−β γεe−sM(γε−β,γε+1;s), ϕ′′ β,ε(s) =β(β+1) γε(γε+1)e−sM(γε−β,γε+2;s). In particular, if 0< β < γ ε, thenϕ′ β,ε(s)andϕ′′ β,ε(s)satisfy −Kβ,ε(1+s)−β−1≤ϕ′ β,ε(s)≤ −kβ,ε(1+s)−β−1, kβ,ε(1+s)−β−2≤ϕ′′ β,ε(s)≤Kβ,ε(1+s)−β−2 with some constants kβ,ε,Kβ,ε>0. Finally, we define the weight function which will be used for our energy method.12 Y. WAKASUGI Definition 2.5. Forβ∈Rand(x,t)∈Rn×[0,∞), we define Φβ,ε(x,t;t0) = (t0+t)−βϕβ,ε(z), z=/tildewideγεAε(x) t0+t, whereε∈(0,1/2),/tildewideγεis the constant given in (2.4),t0≥1,ϕβ,εis the function defined by Definition 2.3, and Aε(x)is the function constructed in Lemma 2.1. Sinceϕ0,ε(s)≡1, we again remark that Φ 0,ε(x,t;t0)≡1. Fort0≥1,t >0, andx∈Rn, we also define Ψ(x,t;t0) :=t0+t+Aε(x). (2.6) Proposition 2.6. The function Φβ,ε(x,t;t0)satisfies the following properties: (i)For every β≥0, we have ∂tΦβ,ε(x,t;t0) =−βΦβ+1,ε(x,t;t0). (ii)Ifβ≥0, then there exists a constant C=C(n,α,β,ε)>0such that |Φβ,ε(x,t;t0)| ≤CΨ(x,t;t0)−β for any(x,t)∈Rn×[0,∞). (iii)If0≤β < γε, then there exists a constant c=c(n,α,β,ε)>0such that Φβ,ε(x,t;t0)≥cΨ(x,t;t0)−β for any(x,t)∈Rn×[0,∞). (iv)Forβ >0, there exists a constant c=c(n,α,β,ε)>0such that a(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0)≥ca(x)Ψ(x,t;t0)−β−1 for any(x,t)∈Rn×[0,∞). Finally, we prepare a useful lemma for our weighted energy method. The proof can be found in [83, Lemma 3.6] or [77, Lemma 2.5]. However, for the c onvenience, we give its proof in the appendix. Lemma 2.7. LetΩ =Rnwithn≥1orΩ⊂Rnwithn≥2be an exterior domain withC2-boundary. Let Φ∈C2(Ω)be a positive function and let δ∈(0,1/2). Then, for anyu∈H2(Ω)∩H1 0(Ω)satisfying suppu∈BR(0) ={x∈Rn;|x|< R}with someR >0, we have /integraldisplay Ω(u∆u)Φ−1+2δdx≤ −δ 1−δ/integraldisplay Ω|∇u|2Φ−1+2δdx+1−2δ 2/integraldisplay Ωu2(∆Φ)Φ−2+2δdx. 3.Proof of Theorem 1.4: first part In this section, we prove Theorem 1.4 (i). First, we note that Propo sition 1.2 implies the existence of the global mild solution u. Following the argument in Sobajima [79], we first prove Theorem 1.4 (i) in the case of compactly supported initial data, and after that, we will tr eat the general case by an approximation argument.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 13 3.1.Proof for the compactly supported initial data. We first consider the case where the initial data are compactly supported, that is, we as sume that suppu0∪suppu1⊂BR0(0) ={x∈Rn;|x|< R0}. Then, by the finite prop- agation property (see Section A.2.7), the corresponding mild solutio nusatisfies suppu(t,·)⊂BR0+t(0). LetT0>0 be arbitrary fixed and let T∈(0,T0). Then, we have supp u(t,·)⊂ BR0+T0(0) for all t∈[0,T]. LetD= Ω∩BR0+T0(0). Then, for t∈[0,T], we can convert the problem (1.1) to the problem in the bounded domain ∂2 tu−∆u+a(x)∂tu+|u|p−1u= 0, t∈(0,T],x∈D, u(t,x) = 0, t ∈(0,T],x∈∂D, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈D with (u0,u1)∈ HD:=H1 0(D)×L2(D). LetADbe the operator AD=/parenleftbigg0 1 ∆−a(x)/parenrightbigg defined on HDwith the domain D(AD) = (H2(D)∩H1 0(D))×H1 0(D). Then, from the argument in Section A.1, there exists λ∗>0 such that for any λ > λ ∗, the resolvent Jλ= (I−λ−1AD)−1is defined as a bounded operator on HD. Take a sequence {λj}∞ j=1such that λj> λ∗forj≥1 and lim j→∞λj=∞, and define /parenleftBigg u(j) 0 u(j) 1/parenrightBigg :=Jλj/parenleftbiggu0 u1/parenrightbigg . Then, we have (u(j) 0,u(j) 1)∈D(AD),lim j→∞(u(j) 0,u(j) 1) = (u0,u1) inHD (3.1) (see e.g. the proof of [19, Theorem 2.18]). Therefore, Proposition 1.2 shows that the mild solution u(j)corresponding to the initial data ( u(j) 0,u(j) 1) becomes a strong solution. Moreover, the continuous dependence on the initial data (see Section A.2.4) implies lim j→∞sup t∈[0,T]/ba∇dbl(u(j)(t),∂tu(j)(t))−(u(t),∂tu(t))/ba∇dblHD= 0. This means that, if we prove the conclusion of Theorem 1.4 (i) for u(j), that is, (1+t)E[u(j)](t)+/integraldisplay Ωa(x)|u(j)(t,x)|2dx≤CI0[u(j) 0,u(j) 1](1+t)−λ fort∈[0,T], where the constant Cis independent of j,T,T0,R0, then letting j→ ∞and also using the Sobolev embedding /ba∇dblu/ba∇dblLp+1(D)≤C/ba∇dblu/ba∇dblH1(D), we have the same estimate for the original mild solution u. Note that (3.1) implies limj→∞I0[u(j) 0,u(j) 1] =I0[u0,u1], sincethe integralis takenoverthe bounded region D. Finally, since TandT0are arbitrary and Cis independent of them, we obtain the desired energy estimate for any t≥0. Therefore, in the following argument, we may further assume ( u0,u1)∈D(AD) anduis the strong solution. This enables us to justify all the computation s in this section.14 Y. WAKASUGI In what follows, we shall use the weight functions Φ β,ε(x,t;t0) and Ψ( x,t;t0) defined by Definition 2.5 and (2.6), respectively. We also recall that t he constant γεis given by (2.4). Then, we define the following energies. Definition 3.1. For a function u=u(t,x),α∈[0,1),δ∈(0,1/2),ε∈(0,1/2), λ∈[0,(1−2δ)γε),β=λ/(1−2δ),ν >0, andt0≥1, we define E1(t;t0,λ) =/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig +1 p+1|u(t,x)|p+1/bracketrightbigg Ψ(t,x;t0)λ+α 2−αdx, E0(t;t0,λ) =/integraldisplay Ω/parenleftbig 2u(t,x)∂tu(t,x)+a(x)|u(t,x)|2/parenrightbig Φβ,ε(t,x;t0)−1+2δdx, E∗(t;t0,λ,ν) =E1(t;t0,λ)+νE0(t;t0,λ), ˜E(t;t0,λ) = (t0+t)/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig +1 p+1|u(t,x)|p+1/bracketrightbigg Ψ(t,x;t0)λdx fort≥0. Since 2u∂tu≤a(x) 2|u|2+2 a(x)|∂tu|2≤a(x) 2|u|2+CΨα 2−α|∂tu|2(3.2) and Φ−1+2δ β,ε≤CΨλ(see (2.2) and Proposition 2.6 (iii)), we see that there exists a small constant ν0=ν0(n,a,δ,ε,λ )>0 such that for any ν∈(0,ν0), E∗(t;t0,λ,ν)≥1 2E1(t;t0,λ)+ν 2/integraldisplay Ωa(x)|u(t,x)|2Ψ(t,x;t0)λdx(3.3) holds. We first prepare the following energy estimates for E1(t;t0,λ) andE0(t;t0,λ). Lemma3.2. Under the assumptions on Theorem 1.4 (i), there exists t1=t1(n,a,λ,ε)≥ 1such that for t0≥t1andt >0, we have d dtE1(t;t0,λ)≤ −1 2/integraldisplay Ωa(x)|∂tu(t,x)|2Ψ(t,x;t0)λ+α 2−αdx +C/integraldisplay Ω/parenleftbig |∇u(t,x)|2+|u(t,x)|p+1/parenrightbig Ψ(t,x;t0)λ+α 2−α−1dx with some constant C=C(n,α,p,λ)>0. Proof.Differentiating E1(t;t0,λ), one has d dtE1(t;t0,λ) =/integraldisplay Ω/bracketleftbig ∂tu∂2 tu+∇u·∇∂tu+|u|p−1u∂tu/bracketrightbig Ψλ+α 2−αdx +/parenleftbigg λ+α 2−α/parenrightbigg/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∇u|2+|∂tu|2/parenrightbig +1 p+1|u|p+1/bracketrightbigg Ψλ+α 2−α−1dx.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 15 The integration by parts and the equation (1.1) imply d dtE1(t;t0,λ) =−/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx −/parenleftbigg λ+α 2−α/parenrightbigg/integraldisplay Ω∂tu(∇u·∇Ψ)Ψλ+α 2−α−1dx +/parenleftbigg λ+α 2−α/parenrightbigg/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∇u|2+|∂tu|2/parenrightbig +1 p+1|u|p+1/bracketrightbigg Ψλ+α 2−α−1dx. (3.4) Let us estimate the right-hand side. First, the Schwarz inequality g ives /vextendsingle/vextendsingle/vextendsingle/vextendsingle−/parenleftbigg λ+α 2−α/parenrightbigg ∂tu(∇u·∇Ψ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤a(x) 4|∂tu|2Ψ+C|∇u|2|∇Ψ|2 a(x)Ψ. Moreover, by (2.3), we have |∇Ψ|2 a(x)Ψ≤|∇Aε(x)|2 a(x)Aε(x)≤2−α n−α+ε. (3.5) Also, from the definition of Ψ, (2.2), and a(x)∼ /a\}b∇acketle{tx/a\}b∇acket∇i}ht−α, one obtains Ψ(t,x;t0)−1≤t−1+α 2−α 0Aε(x)−α 2−α≤Ct−2(1−α) 2−α 0a(x). (3.6) Therefore, taking t1≥1 sufficiently large, we have, for t0≥t1, /parenleftbigg λ+α 2−α/parenrightbigg/integraldisplay Ω|∂tu|2Ψλ+α 2−α−1dx≤1 4/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx. Using the above estimates to (3.4), we deduce d dtE1(t;t0,λ)≤ −1 2/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx +C/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλ+α 2−α−1dx, which completes the proof. /square Lemma 3.3. Under the assumptions on Theorem 1.4 (i), for t0≥1andt >0, we have d dtE0(t;t0,λ)≤ −η/integraldisplay Ω/parenleftbig |∇u(t,x)|2+|u(t,x)|p+1/parenrightbig Ψ(t,x;t0)λdx +C/integraldisplay Ω|∂tu(t,x)|2Ψ(t,x;t0)λdx with some positive constants η=η(n,α,δ,ε,λ )andC=C(n,α,δ,ε,λ ). Proof.Differentiating E0(t;t0,λ) and using the equation (1.1) yield d dtE0(t;t0,λ) =/integraldisplay Ω/parenleftbig 2|∂tu|2+2u∂2 tu+2a(x)u∂tu/parenrightbig Φ−1+2δ β,εdx −(1−2δ)/integraldisplay Ω/parenleftbig 2u∂tu+a(x)|u|2/parenrightbig Φ−2+2δ β,ε∂tΦβ,εdx.16 Y. WAKASUGI Using the equation (1.1), we have d dtE0(t;t0,λ) = 2/integraldisplay Ω|∂tu|2Φ−1+2δ β,εdx+2/integraldisplay Ωu∆uΦ−1+2δ β,εdx −2/integraldisplay Ω|u|p+1Φ−1+2δ β,εdx −(1−2δ)/integraldisplay Ω/parenleftbig 2u∂tu+a(x)|u|2/parenrightbig Φ−2+2δ β,ε∂tΦβ,εdx. Applying Lemma 2.7 with Φ = Φ β,εto the second term of the right-hand side, one obtains d dtE0(t;t0,λ)≤2/integraldisplay Ω|∂tu|2Φ−1+2δ β,εdx−2δ 1−δ/integraldisplay Ω|∇u|2Φ−1+2δ β,εdx −2/integraldisplay Ω|u|p+1Φ−1+2δ β,εdx −2(1−2δ)/integraldisplay Ωu∂tuΦ−2+2δ β,ε∂tΦβ,εdx −(1−2δ)/integraldisplay Ω|u|2Φ−2+2δ β,ε(a(x)∂tΦβ,ε−∆Φβ,ε)dx.(3.7) Next, we estimate the terms in the right-hand side. First, we remar k that if λ= 0 (i.e.,β= 0), then the last two terms in (3.7) vanish, since Φ β,ε≡1. For the case β >0, by Proposition 2.6 (ii) and (iv), we have /integraldisplay Ω|u|2Φ−2+2δ β,ε(a(x)∂tΦβ,ε−∆Φβ,ε)dx≥η1/integraldisplay Ωa(x)|u|2Ψλ−1dx with some constant η1=η1(n,α,δ,ε,λ )>0. Moreover, Proposition 2.6 (i), (ii), and (iii) imply |u∂tuΦ−2+2δ β,ε∂tΦβ,ε| ≤C|u||∂tu||Φ−2+2δ β,ε||Φβ+1,ε| ≤C|u||∂tu|Ψλ−1. This and the Schwarz inequality lead to /vextendsingle/vextendsingle/vextendsingle/vextendsingle2(1−2δ)/integraldisplay Ωu∂tuΦ−2+2δ β,ε∂tΦβ,εdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤C/integraldisplay Ω|u||∂tu|Ψλ−1dx ≤C/parenleftbigg/integraldisplay Ωa(x)|u|2Ψλ−1dx/parenrightbigg1/2/parenleftbigg/integraldisplay Ωa(x)−1|∂tu|2Ψλ−1dx/parenrightbigg1/2 ≤η1 2/integraldisplay Ωa(x)|u|2Ψλ−1dx+C/integraldisplay Ω|∂tu|2Ψλdx with some C=C(n,a,δ,ε,λ )>0. Summarizing the above computations, we see that for both cases λ= 0 and λ >0, the last two terms of (3.7) can be estimatedSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 17 as −2(1−2δ)/integraldisplay Ωu∂tuΦ−2+2δ β,ε∂tΦβ,εdx −(1−2δ)/integraldisplay Ω|u|2Φ−2+2δ β,ε(a(x)∂tΦβ,ε−∆Φβ,ε)dx ≤C/integraldisplay Ω|∂tw|2Ψλdx. Finally, from Proposition 2.6 (ii) and (iii), one obtains 2/integraldisplay Ω|∂tu|2Φ−1+2δ β,εdx≤C/integraldisplay Ω|∂tu|2Ψλdx and 2δ 1−δ/integraldisplay Ω|∇u|2Φ−1+2δ β,εdx+2/integraldisplay Ω|u|p+1Φ−1+2δ β,εdx≥η/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdx with some positive constants C=C(n,α,δ,ε,λ ) andη=η(n,α,δ,ε,λ ). Putting this all together, we deduce from (3.7) that d dtE0(t;t0,λ)≤ −η/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdx +C/integraldisplay Ω|∂tu|2Ψλdx, and the proof is complete. /square Combining Lemmas3.2 and3.3, we havethe followingestimate for E∗(t;t0,λ,ν). Lemma 3.4. Under the assumptions on Theorem 1.4 (i), there exist consta nts ν∗=ν∗(n,a,δ,ε,λ )∈(0,ν0)andt2=t2(n,a,p,δ,ε,λ,ν ∗)≥1such that for t0≥t2 andt >0, we have E∗(t;t0,λ,ν∗)+/integraldisplayt 0/integraldisplay Ωa(x)|∂tu(s,x)|2Ψ(s,x;t0)λ+α 2−αdxds +/integraldisplayt 0/integraldisplay Ω(|∇u(s,x)|2+|u(s,x)|p+1)Ψ(s,x;t0)λdxds ≤CE∗(0;t0,λ,ν∗) with some constant C=C(n,a,δ,ε,λ,ν ∗)>0. Proof.Letν∈(0,ν0), where ν0is taken so that (3.2) holds. From the definition of E∗(t;t0,λ,ν) and Lemmas 3.2 and 3.3, one has d dtE∗(t;t0,λ,ν) =d dtE1(t;t0,λ)+νd dtE0(t;t0,λ) ≤ −1 2/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx +C/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλ+α 2−α−1dx −νη/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdx +Cν/integraldisplay Ω|∂tu|2Ψλdx (3.8)18 Y. WAKASUGI fort0≥t1andt >0, where t1≥1 is determined in Lemma 3.2. Noting that (1.12) and (2.2) imply |∂tu|2Ψλ≤C/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αAε(x)α 2−α|∂tu|2Ψλ≤Ca(x)|∂tu|2Ψλ+α 2−α with some constant C=C(n,a,α,ε)>0, and taking ν=ν∗with sufficiently small ν∗∈(0,ν0), we deduce −1 2/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx+Cν∗/integraldisplay Ω|∂tu|2Ψλdx≤ −1 4/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx. Next, by Ψα 2−α−1≤(t0+t)α 2−α−1and taking t2≥t1sufficiently large depending onν∗, one obtains C/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλ+α 2−α−1dx−ν∗η/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdx ≤ −ν∗η 2/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdx fort0≥t2. Finally, plugging the above estimates into (3.8) with ν=ν∗, we conclude d dtE∗(t;t0,λ,ν∗)≤ −1 4/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx −ν∗η 2/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdx fort0≥t2andt >0. Integrating it over [0 ,t], we have the desired estimate. /square Lemma 3.5. Under the assumptions on Theorem 1.4 (i), there exists a cons tant t2=t2(n,a,p,δ,ε,λ )≥1such that for t0≥t2andt >0, we have ˜E(t;t0,λ)+/integraldisplay Ωa(x)|u(t,x)|2Ψ(t,x;t0)λdx≤CI0[u0,u1] with some constant C=C(n,a,p,δ,ε,λ,ν ∗,t0)>0. Proof.Take the same constants ν∗andt2as in Lemma 3.4. The integration by parts and the equation (1.1) imply d dt˜E(t;t0,λ) =/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∂tu|2+|∇u|2/parenrightbig +1 p+1|u|p+1/bracketrightbigg (Ψ+λ(t0+t))Ψλ−1dx +(t0+t)/integraldisplay Ω/parenleftbig ∂tu∂2 tu+∇u·∇∂tu+|u|p−1u∂tu/parenrightbig Ψλdx =/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∂tu|2+|∇u|2/parenrightbig +1 p+1|u|p+1/bracketrightbigg (Ψ+λ(t0+t))Ψλ−1dx −(t0+t)/integraldisplay Ωa(x)|∂tu|2Ψλdx−λ(t0+t)/integraldisplay Ω∂tu(∇u·∇Ψ)Ψλ−1dx. The last term of the right-hand side is estimated as −λ(t0+t)/integraldisplay Ω∂tu(∇u·∇Ψ)Ψλ−1dx≤η(t0+t)/integraldisplay Ωa(x)|∂tu|2|∇Ψ|2 a(x)Ψλ−1dx +C(t0+t)/integraldisplay Ω|∇u|2Ψλ−1dxSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 19 for anyη >0. Using (3.5) and taking η=η(n,α,ε) sufficiently small, we have d dt˜E(t;t0,λ)≤C/integraldisplay Ω/parenleftbig |∂tu|2+|∇u|2+|u|p+1/parenrightbig (Ψ+(t0+t))Ψλ−1dx −1 2(t0+t)/integraldisplay Ωa(x)|∂tu|2Ψλdx. Notingt0+t≤Ψ anda(x)−1≤CΨα 2−α, we estimate /integraldisplay Ω|∂tu|2(Ψ+λ(t0+t))Ψλ−1dx≤C/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdx. Therefore, integrating over [0 ,t] yield ˜E(t;t0,λ)+1 2/integraldisplayt 0(t0+s)/integraldisplay Ωa(x)|∂tu|2Ψλdxds ≤˜E(0;t0,λ)+C/integraldisplayt 0/integraldisplay Ωa(x)|∂tu|2Ψλ+α 2−αdxds+C/integraldisplayt 0/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdxds. Now, we multiply the both sides of above inequality by a sufficiently small constant µ >0, and add it and the conclusion of Lemma 3.4. Then, we obtain µ˜E(t;t0,λ)+E∗(t;t0,λ,ν∗) +/integraldisplayt 0/integraldisplay Ωa(x)|∂tu|2/bracketleftBigµ 2(t0+s)+(1−Cµ)Ψα 2−α/bracketrightBig Ψλdxds +(1−Cµ)/integraldisplayt 0/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdxds ≤µ˜E(0;t0,λ)+CE∗(0;t0,λ,ν∗) (3.9) fort0≥t2andt >0. Let us take µsufficiently small so that 1 −Cµ >0. Then, the last three terms in the left-hand side can be dropped. Finally, from t he definitions ofE∗(t;t0,λ) and˜E(t;t0,λ), we can easily verify µ˜E(0;t0,λ)+E∗(0;t0,λ,ν∗)≤CI0[u0,u1] with some constant C=C(a,p,λ,t 0)>0. Thus, we conclude ˜E(t;t0,λ)+E∗(t;t0,λ,ν∗)≤CI0[u0,u1] fort0≥t2andt >0. This and the lower bound (3.3) of E∗(t;t0,λ,ν∗) give the desired estimate. /square Proof of Theorem 1.4 (i) for compactly supported initial dat a.Takeλ∈[0,n−α 2−α) asin theassumption(1.13), andthen choose δ,ε∈(0,1/2)sothat λ∈[0,(1−2δ)γε) holds. Moreover, take the same constants ν∗andt2as in Lemmas 3.4 and 3.5. By (3.3), Lemmas 3.4 and 3.5, Definition 3.1, and ( t0+t)λ≤Ψλ, we have (t0+t)λ+1E[u](t)+(t0+t)λ/integraldisplay Ωa(x)|u(t,x)|2dx≤CI0[u0,u1] (3.10) fort0≥t2andt >0 with some constant C=C(n,a,p,δ,ε,λ,ν ∗,t0)>0. This completes the proof. /square20 Y. WAKASUGI Remark 3.6. From(3.9), we have a slightly more general estimate /integraldisplay Ω/parenleftbig |∂tu|2+|∇u|2+|u|p+1/parenrightbig/bracketleftbig (t0+t)+Ψα 2−α/bracketrightbig Ψλ+/integraldisplay Ωa(x)|u|2Ψλdx +/integraldisplayt 0/integraldisplay Ωa(x)|∂tu|2/bracketleftbig (t0+s)+Ψα 2−α/bracketrightbig Ψλdxds +/integraldisplayt 0/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Ψλdxds ≤CI0[u0,u1] fort0≥t2andt >0. Moreover, from the proof of Lemma 3.3, we can add the term/integraltextt 0/integraltext Ωa(x)|u|2Ψλ−1dxdsto the left-hand side when λ >0. 3.2.Proof for the general case. Here, we give a proof of Theorem 1.4 (i) for non-compactly supported initial data. Let (u0,u1)∈H1 0(Ω)×L2(Ω) satisfy I0[u0,u1]<∞and letube the corre- sponding mild solution to (1.1). We take a cut-off function χ∈C∞ 0(Rn) such that 0≤χ(x)≤1 (x∈Rn), χ(x) =/braceleftBigg 1 (|x| ≤1), 0 (|x| ≥2). For each j∈N, we define χj(x) =χ(x/j). Then, we have 0≤χj(x)≤1 (x∈Rn), χj(x) =/braceleftBigg 1 (|x| ≤j), 0 (|x| ≥2j), |∇χj(x)| ≤C j(x∈Rn),supp∇χj⊂B2j(0)\Bj(0), where the constant Cis independent of j. Let (u(j) 0,u(j) 1) = (χju0,χju1) and let u(j)be the corresponding mild solution to (1.1). First, by definition, it is easily seen that lim j→∞(u(j) 0,u(j) 1) = (u0,u1) inH1 0(Ω)×L2(Ω). Therefore, the continuous dependence on the initial data (see Se ction A.2.4) yields lim j→∞(u(j)(t),∂tu(j)(t)) = (u(t),∂tu(t)) inC([0,T];H1 0(Ω))∩C1([0,T];L2(Ω)) for any fixed T >0. From this and the Sobolev embedding, we deduce lim j→∞E[u(j)](t) =E[u](t) (3.11) for anyt≥0. We next show lim j→∞I0[u(j) 0,u(j) 1] =I0[u0,u1]. (3.12) To prove this, we use the notation I0[u0,u1;D] :=/integraldisplay D/bracketleftbig (|u1(x)|2+|∇u0(x)|2+|u0(x)|p+1)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+|u0(x)|2/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α/bracketrightbig /a\}b∇acketle{tx/a\}b∇acket∇i}htλ(2−α)dxSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 21 for a region D⊂Ω. Using the properties of χjdescribed above and |∇(χju0)|2=χ2 j|∇u0|2+2(∇χj·∇u0)χju0+|∇χj|2|u0|2, we calculate |I0[u0,u1]−I0[u(j) 0,u(j) 1]| ≤I0[u0,u1;Ω\Bj(0)] +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay B2j(0)\Bj(0)2(∇χj·∇u0)χju0/a\}b∇acketle{tx/a\}b∇acket∇i}htα+λ(2−α)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +/integraldisplay B2j(0)\Bj(0)|∇χj|2|u0|2/a\}b∇acketle{tx/a\}b∇acket∇i}htα+λ(2−α)dx.(3.13) The Schwarz inequality gives /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay B2j(0)\Bj(0)2(∇χj·∇u0)χju0/a\}b∇acketle{tx/a\}b∇acket∇i}htα+λ(2−α)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤I0[u0,u1;Ω\Bj(0)]+/integraldisplay B2j(0)\Bj(0)|∇χj|2|u0|2/a\}b∇acketle{tx/a\}b∇acket∇i}htα+λ(2−α)dx. Furthermore, using the estimate of ∇χj, one sees that /integraldisplay B2j(0)\Bj(0)|∇χj|2|u0|2/a\}b∇acketle{tx/a\}b∇acket∇i}htα+λ(2−α)dx ≤Cj−2(1+|2j|2)α/integraldisplay B2j(0)\Bj(0)|u0|2/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α+λ(2−α)dx ≤CI0[u0,u1;Ω\Bj(0)], where the constant Cis independent of j. Putting this all together into (3.13), we have |I0[u0,u1]−I0[u(j) 0,u(j) 1]| ≤CI0[u0,u1;Ω\Bj(0)]. SinceI0[u0,u1]<∞, the right-hand side tends to zero as j→ ∞. This proves (3.12). Now we are at the position to proof Theorem 1.4 (i). Proof of Theorem 1.4 (i) for the general case. Takethesameconstant t2asinLem- mas 3.4 and 3.5. Let {(u(j) 0,u(j) 1)}∞ j=1be the sequence defined above and let u(j) be the corresponding mild solution to (1.1) with the initial data ( u(j) 0,u(j) 1). Since each (u(j) 0,u(j) 1) has the compact support, one can apply the result (3.10) in the previous subsection to obtain (t0+t)λ+1E[u(j)](t)+(t0+t)λ/integraldisplay Ωa(x)|u(j)(t,x)|2dx≤CI0[u(j) 0,u(j) 1] fort0≥t2andt >0. Finally, using (3.11) and (3.12), we have (t0+t)λ+1E[u](t)+(t0+t)λ/integraldisplay Ωa(x)|u(t,x)|2dx≤CI0[u0,u1] fort0≥t2andt >0, which completes the proof. /square22 Y. WAKASUGI 4.Proof of Theorem 1.4: second part In this section, we prove Theorem 1.4 (ii). By the same approximation argument described in Section 3, we may assume ( u0,u1)∈D(AD) and consider the strong solution u. First, we note that, since the larger λis, the stronger the assumption on the initial data is. Thus, without loss of generality, we may assume that λalways satisfies λ <min/braceleftbigg2 p−1,4 2−α/parenleftbigg1 p−1−n−α 4/parenrightbigg/bracerightbigg +ε, (4.1) whereε >0 is a sufficiently small constant specified later. This will be used for th e estimate of the remainder term. In contrast to the previous section, in the following, we shall use on ly Θ(x,t;t0) :=t0+t+/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α(4.2) as a weight function, and we define the following energies. Definition 4.1. For a function u=u(t,x),α∈[0,1),λ∈[0,∞),ν >0, and t0≥1, we define E1(t;t0,λ) =/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig +1 p+1|u(t,x)|p+1/bracketrightbigg Θ(t,x;t0)λ+α 2−αdx, E0(t;t0,λ) =/integraldisplay Ω/parenleftbig 2u(t,x)∂tu(t,x)+a(x)|u(t,x)|2/parenrightbig Θ(t,x;t0)λdx, E∗(t;t0,λ,ν) =E1(t;t0,λ)+νE0(t;t0,λ), ˜E(t;t0,λ) = (t0+t)/integraldisplay Ω/bracketleftbigg1 2/parenleftbig |∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig +1 p+1|u(t,x)|p+1/bracketrightbigg Θ(t,x;t0)λdx fort≥0. Similarly to (3.2) and (3.3), we can prove the lower bound E∗(t;t0,λ,ν)≥1 2E1(t;t0,λ)+ν 2/integraldisplay Ωa(x)|u(t,x)|2Θ(t,x;t0)λdx,(4.3) provided that ν∈(0,ν0) with some constant ν0>0. We start with the following simple estimates for E1(t;t0,λ) andE0(t;t0,λ). Lemma4.2. Under the assumptions on Theorem 1.4 (ii), there exists t1=t1(n,α,a 0,λ,ε)≥ 1such that for t0≥t1andt >0, we have d dtE1(t;t0,λ)≤ −1 2/integraldisplay Ωa(x)|∂tu(t,x)|2Θ(t,x;t0)λ+α 2−αdx +C/integraldisplay Ω/parenleftbig |∇u(t,x)|2+|u(t,x)|p+1/parenrightbig Θ(t,x;t0)λ+α 2−α−1dx with some constant C=C(n,α,a 0,p,λ)>0. Proof.The proof is almost the same as that of Lemma 3.2. The only difference s are the use of |∇Θ|2 a(x)Θ= (2−α)2/a\}b∇acketle{tx/a\}b∇acket∇i}ht−2α|x|2 a(x)(t0+t+/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α)≤(2−α)2 a0(4.4)SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 23 and Θ(t,x;t0)−1≤t−1+α 2−α 0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α≤1 a0t−1+α 2−α 0a(x) instead of (3.5) and (3.6), respectively. Thus, we omit the detail. /square Lemma 4.3. Under the assumptions on Theorem 1.4 (ii), for t0≥1andt >0, we have d dtE0(t;t0,λ)≤ −/integraldisplay Ω|∇u(t,x)|2Θ(t,x;t0)λdx−2/integraldisplay Ω|u(t,x)|p+1Θ(t,x;t0)λdx +C/integraldisplay Ωa(x)|∂tu(t,x)|2Θ(t,x;t0)λ+α 2−αdx+C/integraldisplay Ωa(x)|u(t,x)|2Θ(t,x;t0)λ−1dx with some constant C=C(n,α,a 0,λ)>0. Proof.The equation (1.1) and the integration by parts imply d dtE0(t;t0,λ) = 2/integraldisplay Ω|∂tu|2Θλdx+2/integraldisplay Ω/parenleftbig ∂2 tu+a(x)∂tu/parenrightbig Θλdx +λ/integraldisplay Ω/parenleftbig 2u∂tu+a(x)|u|2/parenrightbig Θλ−1dx = 2/integraldisplay Ω|∂tu|2Θλdx+2/integraldisplay Ω/parenleftbig ∆u−|u|p−1u/parenrightbig uΘλdx +λ/integraldisplay Ω/parenleftbig 2u∂tu+a(x)|u|2/parenrightbig Θλ−1dx =−2/integraldisplay Ω|∇u|2Θλdx−2/integraldisplay Ω|u|p+1Θλdx +2/integraldisplay Ω|∂tu|2Θλdx−2λ/integraldisplay Ω(∇u·∇Θ)uΘλ−1dx +λ/integraldisplay Ω/parenleftbig 2u∂tu+a(x)|u|2/parenrightbig Θλ−1dx. (4.5) Let us estimates the right-hand side. Applying the Schwarz inequalit y and (4.4), we obtain −2λ/integraldisplay Ω(∇u·∇Ψ)uΘλ−1dx≤1 2/integraldisplay Ω|∇u|2Θλdx+C/integraldisplay Ω|u|2|∇Θ|2Θλ−2dx ≤1 2/integraldisplay Ω|∇u|2Θλdx+C/integraldisplay Ωa(x)|u|2Θλ−1dx. Moreover, the Schwarz inequality and Θ−1≤1 a0a(x) imply λ/integraldisplay Ω2u(t,x)∂tu(t,x)Θλ−1dx≤1 2/integraldisplay Ω|∇u|2Θλdx+C/integraldisplay Ω|u|2Θλ−2dx ≤1 2/integraldisplay Ω|∇u|2Θλdx+C/integraldisplay Ωa(x)|u|2Θλ−1dx. From 1≤1 a0a(x)Θα 2−α, we also obtain 2/integraldisplay Ω|∂tu|2Θλdx≤C/integraldisplay Ωa(x)|∂tu|2Θλ+α 2−αdx.24 Y. WAKASUGI Putting them all together into (4.5), we conclude d dtE0(t;t0,λ)≤ −/integraldisplay Ω|∇u|2Θλdx−2/integraldisplay Ω|u|p+1Θλdx +C/integraldisplay Ωa(x)|∂tu|2Θλ+α 2−αdx+C/integraldisplay Ωa(x)|u|2Θλ−1dx. This completes the proof. /square Combining Lemmas 4.2 and 4.3, we have the following. Lemma 4.4. Under the assumptions on Theorem 1.4 (ii), there exist const ants ν∗=ν∗(n,α,a 0,λ)∈(0,ν0)andt2=t2(n,α,a 0,p,λ,ν ∗)≥1such that for t0≥t2, andt >0, we have E∗(t;t0,λ,ν∗)+/integraldisplayt 0/integraldisplay Ωa(x)|∂tu(s,x)|2Θ(s,x;t0)λ+α 2−αdxds +/integraldisplayt 0/integraldisplay Ω/parenleftbig |∇u(s,x)|2+|u(s,x)|p+1/parenrightbig Θ(s,x;t0)λdxds ≤CE∗(0;t0,λ,ν)+C/integraldisplayt 0/integraldisplay Ωa(x)|u(s,x)|2Θ(s,x;t0)λ−1dxds with some constant C=C(n,α,a 0,p,λ,ν ∗)>0. Proof.Letν∈(0,ν0), where ν0is taken so that (4.3) holds. Let t1be the constant determined by Lemma 4.2. Then, by Lemmas 4.2 and 4.3, we obtain for t0≥t1 andt >0, d dtE∗(t;t0,λ,ν) =d dtE1(t;t0,λ)+νd dtE0(t;t0,λ) ≤ −1 2/integraldisplay Ωa(x)|∂tu|2Θλ+α 2−αdx +C/integraldisplay Ω|∇u|2Θλ+α 2−α−1dx+C/integraldisplay Ω|u|p+1Θλ+α 2−α−1dx −ν/integraldisplay Ω|∇u|2Θλdx−2ν/integraldisplay Ω|u|p+1Θλdx +Cν/integraldisplay Ωa(x)|∂tu|2Θλ+α 2−αdx+Cν/integraldisplay Ωa(x)|u|2Θλ−1dx. We take ν=ν∗with sufficiently small ν∗∈(0,ν0) such that the constants in front of the last two terms satisfy Cν∗<1 2. Moreover, taking t2>0 sufficiently large depending on ν∗so thatCΘα 2−α−1< ν∗fort0≥t2, we conclude d dtE∗(t;t0,λ,ν)≤ −η/integraldisplay Ωa(x)|∂tu|2Θλ+α 2−αdx−η/integraldisplay Ω|∇u|2Θλdx −η/integraldisplay Ω|u|p+1Θλdx+Cν/integraldisplay Ωa(x)|u|2Θλ−1dx with some constant η=η(n,α,a 0,p,λ,ν ∗)>0. Finally, integrating the above inequality over [0 ,t] gives the desired estimate. /square Besed on Lemma 4.4, we show the following estimate for ˜E(t;t0,λ).SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 25 Lemma 4.5. Under the assumptions on Theorem 1.4 (ii), there exists a con stant t2=t2(n,α,a 0,p,λ)≥1such that for t0≥t2andt >0, we have ˜E(t;t0,λ)+/integraldisplay Ωa(x)|u(t,x)|2Θ(t,x;t0)λdx +/integraldisplayt 0/integraldisplay Ωa(x)|∂tu(s,x)|2/bracketleftbig (t0+s)+Θ(s,x;t0)α 2−α/bracketrightbig Θ(s,x;t0)λdxds +/integraldisplayt 0/integraldisplay Ω/parenleftbig |∇u(s,x)|2+|u(s,x)|p+1/parenrightbig Θ(s,x;t0)λdxds ≤CI0[u0,u1]+C/integraldisplayt 0/integraldisplay Ωa(x)p+1 p−1Θ(s,x;t0)λ−p+1 p−1dxds with some constant C=C(n,α,a 0,a1,p,λ,t 0)>0. Proof.Take the same constants ν∗andt2as in Lemma 4.4. By the same compu- tation as in Lemma 3.5, we can obtain ˜E(t;t0,λ)+1 2/integraldisplayt 0(t0+s)/integraldisplay Ωa(x)|∂tu|2Θλdxds ≤˜E(0;t0,λ)+C/integraldisplayt 0/integraldisplay Ωa(x)|∂tu|2Θλ+α 2−αdxds+C/integraldisplayt 0/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Θλdxds. We multiply the both sides by a sufficiently small constant µ >0, and add it and the conclusion of Lemma 4.4. Then, we obtain µ˜E(t;t0,λ)+E∗(t;t0,λ,ν∗) +/integraldisplayt 0/integraldisplay Ωa(x)|∂tu|2/bracketleftBigµ 2(t0+s)+(1−Cµ)Θα 2−α/bracketrightBig Θλdxds +(1−Cµ)/integraldisplayt 0/integraldisplay Ω/parenleftbig |∇u|2+|u|p+1/parenrightbig Θλdxds ≤µ˜E(0;t0,λ)+CE∗(0;t0,λ,ν∗) fort0≥t2andt >0. By taking µsufficiently small so that 1 −Cµ >0 holds, the terms including |∂tu|2and|∇u|2in the left-hand side can be dropped. Since both˜E(0;t0,λ) andE∗(0;t0,λ,ν∗) are bounded by CI0[u0,u1] with some constant C=C(a1,p,λ,t 0)>0, one obtains ˜E(t;t0,λ)+/integraldisplay Ωa(x)|u(t,x)|2Θ(t,x;t0)λdx+/integraldisplayt 0/integraldisplay Ω|u|p+1Θλdxds ≤CI0[u0,u1]+C/integraldisplayt 0/integraldisplay Ωa(x)|u|2Θλ−1dxds (4.6) with some C=C(n,α,a 0,a1,p,λ,t 0)>0. Finally, applying the Young inequality to the last term of the right-hand side, we deduce C/integraldisplayt 0/integraldisplay Ωa(x)|u|2Θλ−1dxds=C/integraldisplayt 0/integraldisplay Ω|u|2Θ2 p+1λ·a(x)Θλ(1−2 p+1)−1dxds ≤1 2/integraldisplayt 0/integraldisplay Ω|u|p+1Θλdxds+C/integraldisplayt 0/integraldisplay Ωa(x)p+1 p−1Θλ−p+1 p−1dxds.26 Y. WAKASUGI This and (4.6) give the conclusion. /square By virtue of Lemma 4.5, it suffices to estimate the term C/integraldisplayt 0/integraldisplay Ωa(x)p+1 p−1Θ(s,x;t0)λ−p+1 p−1dxds. For this, we have the following lemma. Lemma 4.6. Under the assumptions on Theorem 1.4 (ii) and (4.1), we have for anyt0>0andt≥0, /integraldisplayt 0/integraldisplay Ωa(x)p+1 p−1Θ(s,x;t0)λ−p+1 p−1dxds ≤C 1 ( λ <min{4 2−α(1 p−1−n−α 4),2 p−1}), log(t0+t) ( λ= min{4 2−α(1 p−1−n−α 4),2 p−1}, p/\e}atio\slash=psubc(n,α)), (log(t0+t))2(λ=4 2−α(1 p−1−n−α 4) =2 p−1,i.e., p=psubc(n,α)), (1+t)λ−4 2−α(1 p−1−n−α 4)(λ >4 2−α(1 p−1−n−α 4), p > p subc(n,α)), (1+t)λ−2 p−1log(t0+t) (λ >2 p−1, p=psubc(n,α)), (1+t)λ−2 p−1 (λ >2 p−1, p < p subc(n,α)) with some constant C=C(n,α,a 1,p,λ)>0. Proof.Lets∈(0,t). First, we divide Ω into Ω = Ω 1(s)∪Ω2(s), where Ω1(s) =/braceleftbig x∈Ω;/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α≤t0+s/bracerightbig , Ω2(s) = Ω\Ω1(s) =/braceleftbig x∈Ω;/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α> t0+s/bracerightbig . The corresponding integral is also decomposed into /integraldisplay Ωa(x)p+1 p−1Θ(s,x;t0)λ−p+1 p−1dx=/integraldisplay Ω1(s)a(x)p+1 p−1Θ(s,x;t0)λ−p+1 p−1dx +/integraldisplay Ω2(s)a(x)p+1 p−1Θ(s,x;t0)λ−p+1 p−1dx =:I(s)+II(s). Note that, in Ω 1(s), the function Θ( s,x;t0) =t0+s+/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−αis bounded from both above and below by t0+s. Therefore, we estimate I(s)≤C(t0+s)λ−p+1 p−1/integraldisplay Ω1(s)a(x)p+1 p−1dx ≤C(t0+s)λ−p+1 p−1/integraldisplay Ω1(s)/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αp+1 p−1dx ≤C(t0+s)λ−p+1 p−1h(s), (4.7) where h(s) = 1 ( p < psubc(n,α)), log(t0+s) ( p=psubc(n,α)), (t0+s)1 2−α(n−αp+1 p−1)(p > psubc(n,α)).(4.8)SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 27 On the other hand, in Ω 2(s), the function Θ is bounded from both above and below by/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α. Thus, we have II(s)≤C/integraldisplay Ω2(s)/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αp+1 p−1+(2−α)(λ−p+1 p−1)dx. Here, we remarkthat the condition(4.1) ensuresthe finiteness of the aboveintegral, provided that εis taken sufficiently small depending on nandα. A straightforward computation shows II(s)≤C(t0+s)λ−p+1 p−1+1 2−α(n−αp+1 p−1). Since the above estimate is better than (4.7) if p≤psubc(n,α) and is the same if p > psubc(n,α), we conclude /integraldisplay Ωa(x)p+1 p−1Θ(s,x;t0)λ−p+1 p−1dx≤C(t0+s)λ−p+1 p−1h(s). Next, we compute the integral of the function ( t0+s)λ−p+1 p−1h(s) over [0,t]. From the definition (4.8) of h(s), one has the following: If p < psubc(n,α), then /integraldisplayt 0(t0+s)λ−p+1 p−1h(s)ds≤C 1/parenleftbigg λ <2 p−1/parenrightbigg , log(t0+t)/parenleftbigg λ=2 p−1/parenrightbigg , (t0+t)λ−2 p−1/parenleftbigg λ >2 p−1/parenrightbigg ; Ifp=psubc(n,α), then /integraldisplayt 0(t0+s)λ−p+1 p−1h(s)ds≤C 1/parenleftbigg λ <2 p−1/parenrightbigg , (log(t0+t))2/parenleftbigg λ=2 p−1/parenrightbigg , (t0+t)λ−2 p−1log(t0+t)/parenleftbigg λ >2 p−1/parenrightbigg ; Ifp > psubc(n,α), then /integraldisplayt 0(t0+s)λ−p+1 p−1h(s)ds≤C 1/parenleftbigg λ <4 2−α/parenleftbigg1 p−1−n−α 4/parenrightbigg/parenrightbigg , log(t0+t)/parenleftbigg λ=4 2−α/parenleftbigg1 p−1−n−α 4/parenrightbigg/parenrightbigg , (t0+t)λ−4 2−α(1 p−1−n−α 4)/parenleftbigg λ >4 2−α/parenleftbigg1 p−1−n−α 4/parenrightbigg/parenrightbigg . This completes the proof. /square We are now at the position to prove Theorem 1.4 (ii):28 Y. WAKASUGI Proof of Theorem 1.4 (ii). By Lemmas 4.5 and 4.6 with the constant t2≥1 deter- mined in Lemma 4.5, we have ˜E(t;t0,λ)+/integraldisplay Ωa(x)|u(t,x)|2Θ(t,x;t0)λdx ≤CI0[u0,u1]+C 1 ( λ <min{4 2−α(1 p−1−n−α 4),2 p−1}), log(t0+t) ( λ= min{4 2−α(1 p−1−n−α 4),2 p−1}, p/\e}atio\slash=psubc(n,α)), (log(t0+t))2(λ=4 2−α(1 p−1−n−α 4) =2 p−1,i.e., p=psubc(n,α)), (1+t)λ−4 2−α(1 p−1−n−α 4)(λ >4 2−α(1 p−1−n−α 4), p > p subc(n,α)), (1+t)λ−2 p−1log(t0+t) (λ >2 p−1, p=psubc(n,α)), (1+t)λ−2 p−1 (λ >2 p−1, p < p subc(n,α)) fort0≥t2andt≥0. On the other hand, the definition (4.2) of Θ immediately gives the lower bound ˜E(t;t0,λ)+/integraldisplay Ωa(x)|u(t,x)|2Θ(t,x;t0)λdx ≥(t0+t)λ+1E[u](t)+(t0+t)λ/integraldisplay Ωa(x)|u(t,x)|2dx, whereE(t) is defined by (1.2). Combining them, we have the desired estimate. /square Appendix A.Outline of the proof of Proposition 1.2 In this section, we give a proof of Proposition 1.2. The solvability and b asic properties of the solution of the linear problem (A.1) below can be fou nd in, for example, [8, 19, 25, 68]. Here, we give an outline of the argument alon g with [19]. The existence of the unique mild solution of the semilinear problem ( 1.1) is proved by the contraction mapping principle. This argument can be f ound in, e.g., [6, 25, 36, 85]. Here, we will give a proof based on [6]. A.1.Linear problem. Letn∈N, and let Ω be an open set in Rnwith a compact C2-boundary ∂Ω or Ω = Rn. We discuss the linear problem ∂2 tu−∆u+a(x)∂tu= 0, t > 0,x∈Ω, u(x,t) = 0, t > 0,x∈∂Ω, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(A.1) The function a(x) is nonnegative, bounded, and continuous in Rn. LetH:= H1 0(Ω)×L2(Ω) be the real Hilbert space equipped with the inner product /parenleftbigg/parenleftbigg u v/parenrightbigg ,/parenleftbigg w z/parenrightbigg/parenrightbigg H= (u,w)H1+(v,z)L2. LetAbe the operator A=/parenleftbigg0 1 ∆−a(x)/parenrightbigg defined on Hwith the domain D(A) = (H2(Ω)∩H1 0(Ω))×H1 0(Ω), which is dense inH.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 29 We first show the estimate/parenleftbigg A/parenleftbigg u v/parenrightbigg ,/parenleftbigg u v/parenrightbigg/parenrightbigg H≤ /ba∇dbl(u,v)/ba∇dbl2 H for (u,v)∈D(A). Indeed, we calculate /parenleftbigg A/parenleftbiggu v/parenrightbigg ,/parenleftbiggu v/parenrightbigg/parenrightbigg H=/parenleftbigg/parenleftbiggv ∆u−a(x)v/parenrightbigg ,/parenleftbiggu v/parenrightbigg/parenrightbigg H = (v,u)H1+(∆u−a(x)v,v)L2 = (∇v,∇u)L2+(v,u)L2−(∇v,∇u)L2−(a(x)v,v)L2 ≤(v,u)L2≤ /ba∇dbl(u,v)/ba∇dbl2 H. Next, we prove that there exists λ0∈Rsuch that for any λ≥λ0, the operator λ−Ais invertible, that is, for any ( f,g)∈ H, we can find a unique ( u,v)∈D(A) satisfying (λ−A)/parenleftbigg u v/parenrightbigg =/parenleftbigg f g/parenrightbigg . (A.2) Indeed, the above equation is equivalent with /braceleftBigg λu−v=f, λv−∆u+a(x)v=g. We remark that the first equation implies v=λu−f. Substituting this into the second equation, one has (λ2+λa(x))u−∆u=h, (A.3) whereh=g+ (λ+a(x))f∈L2(Ω). Take an arbitrary constant λ0>0 and letλ≥λ0be fixed. Associated with the above equation, we define the bilinear functional a(z,w) = ((λ2+λa(x))z,w)L2+(∇z,∇w)L2 forz,w∈H1 0(Ω). Since λ >0 anda(x) is nonnegative and bounded, ais bounded: a(z,w)≤C/ba∇dblz/ba∇dblH1/ba∇dblw/ba∇dblH1, and coercive: a(z,z)≥C/ba∇dblz/ba∇dbl2 H1. Therefore, by the Lax– Milgram theorem (see, e.g., [6, Theorem 1.1.4]), there exists a unique u∈H1 0(Ω) satisfying a(u,ϕ) = (h,ϕ)H1for anyϕ∈H1 0(Ω). In particular, usatisfies the equation (A.3) in the distribution sense. This shows ∆ u∈L2(Ω), and hence, a standard elliptic estimate implies u∈H2(Ω) (see, for example, Brezis [4, Theorem 9.25]). Defining vbyv=λu−f∈H1 0(Ω), we find the solution ( u,v)∈D(A) to the equation (A.2). The above properties enable us to apply the Hille–Yosida theorem (se e, e.g., [19, Theorem 2.18]), and there exists a C0-semigroup U(t) onHsatisfying the estimate /vextenddouble/vextenddouble/vextenddouble/vextenddoubleU(t)/parenleftbigg u0 u1/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble H≤eCt/ba∇dbl(u0,u1)/ba∇dblH (A.4) with some constant C >0. Moreover, if ( u0,u1)∈D(A), thenU(t) :=U(t)/parenleftbiggu0 u1/parenrightbigg satisfies d dtU(t) =AU(t), t >0. (A.5)30 Y. WAKASUGI Therefore, the first component u(t) ofU(t) satisfies u∈C([0,∞);H2(Ω))∩C1([0,∞);H1 0(Ω))∩C2([0,∞);L2(Ω)) and the equation (A.1) in C([0,∞);L2(Ω)). For (u0,u1)∈ H, letU(t) =/parenleftbiggu(t) v(t)/parenrightbigg :=U(t)/parenleftbiggu0 u1/parenrightbigg . We next show that usatisfies u∈C([0,∞);H1 0(Ω))∩C1([0,∞);L2(Ω)). (A.6) The property u∈C([0,∞);H1 0(Ω)) is obvious from U ∈C([0,∞);H). In or- der to prove u∈C1([0,∞);L2(Ω)), we employ an approximation argument. Let {(u(j) 0,u(j) 1)}∞ j=1be a sequence in D(A) such that lim j→∞(u(j) 0,u(j) 1) = (u0,u1) in H, and let U(j)(t) =/parenleftbiggu(j) v(j)/parenrightbigg :=U(t)/parenleftBigg u(j) 0 u(j) 1/parenrightBigg . From ( u(j) 0,u(j) 1)∈D(A),U(j)satis- fies the equation (A.5), and hence, one obtains v(j)=∂tu(j). For any fixed T >0, the estimate (A.4) implies sup t∈[0,T]/ba∇dblu(j)(t)−u(t)/ba∇dblL2≤eCT/ba∇dbl(u(j) 0−u0,u(j) 1−u1)/ba∇dblH→0, sup t∈[0,T]/ba∇dbl∂tu(j)(t)−v(t)/ba∇dblL2≤eCT/ba∇dbl(u(j) 0−u0,u(j) 1−u1)/ba∇dblH→0 asj→ ∞. This shows u∈C1([0,T];L2(Ω)) and ∂tu=v. SinceT >0 is arbitrary, we obtain (A.6). A.2.Semilinear problem. Let us turn to study the semilinear problem (1.1). A.2.1.Uniqueness of the mild solution. We first show the uniqueness of the mild solution of the integral equation U(t) =/parenleftbigg u(t) v(t)/parenrightbigg =U(t)/parenleftbigg u0 u1/parenrightbigg +/integraldisplayt 0U(t−s)/parenleftbigg 0 −|u(s)|p−1u(s)/parenrightbigg ds(A.7) inC([0,T0);H) for arbitrary fixed T0>0. Hereafter, as long as there is no risk of confusion, we call both Uand the first component uofUmild solutions. Let T0>0 andC0=eCT0, whereCis the constant in (A.4). Let U(t) =/parenleftbiggu v/parenrightbigg and W(t) =/parenleftbigg w z/parenrightbigg be two solutions to (A.7) in C([0,T0);H). TakeT∈(0,T0) arbitrary and put K:= supt∈[0,T](/ba∇dblU(t)/ba∇dblH+/ba∇dblW(t)/ba∇dblH. Then, the estimate (A.4) implies /ba∇dblU(t)−W(t)/ba∇dblH≤C0/integraldisplayt 0/ba∇dbl|w(s)|p−1w(s)−|u(s)|p−1u(s)/ba∇dblL2ds. Since the nonlinearity satisfies ||w|p−1w−|u|p−1u| ≤C(|w|+|u|)p−1|u−w|SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 31 andpfulfills the condition(1.11), weapply the H¨ olderandthe Gagliardo–Nir enberg inequality /ba∇dblu/ba∇dblL2p≤C/ba∇dblu/ba∇dblH1to obtain /ba∇dblU(t)−W(t)/ba∇dblH≤C0/integraldisplayt 0/ba∇dbl|u(s)|p−1u(s)−|w(s)|p−1w(s)/ba∇dblL2ds ≤C0C/integraldisplayt 0(/ba∇dblu(s)/ba∇dblL2p+/ba∇dblw(s)/ba∇dblL2p)p−1/ba∇dblu(s)−w(s)/ba∇dblL2pds ≤C0C/integraldisplayt 0(/ba∇dblu(s)/ba∇dblH1+/ba∇dblw(s)/ba∇dblH1)p−1/ba∇dblu(s)−w(s)/ba∇dblH1ds ≤C0CKp−1/integraldisplayt 0/ba∇dblU(s)−W(s)/ba∇dblHds (A.8) fort∈[0,T]. Therefore, by the Gronwall inequality, we have /ba∇dblU(t)−W(t)/ba∇dblH= 0 fort∈[0,T]. Since T∈(0,T0) is arbitrary, we conclude U(t) =W(t) for all t∈[0,T0). A.2.2.Existence of the mild solution. Here, we show the existence of the mild solution. LetT0>0 be arbitrarily fixed. For T∈(0,T0) andU=/parenleftbigg u v/parenrightbigg ∈C([0,T];H), we define the mapping Φ(U)(t) =U(t)/parenleftbigg u0 u1/parenrightbigg +/integraldisplayt 0U(t−s)/parenleftbigg 0 −|u(s)|p−1u(s)/parenrightbigg ds. LetC0=eCT0, whereCis the constant in (A.4). Then, we have/vextenddouble/vextenddouble/vextenddouble/vextenddoubleU(t)/parenleftbiggu0 u1/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble H≤C0/ba∇dbl(u0,u1)/ba∇dblH fort∈(0,T0). LetK= 2C0/ba∇dbl(u0,u1)/ba∇dblHand define MT,K:=/braceleftBigg U=/parenleftbigg u v/parenrightbigg ∈C([0,T];H); sup t∈[0,T]/ba∇dbl(u(t),v(t))/ba∇dblH≤K/bracerightBigg . MT,Kis a complete metric space with respect to the metric d(U,W) = sup t∈[0,T]/ba∇dbl(u(t)−w(t),v(t)−z(t))/ba∇dblH forU=/parenleftbiggu v/parenrightbigg andW=/parenleftbiggw z/parenrightbigg . We shall prove that Φ is the contraction mapping on MT,R, provided that Tis sufficiently small. First, we show that Φ(U)∈MT,KforU ∈MT,K. By the estimate (A.4) and the Gagliardo–Nirenberg inequality, we obtain for t∈[0,T], /ba∇dblΦ(U)(t)/ba∇dblH≤K 2+C0/integraldisplayt 0/ba∇dbl|u(s)|p−1u(s)/ba∇dblL2ds ≤K 2+C0/integraldisplayt 0/ba∇dblu(s)/ba∇dblp L2pds ≤K 2+C0C/integraldisplayt 0/ba∇dblu(s)/ba∇dblp H1ds ≤K 2+C0CTKp. (A.9)32 Y. WAKASUGI Therefore, taking Tsufficiently small so that K 2+C0CTKp≤K holds, we see that Φ(U)∈MT,K. Moreover, for U=/parenleftbigg u v/parenrightbigg ,W=/parenleftbigg w z/parenrightbigg ∈MT,R, the same computation as in (A.8) yields for t∈[0,T], d(Φ(U),Φ(W))≤C0CTKp−1d(U,W). Thus, retaking Tsmaller if needed so that C0CTKp−1≤1 2, we have the contractivity of Φ. Thus, by the contraction mapping principle, we see that there exists a fixed point U=/parenleftbiggu v/parenrightbigg ∈MT,K, that is, Usatisfies the integral equation (A.7). We postpone to verify u∈C1([0,T];L2(Ω)) and ∂tu=vafter proving the approximation property below. A.2.3.Blow-up alternative. LetTmax=Tmax(u0,u1)bethemaximalexistencetime of the mild solution defined by Tmax= sup/braceleftbigg T∈(0,∞];∃U=/parenleftbigg u v/parenrightbigg ∈C([0,T);H) satisfies (A.7)/bracerightbigg . We show that if Tmax<∞, the corresponding unique mild solution U=/parenleftbiggu v/parenrightbigg must satisfy lim t→Tmax−0/ba∇dblU(t)/ba∇dblH=∞. (A.10) Indeed, if m:= liminf t→Tmax−0/ba∇dblU(t)/ba∇dblH<∞, thenthereexistsamonotoneincreas- ingsequence {tj}∞ j=1in(0,Tmax)suchthatlim j→∞tj=Tmaxandlim j→∞/ba∇dblU(tj)/ba∇dblH= m. LetT0> Tmaxbe arbitrary fixed and let C0=eCT0as in Section A.2.2. Ap- plying the same argument as in Section A.2.2 with replacement ( u0,u1) byU(tj), one can find there exists Tdepending only on p,m, andC0such that there exists a mild solution on the interval [ tj,tj+T]. However, this contradicts the definition ofTmaxwhenjis large. Thus, we have (A.10). A.2.4.Continuous dependence on the initial data. Let (u0,u1)∈ HandT < T 0< Tmax(u0,u1). We take C0=eCT0as in Section A.2.2. Let {(u(j) 0,u(j) 1)}∞ j=1be a sequence in Hsuch that ( u(j) 0,u(j) 1)→(u0,u1) inHasj→ ∞. Then, we will prove that, for sufficiently large j,Tmax(u(j) 0,u(j) 1)> Tand the corresponding solution U(j)with the initial data ( u(j) 0,u(j) 1) satisfies lim j→∞sup t∈[0,T]/ba∇dblU(j)(t)−U(t)/ba∇dblH= 0. (A.11) LetC1= 2supt∈[0,T]/ba∇dblU(t)/ba∇dblHand let τj:= sup/braceleftBigg t∈[0,Tmax(u(j) 0,u(j) 1)); sup t∈[0,T]/ba∇dblU(j)(t)/ba∇dblH≤2C1/bracerightBigg .SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 33 Since (u(j) 0,u(j) 1)→(u0,u1) inHasj→ ∞, we have /ba∇dbl(u(j) 0,u(j) 1)/ba∇dblH≤C1for large j, which ensures τj>0 for such j. Moreover, the same computation as in (A.8) and the Gronwall inequality imply, for t∈[0,min{τj,T}], /ba∇dblU(j)(t)−U(t)/ba∇dblH≤C0/ba∇dblU(j)(0)−U(0)/ba∇dblHexp/parenleftBig CCp−1 1T/parenrightBig .(A.12) Note that the right-hand side tends to zero as j→ ∞. From this and the definition ofC1, we obtain /ba∇dblU(j)(t)/ba∇dblH≤C1(t∈[0,min{τj,T}]) for large j. By the definition of τj, the above estimate implies τj> T, and hence, Tmax(u(j) 0,u(j) 1)> T. From this, the estimate (A.12) holds for t∈[0,T]. Letting j→ ∞in (A.12) gives (A.11). A.2.5.Regularity of solution. Next, we discuss the regularity of the solution. Let (u0,u1)∈D(A) andTmax=Tmax(u0,u1). Then, we will show that the correspond- ing mild solution Usatisfies U ∈C([0,Tmax);D(A))∩C1([0,Tmax);H). TakeT∈(0,Tmax) arbitrary. First, from Section A.1, the linear part ofthe mild so- lution satisfies UL(t) =U(t)/parenleftbiggu0 u1/parenrightbigg ∈C([0,∞);D(A))∩C1([0,∞);H). This implies, forh >0 andt∈[0,T−h], /ba∇dblUL(t+h)−UL(t)/ba∇dblH≤Ch. (A.13) Thus, it suffices to show UNL(t) :=/integraldisplayt 0U(t−s)/parenleftbigg0 −|u(s)|p−1u(s)/parenrightbigg ds ∈C([0,T];D(A))∩C1([0,T];H). (A.14) By the changing variable t+h−s/mapsto→s, we calculate UNL(t+h)−UNL(t) =/integraldisplayt+h 0U(t−s)/parenleftbigg0 −|u(s)|p−1u(s)/parenrightbigg ds −/integraldisplayt 0U(t−s)/parenleftbigg 0 −|u(s)|p−1u(s)/parenrightbigg ds =/integraldisplayt 0U(s)/parenleftbigg0 −|u|p−1u(t+h−s)+|u|p−1u(t−s)/parenrightbigg ds +/integraldisplayt+h tU(s)/parenleftbigg0 −|u|p−1u(t+h−s)/parenrightbigg ds. Therefore, the same computation as in (A.8) and (A.9) implies /ba∇dblUNL(t+h)−UNL(t)/ba∇dblH≤C/integraldisplayt 0/ba∇dblu(s+h)−u(s)/ba∇dblH1ds+Ch. Combining this with (A.13), one obtains /ba∇dblU(t+h)−U(t)/ba∇dblH≤Ch+/integraldisplayt 0/ba∇dblU(s+h)−U(s)/ba∇dblHds.34 Y. WAKASUGI The Gronwall inequality implies /ba∇dblU(t+h)−U(t)/ba∇dblH≤Ch. This further yields /ba∇dbl−|u|p−1u(t+h)+|u|p−1u(t)/ba∇dblH1≤Ch, that is, the nonlinearity is Lipschitz continuous in H1 0(Ω). From this, we can see−|u|p−1u∈W1,∞(0,T;H1 0(Ω)) (see e.g. [6, Corollary 1.4.41]). Thus, we can differentiate the expression /integraldisplayt 0U(t−s)/parenleftbigg 0 −|u|p−1u(s)/parenrightbigg ds=/integraldisplayt 0U(s)/parenleftbigg 0 −|u|p−1u(t−s)/parenrightbigg ds with respect to tinH, and it implies UNL∈C1([0,T];H). Finally, for h >0 and t∈[0,T−h], we have 1 h(U(t)−I)UNL(t) =1 h/integraldisplayt 0U(t+h−s)/parenleftbigg0 −|u|p−1u(s)/parenrightbigg ds−1 h/integraldisplayt 0U(t−s)/parenleftbigg0 −|u|p−1u(s)/parenrightbigg ds =1 h(UNL(t+h)−UNL(t))−1 h/integraldisplayt+h tU(t+h−s)/parenleftbigg 0 −|u|p−1u(s)/parenrightbigg ds. This implies U(t)∈D(A) and d dtUNL(t) =AUNL(t)+/parenleftbigg 0 −|u|p−1u(t)/parenrightbigg . Moreover, the above equation and U ∈C1([0,T];H) lead to U ∈C([0,T];D(A)). This proves the property (A.14). We also remark that the first com ponentuofU is a strong solution to (1.1). A.2.6.Approximation of the mild solution by strong solutions. Let (u0,u1)∈ H andTmax=Tmax(u0,u1). Let{(u(j) 0,u(j) 1)}∞ j=1be a sequence in D(A) satisfying limj→∞(u(j) 0,u(j) 1) = (u0,u1) inH. TakeT∈(0,Tmax) arbitrary. Then, the results of Sections A.2.4 and A.2.5 imply that Tmax(u(j) 0,u(j) 1)> Tfor large j, and the corresponding mild solution U(j)=/parenleftbiggu(j) v(j)/parenrightbigg with the initial data ( u(j) 0,u(j) 1) satisfies U(j)∈C([0,T];D(A))∩C1([0,T];H). Moreover, ∂tu(j)=v(j)holds and u(j)is a strong solution to (1.1). By the result of Section A.2.4, we see that lim j→∞sup t∈[0,T]/ba∇dblu(j)(t)−u(t)/ba∇dblH1= 0, lim j→∞sup t∈[0,T]/ba∇dbl∂tu(j)(t)−v(t)/ba∇dblL2= 0, which yields u∈C1([0,T];L2(Ω)) and ∂tu=v. Namely, we have the property stated at the end of Section A.2.2.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 35 A.2.7.Finite propagation property. Here, we show the finite propagation property for the mild solution. In what follows, we use the notations BR(x0) :={x∈ Rn;|x−x0|< R}forx0∈RnandR >0. LetT∈(0,Tmax(u0,u1)) andR > 0. Assume that ( u0,u1)∈ Hsatisfies supp u0∪suppu1⊂BR(0)∩Ω. Letu∈ C([0,T];H1 0(Ω))∩C1([0,T];L2(Ω)) be the mild solution of (1.1). Then, we have suppu(t,·)⊂Bt+R(0)∩Ω (t∈[0,T]). (A.15) To prove this, we modify the argument of [39] in which the classical so lution is treated. Let ( t0,x0)∈[0,T]×Ω be a point such that |x0|> t0+Rand define Λ(t0,x0) ={(t,x)∈(0,T)×Ω; 0< t < t 0,|x−x0|< t0−t} =/uniondisplay t∈(0,t0)({t}×(Bt0−t(x0)∩Ω))). It suffices to show u= 0 in Λ( t0,x0). We also put St0−t:=∂Bt0−t(x0)∩Ω and Sb,t0−t:=Bt0−t(x0)∩∂Ω. Note that ∂(Bt0−t(x0)∩Ω) =St0−t∪Sb,t0−tholds. First, we further assume ( u0,u1)∈D(A). Then, by the result of Section A.2.5, ubecomes the strong solution. This ensures that the following compu tations make sense. Define E(t;t0,x0) :=1 2/integraldisplay Bt0−t(x0)∩Ω(|∂tu(t,x)|2+|∇u(t,x)|2+|u(t,x)|2)dx fort∈[0,t0]. By differentiating in tand applying the integration by parts, we have d dtE(t;t0,x0) =/integraldisplay Bt0−t(x0)∩Ω/parenleftbig ∂2 tu−∆u+u/parenrightbig ∂tudx −1 2/integraldisplay St0−t∪Sb,t0−t(|∂tu|2+|∇u|2+|u|2−2(n·∇u)∂tu)dS, wherenis the unit outward normal vector of St0−t∪Sb,t0−tanddSdenotes the surface measure. The Schwarz inequality implies the second term of the right- hand side is nonpositive, and hence, we can omit it. Using the equation (1.1) to the first term and the Gagliardo–Nirenberg inequality /ba∇dblu(t)/ba∇dblL2p(Bt0−t(x0)∩Ω)≤ C/ba∇dblu(t)/ba∇dblH1(Bt0−t(x0)∩Ω), we can see that d dtE(t;t0,x0)≤C/parenleftBig /ba∇dblu(t)/ba∇dbl2p H1(Bt0−t(x0)∩Ω)+/ba∇dbl∂tu(t)/ba∇dbl2 L2(Bt0−t(x0)∩Ω)+/ba∇dblu(t)/ba∇dbl2 L2(Bt0−t(x0)∩Ω)/parenrightBig ≤CE(t;t0,x0), where we have also used /ba∇dblu(t)/ba∇dblH1(Bt0−t(x0)∩Ω)is bounded for t∈(0,t0). Noting that the support condition of the initial data implies E(0;t0,x0) = 0, we obtain from the above inequality that E(t;t0,x0) = 0 for t∈[0,t0]. This yields u= 0 in Λ(t0,x0). Finally, for the general case ( u0,u1)∈ H, we take an arbitrary small ε >0 and a sequence {(u(j) 0,u(j) 1)}∞ j=1inD(A) such that supp u(j) 0∪suppu(j) 1⊂BR+ε(0)∩Ω and lim j→∞(u(j) 0,u(j) 1) = (u0,u1) inH. Here, we remark that such a sequence can be constructed by the form ( u(j) 0,u(j) 1) = (φε˜u(j) 0,φε˜u(j) 1), where {(˜u(j) 0,˜u(j) 1)}is a sequencein D(A) which convergesto( u0,u1) inHasj→ ∞, andφε∈C∞ 0(Rn) is a cut-off function satisfy 0 ≤φε≤1,φε= 1 onBR(0), and φε= 0 onRn\BR+ε(0).36 Y. WAKASUGI Then, the result of Section A.2.5 shows that the corresponding str ong solution u(j)to (u(j) 0,u(j) 1) satisfies supp u(j)(t,·)⊂BR+ε+t(0). Moreover, the result of Section A.2.6 leads to lim j→∞u(j)=uinC([0,T];H1 0(Ω)). Hence, we conclude suppu(t,·)⊂BR+ε+t(0). Since εis arbitrary, we have (A.15). A.2.8.Existence of the global solution. Finally, we show the existence of the global solution to (1.1). Let ( u0,u1)∈ Hand suppose that Tmax(u0,u1) is finite. Then, by the blow-up alternative (Section A.2.3), the corresponding mild so lutionumust satisfy lim t→Tmax−0/ba∇dbl(u(t),∂tu(t))/ba∇dblH=∞. (A.16) Let{(u(j) 0,u(j) 1)}∞ j=1be a sequence in D(A) such that lim j→∞(u(j) 0,u(j) 1) = (u0,u1) inH, andletu(j)bethecorrespondingstrongsolutionwiththeinitialdata( u(j) 0,u(j) 1). Using the integration by parts and the equation (1.1), we calculate d dt/bracketleftbigg1 2/parenleftBig /ba∇dbl∂tu(j)(t)/ba∇dbl2 L2+/ba∇dbl∇u(j)(t)/ba∇dbl2 L2/parenrightBig +1 p+1/ba∇dblu(j)(t)/ba∇dblp+1 Lp+1/bracketrightbigg =−/ba∇dbl∂tu(j)(t)/ba∇dbl2 L2. This and the Gagliardo–Nirenberg inequality imply /ba∇dbl∂tu(j)(t)/ba∇dbl2 L2+/ba∇dbl∇u(j)(t)/ba∇dbl2 L2≤C/parenleftBig /ba∇dblu(j) 1/ba∇dbl2 L2+/ba∇dbl∇u(j) 0/ba∇dbl2 L2+/ba∇dblu(j) 0/ba∇dblp+1 H1/parenrightBig . Moreover, by u(t) =u0+/integraldisplayt 0∂tu(s)ds, one obtains the bound /ba∇dbl(u(j)(t),∂tu(j)(t))/ba∇dbl2 H≤C(1+T)2/parenleftBig /ba∇dblu(j) 1/ba∇dbl2 L2+/ba∇dbl∇u(j) 0/ba∇dbl2 L2+/ba∇dblu(j) 0/ba∇dblp+1 H1/parenrightBig (A.17) fort∈[0,T]. Thisandtheblow-upalternative(SectionA.2.3)show Tmax(u(j) 0,u(j) 1) = ∞for allj. The bound (A.17) with T=Tmax(u0,u1) also yields that sup j∈Nsup t∈[0,Tmax(u0,u1)]/ba∇dbl(u(j)(t),∂tu(j)(t))/ba∇dbl2 H<∞. (A.18) On the other hand, from the result of Section A.2.6, we have lim j→∞sup t∈[0,T]/ba∇dbl(u(j)(t)−u(t),∂tu(j)(t)−∂tu(t))/ba∇dblH= 0 (A.19) for anyT∈(0,Tmax(u0,u1)). However, (A.18) and (A.19) contradict (A.16). Thus, we conclude Tmax(u0,u1) =∞. Appendix B.Proof of Preliminary lemmas B.1.Proof of Lemma 2.1. Proof of Lemma 2.1. We define b1(x) = ∆/parenleftbigga0 (n−α)(2−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α/parenrightbigg =a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α+a0α n−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α−2 andb2(x) =a(x)−b1(x). By b2(x) a(x)=1 /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)/parenleftbigg /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)−a0−a0α n−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−2/parenrightbiggSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 37 and the assumption (1.12), there exists a constant Rε>0 such that |b2(x)| ≤εa(x) holds for |x|> Rε. Letηε∈C∞ 0(Rn) satisfy 0 ≤ηε(x)≤1 forx∈Rnand ηε(x) = 1 for |x|< Rε. LetN(x) denote the Newton potential, that is, N(x) = |x| 2(n= 1), 1 2πlog1 |x|(n= 2), Γ(n/2+1) n(n−2)πn/2|x|2−n(n≥3). We define Aε(x) =A0+a0 (n−α)(2−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α−N∗(ηεb2), whereA0>0 is a sufficiently large constant determined later. We show that the aboveAε(x) has the desired properties. First, we compute ∆Aε(x) =b1(x)+ηε(x)b2(x) =a(x)−(1−ηε)b2(x), which implies (2.1). Next, since ηεb2has the compact support, N∗(ηεb2) satisfies |N∗(ηεb2)(x)| ≤C/braceleftBigg 1+log/a\}b∇acketle{tx/a\}b∇acket∇i}ht(n= 2) /a\}b∇acketle{tx/a\}b∇acket∇i}ht2−n(n= 1,n≥3),|∇N∗(ηεb2)(x)| ≤C/a\}b∇acketle{tx/a\}b∇acket∇i}ht1−n with some constant C=C(n,Rε,/ba∇dbla/ba∇dblL∞,α,a0,ε)>0, and the former estimate leads to (2.2), provided that A0is sufficiently large. Moreover, the latter estimate shows lim |x|→∞|∇Aε(x)|2 a(x)Aε(x)= lim |x|→∞1 /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)·1 /a\}b∇acketle{tx/a\}b∇acket∇i}htα−2Aε(x)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea0 n−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1x−/a\}b∇acketle{tx/a\}b∇acket∇i}htα−1∇N∗(ηεb2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =2−α n−α, which implies the inequality (2.3) for sufficiently large x. Finally, taking A0suffi- ciently large, we have (2.3) for any x∈Rn. /square B.2.Properties of Kummer’s function. To proveLemma 2.4, we preparesome properties of Kummer’s function. Lemma B.1. Kummer’s confluent hypergeometric function M(b,c;s)satisfies the properties listed as follows. (i)M(b,c;s)satisfies Kummer’s equation su′′(s)+(c−s)u′(s)−bu(s) = 0. (ii)Ifc≥b >0, thenM(b,c;s)>0fors≥0and lim s→∞M(b,c;s) sb−ces=Γ(c) Γ(b). (B.1) In particular, M(b,c;s)satisfies C(1+s)b−ces≤M(b,c;s)≤C′(1+s)b−ces(B.2) with some positive constants C=C(b,c)andC′=C(b,c)′.38 Y. WAKASUGI (iii)More generally, if −c /∈N∪{0}andc≥b, then, while the sign of M(b,c;s) is indefinite, it still has the asymptotic behavior lim s→∞M(b,c;s) sb−ces=Γ(c) Γ(b), (B.3) where we interpret that the right-hand side is zero if −b∈N∪ {0}. In particular, M(b,c;s)has a bound |M(b,c;s)| ≤C(1+s)b−ces(B.4) with some positive constant C=C(b,c). (iv)M(b,c;s)satisfies the relations sM(b,c;s) =sM′(b,c;s)+(c−b)M(b,c;s)−(c−b)M(b−1,c;s), cM′(b,c;s) =cM(b,c;s)−(c−b)M(b,c+1;s). Proof.The property (i) is directly obtained from the definition of M(b,c;s). When c=b >0, (ii) is obviousfrom M(b,b;s) =es. Whenc > b >0, we havethe integral representation (see [3, (6.1.3)]) M(b,c;s) =Γ(c) Γ(b)Γ(c−b)/integraldisplay1 0tb−1(1−t)c−b−1etsdt, which implies M(b,c;s)>0. Moreover, [3, (6.1.8)] shows the asymptotic behavior (B.1). The estimate (B.2) is obvious, since the right-hand side of (B.1 ) is positive andM(b,c;s)>0 fors≥0. Next, the property (iii) clearly holds if c=bor −b∈N∪{0}, sinceM(b,c;s) is a polynomial of order −bif−b∈N∪{0}. For the casesc > band−b /∈N∪{0}, note that for any m∈N∪{0}we have dm dsmM(b,c;s) =(b)m (c)mM(b+m,c+m;s), which implies |dm dsmM(b,c;s)| → ∞ass→ ∞. By taking m∈N∪ {0}so that b+m >0 and applying l’Hˆ opital theorem we deduce lim s→∞M(b,c;s) sb−ces= lim s→∞dm dsmM(b,c;s) dm dsm(sb−ces)=(b)m (c)mlim s→∞M(b+m,c+m;s) sb−ces+o(sb−ces) =(b)mΓ(c+m) (c)mΓ(b+m)=Γ(c) Γ(b). The estimate (B.4) is easily follows from the asymptotic behavior (B.3) and we have (iii). Finally, the property (iv) can be found in [3, p.200]. /square B.3.Proof of Lemma 2.4. Proof of Lemma 2.4. The property (i) is directly follows from Lemma B.1 (i). For (ii), noting that 0 ≤β < γ εand applying Lemma B.1 (ii) with b=γε−βand c=γε, we have ϕβ(s)>0 fors≥0 and lim s→∞sβϕβ,ε(s) =Γ(γε) Γ(γε−β). This proves the property (ii). Next, by Lemma B.1 (iii) with b=γε−βandc=γε, one still obtains lim s→∞sβϕε(s) = Γ(γε)/Γ(γε−β), where the right-hand side isSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 39 interpreted as zero if β−γε∈N∪{0}. In particular, this (or the estimate (B.4)) gives |ϕβ,ε(s)| ≤Kβ,ε(1+s)−β with some constant Kβ,ε>0. Thus, we have (iii). Noting that ϕ′ β,ε(s) =e−s[−M(γε−β,γε;s)+M′(γε−β,γε;s)] (B.5) and applying the first assertion of Lemma B.1 (iv), we have the prope rty (iv). Finally, from (B.5) and the second assertion of Lemma B.1 (iv), we obt ain γεϕ′ β,ε(s) =−βe−sM(γε−β,γε+1;s). Differentiating again the above identity gives γεϕ′′ β,ε(s) =−βe−s[−M(γε−β,γε+1;s)+M′(γε−β,γε+1;s)]. Therefore, the second assertion of Lemma B.1 (iv) implies γε(γε+1)ϕ′′ β,ε(s) =β(β+1)e−sM(γε−β,γε+2;s). In particular, if 0 < β < γ ε, then Lemma B.1 (ii) shows that M(γε−β,γε+1;s) (resp.M(γε−β,γε+ 2;s) ) is bounded from above and below by (1 + s)−β−1es (resp. (1+ s)−β−2es), and hence, we have the assertions of (v). /square B.4.Proof of Proposition 2.6. We are now in a position to prove Proposition 2.6. Proof of Proposition 2.6. Letz=/tildewideγεAε(x)/(t0+t). FromDefinition2.5andLemma 2.4 (iv), one obtains ∂tΦβ,ε(t,x;t0) =−(t0+t)−β−1/bracketleftbig βϕβ,ε(z)+zϕ′ β,ε(z)/bracketrightbig =−(t0+t)−β−1βϕβ+1,ε(z) =−βΦβ+1,ε(t,x;t0), which proves (i). Applying Lemma 2.4 (iii), we have |Φβ,ε(t,x;t0)| ≤Kβ,ε(t0+t)−β/parenleftbigg 1+/tildewideγεAε(x) t0+t/parenrightbigg−β ≤C(t0+t+Aε(x))−β =CΨ(t,x;t0)−β with some constant C=C(n,α,β,ε)>0. This implies (ii). Next, by Lemma 2.4 (ii), Φ β,ε(t,x;t0) satisfies Φβ,ε(t,x;t0)≥kβ,ε(t0+t)−β/parenleftbigg 1+/tildewideγεAε(x) t0+t/parenrightbigg−β ≥c(t0+t+Aε(x))−β =cΨ(t,x;t0)−β40 Y. WAKASUGI with some constant c=c(n,α,β,ε)>0, and (iii) is verified. For (iv), we again put z= ˜γεAε(x)/(t0+t) and compute a(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0) =−a(x)(t0+t)−β−1 ×/parenleftbigg βϕβ,ε(z)+zϕ′ β,ε(z)+ ˜γε∆Aε(x) a(x)ϕ′ β,ε(z)+ ˜γε|∇Aε(x)|2 a(x)Aε(x)zϕ′′ β,ε(z)/parenrightbigg . Using the equation (2.5) and the definition (2.4), we rewrite the right -hand side as ˜γεa(x)(t0+t)−β−1/parenleftbigg 1−2ε−∆Aε(x) a(x)/parenrightbigg ϕ′ β,ε(z) +a(x)(t0+t)−β−1/parenleftbigg 1−˜γε|∇Aε(x)|2 a(x)Aε(x)/parenrightbigg ϕ′′ β,ε(z). By (2.1) and (2.3) in Lemma 2.1, we have 1−2ε−∆Aε(x) a(x)≤ −ε, 1−˜γε|∇Aε(x)|2 a(x)Aε(x)≥ε/parenleftbigg2−α n−α+2ε/parenrightbigg−1 >0. From them and the property (v) of Lemma 2.4, we conclude a(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0)≥ −ε˜γεa(x)(t0+t)−β−1ϕ′ β,ε/parenleftbigg˜γεAε(x) t0+t/parenrightbigg ≥εkβ,εa(x)(t0+t)−β−1/parenleftbigg 1+˜γεAε(x) t0+t/parenrightbigg−β−1 ≥ca(x)(t0+t+Aε(x))−β−1 =ca(x)Ψ(x,t;t0)−β−1 with some constant c=c(n,α,β,ε)>0, which completes the proof. /square B.5.Proof of Lemma 2.7. Proof of Lemma 2.7. Putting v= Φ−1+δu, noting ∇u= (1−δ)Φ−δ(∇Φ)v+ Φ1−δ∇v, and applying integration by parts imply /integraldisplay Ω|∇u|2Φ−1+2δdx =/integraldisplay Ω|∇v|2Φdx+2(1−δ)/integraldisplay Ωv(∇v·∇Φ)dx+(1−δ)2/integraldisplay Ω|v|2|∇Φ|2 Φdx =/integraldisplay Ω|∇v|2Φdx−(1−δ)/integraldisplay Ω|v|2∆Φdx+(1−δ)2/integraldisplay Ω|v|2|∇Φ|2 Φdx ≥ −(1−δ)/integraldisplay Ω|u|2(∆Φ)Φ−2+2δdx+(1−δ)2/integraldisplay Ω|u|2|∇Φ|2Φ−3+2δdx.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 41 Byu∆u=−|∇u|2+∆(u2 2), integration by parts, and applying the above estimate, we have/integraldisplay Ωu∆uΦ−1+2δdx =−/integraldisplay Ω|∇u|2Φ−1+2δdx+1 2/integraldisplay Ω|u|2∆(Φ−1+2δ)dx =−/integraldisplay Ω|∇u|2Φ−1+2δdx−1−2δ 2/integraldisplay Ω|u|2(∆Φ)Φ−2+2δdx +(1−δ)(1−2δ)/integraldisplay Ω|u|2|∇Φ|2Φ−3+2δdx ≤ −δ 1−δ/integraldisplay Ω|∇u|2Φ−1+2δdx+1−2δ 2/integraldisplay Ω|u|2(∆Φ)Φ−2+2δdx. This completes the proof. /square Acknowledgements This work was supported by JSPS KAKENHI Grant Numbers JP18H01 132 and JP20K14346. 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Laboratory of Mathematics, Graduate School of Engineering , Hiroshima University, Higashi-Hiroshima, 739-8527, Japan Email address :wakasugi@hiroshima-u.ac.jp

1209.3120v1.Skyrmion_Dynamics_in_Multiferroic_Insulator.pdf

arXiv:1209.3120v1 [cond-mat.str-el] 14 Sep 2012Skyrmion Dynamics in Multiferroic Insulator Ye-Hua Liu,1You-Quan Li,1and Jung Hoon Han2,3,∗ 1Zhejiang Institute of Modern Physics and Department of Physi cs, Zhejiang University, Hangzhou 310027, People’s Republic of China 2Department of Physics and BK21 Physics Research Division, Sungkyunkwan University, Suwon 440-746, Korea 3Asia Pacific Center for Theoretical Physics, Pohang, Gyeong buk 790-784, Korea (Dated: October 31, 2018) Recent discovery of Skyrmion crystal phase in insulating mu ltiferroic compound Cu 2OSeO 3calls for new ways and ideas to manipulate the Skyrmions in the abse nce of spin transfer torque from the conduction electrons. It is shown here that the position -dependent electric field, pointed along the direction of the average induced dipole moment of the Sky rmion, can induce the Hall motion of Skyrmion with its velocity orthogonal to the field gradien t. Finite Gilbert damping produces longitudinal motion. We find a rich variety of resonance mode s excited by a.c. electric field. PACS numbers: 75.85.+t, 75.70.Kw, 76.50.+g Skyrmions are increasingly becoming commonplace sightings among spiral magnets including the metallic B20 compounds[1–5] and most recently, in a multiferroic insulator Cu 2OSeO3[6]. Both species of compounds dis- playsimilarthickness-dependentphasediagrams[5,6] de- spitetheircompletelydifferentelectricalproperties,high- lighting the generality of the Skyrmion phase in spiral magnets. Along with the ubiquity of Skyrmion matter comes the challenge of finding means to control and ma- nipulatethem, inadevice-orientedmannerakintoefforts in spintronics community to control the domain wall and vortex motion by electrical current. Spin transfer torque (STT) is a powerful means to induce fast domain wall motion in metallic magnets[7, 8]. Indeed, current-driven Skyrmion rotation[9] and collective drift[10], originating from STT, have been demonstrated in the case of spiral magnets. Theory of current-induced Skyrmion dynam- ics has been worked out in Refs. [11, 12]. In insulating compounds such as Cu 2OSeO3, however, the STT-driven mechanism does not work due to the lack of conduction electrons. As with other magnetically driven multiferroic compounds[13], spiral magnetic order in Cu 2OSeO3is accompanied by finite electric dipole moment. Recent work by Seki et al.[14] further confirmed the mecha- nism of electric dipole moment induction in Cu 2OSeO3 to be the so-called pd-hybridization[15–17]. In short, the pd-hybridization mechanism claims the dipole moment Pijfor every oxygen-TM(transition metal) bond propor- tional to ( Si·ˆeij)2ˆeijwhereiandjstand for TM and oxygen sites, respectively, and ˆ eijis the unit vector con- necting them. Carefully summing up the contributions of such terms over a unit cell consisting of many TM- O bonds, Seki et al.were able to deduce the dipole moment distribution associated with a given Skyrmionic spin configuration[14]. It is interesting to note that the numerical procedure performed by Seki et al.is pre- cisely the coarse-graining procedure which, in the text-book sense of statistical mechanics, is tantamount to the Ginzburg-Landau theory of order parameters. Indeed we can show that Seki et al.’s result for the dipole moment distribution is faithfully reproduced by the assumption that the local dipole moment Piis related to the local magnetization Siby Pi=λ(Sy iSz i,Sz iSx i,Sx iSy i) (1) with some coupling λ. A similar expression was pro- posed earlier in Refs. [18, 19] as the GL theory of Ba 2CoGe2O7[20], another known pd-hybridization- originated multiferroic material with cubic crystal struc- ture. Each site icorresponds to one cubic unit cell of Cu2OSeO3with linear dimension a∼8.9˚A, and we have normalized Sito have unit magnitude. The dimension of the coupling constant is therefore [ λ] = C·m. Having obtained the proper coupling between dipole moment and the magnetizaiton vector in Cu 2OSeO3one can readily proceed to study the spin dynamics by solv- ing Landau-Lifshitz-Gilbert (LLG) equation. Very small values of Gilbert damping parameter are assumed in the simulation as we are dealing with an insulating magnet. A new, critical element in the simulation is the term aris- ing from the dipolar coupling HME=−/summationdisplay iPi·Ei=−λ 2/summationdisplay iSi 0Ez iEy i Ez i0Ex i Ey iEx i0 Si,(2) where we have used the magneto-electric coupling ex- pression in Eq. (1). In essence this is a field-dependent (voltage-dependent) magnetic anisotropy term. The to- tal Hamiltonian for spin is given by H=HHDM+ HME, whereHHDMconsists of the Heisenberg and the Dzyaloshinskii-Moriya (DM) exchange and a Zeeman field term. Earlier theoretical studies showed HHDMto stabilize the Skyrmion phase[1, 21–24].2 Two field orientations can be chosen independently in experiments performed on insulating magnets. First, the direction of magnetic field Bdetermines the plane, or- thogonal to B, in which Skyrmions form. Second, the electric field Ecan be applied to couple to the induced dipole moment of the Skyrmion and used as a “knob” to move it around. Three field directions used in Ref. [14] and the induced dipole moment in eachcase areclassified as (I)B/bardbl[001],P= 0, (II) B/bardbl[110],P/bardbl[001], and (III)B/bardbl[111],P/bardbl[111]. One can rotate the spin axis appearing in Eq. (1) accordingly so that the z-direction coincides with the magnetic field orientation in a given setup and the x-direction with the crystallographic[ 110]. In each of the cases listed above we obtain the magneto- electric coupling, after the rotation, H(I) ME=−λ 2/summationdisplay iEi([Sy i]2−[Sx i]2), H(II) ME=−λ 2/summationdisplay iEi([Sz i]2−[Sx i]2), H(III) ME=−λ 2√ 3/summationdisplay iEi(3[Sz i]2−1).(3) In cases (II) and (III) the E-field is chosen parallel to the induced dipole moment P,Ei=EiˆP, to maximize the effect of dipolar coupling. In case (I) where there is no net dipole moment for Skyrmions we chose E/bardbl[001] to arrive at a simple magneto-electric coupling form shown above. Suppose now that the E-field variation is sufficiently slow on the scale of the lattice constant ato allow the writing of the continuum energy, HME=−λd a/integraldisplay d2rE(r)ρD(r). (4) It is assumed that all variables behave identically along the thickness direction, oflength d. The “dipolarcharge” densityρD(r)couplestotheelectricfield E(r)inthesame wayasthe conventionalelectric chargedoes tothe poten- tial field in electromagnetism. The analogy is also useful in thinking about the Skyrmion dynamics under the spa- tially varying E-field as we will show. The continuum form of dipolar charge density in Eq. 4 is ρ(I) D(ri) =1 2a2([Sy i]2−[Sx i]2), ρ(II) D(ri) =1 2a2([Sz i]2−[Sx i]2−1), ρ(III) D(ri) =√ 3 2a2([Sz i]2−1). (5) Division by the unit cell area a2ensures that ρD(r) has the dimension of areal density. Values for the ferromag- netic case, Sz i= 1, is subtracted in writing down the def- inition (5) in order to isolate the motion of the Skyrmion FIG. 1: (color online) (a) Skyrmion configuration and (b)-(d ) the corresponding distribution of dipolar charge density f or three magnetic field orientations as in Ref. 14. (b) B/bardbl[001] (c)B/bardbl[110] (d) B/bardbl[111]. For each case, electric field is chosen as E/bardbl[001],E/bardbl[001] and E/bardbl[111], respectively. See text for the definition of dipolar charge density. As schemat i- cally depicted in (a), the Skyrmion executes a Hall motion in response to electric field gradient. relative to the ferromagnetic background. Due to the subtraction, the dipolar charge is no longer equivalent to the dipole moment of the Skyrmion. The distribution of dipolar charge density for the Skyrmion spin configura- tion in the three cases are plotted in Fig. 1. In case (I) the total dipolar charge is zero. In cases (II) and (III) the net dipolar charges are both negative with the re- lation,Q(II) D/Q(III) D=√ 3/2, where QD, of order unity, is obtained by integrating ρD(r) over the space of one Skyrmion and divide the result by the number of spins NSkinside the Skyrmion. If the field variation is slow on the scale of the Skyrmion, then the point-particle limit is reached by writing ρD(r) =QDNSk/summationtext jδ(r−rj) whererj spans the Skyrmion positions, and identical charge QDis assumed for all the Skyrmions. We arrive at the “poten- tial energy” of the collection of Skyrmion particles, HME=−λQDNSkd a/summationdisplay jE(rj). (6) A force acting on the Skyrmion will be given as the gra- dientFi=−∇iHME. Inter-Skyrmion interaction is ig- nored. The response of Skyrmions to a given force, on the other hand, is that of an electric charge in strong magnetic field, embodied in the Berry phase action3 (−2πS¯hQSkd/a3)/summationtext j/integraltext dt(rj×˙rj)·ˆz, whereQSkis the quantized Skyrmion charge[12, 25], and Sis the size of spin. Equation of motion follows from the combination of the Berry phase action and Eq. (6), vj=λ 4πS¯ha2NSkQD QSkˆz×∇jE(rj), (7) wherevjis thej-th Skyrmion velocity. Typical Hall ve- locity can be estimated by replacing |∇E|with ∆E/lSk, where ∆Eis the difference in the field strength between the left and the right edge of the Skyrmion and lSkis its diameter. Taking a2NSk∼l2 Skwe find the velocity λlSk 4πS¯h∆E∼10−6∆E[m2/V·s], (8) which gives the estimated drift velocity of 1 mm/s for the field strength difference of 103V/m across the Skyrmion. Experimental input parameters of lSk= 10−7m, and λ= 10−32C·m were taken from Ref. [14] in arriving at the estimation, as well as the dipolar and the Skyrmion chargesQD≈ −1 andQSk=−1. We may estimate the maximum allowed drift velocity by equating the dipolar energy difference λ∆Eacross the Skyrmion to the ex- change energy J, also corresponding to the formation en- ergy of one Skyrmion[24]. The maximum expected veloc- ity thus obtained is enormous, ∼104m/s forJ∼1meV, implying that with the right engineering one can achieve rather high Hall velocity of the Skyrmion. In an encour- aging step forward, electric field control of the Skyrmion lattice orientation in the Cu 2OSeO3crystal was recently demonstrated[26]. Results of LLG simulation is discussed next. To start, a sinusoidal field configuration Ei=E0sin(2πxi/Lx) is imposed on a rectangular Lx×Lysimulation lattice with Lxmuch larger than the Skyrmion size. In the absence of Gilbert damping, a single Skyrmion placed in such an environment moved along the “equi-potential line” in they-direction as expected from the guiding-center dy- namics of Eq. (7). In cases (II) and (III) where the dipolar charges are nonzero the velocity of the Skyrmion drift is found to be proportional to their respective dipo- lar charges QDas shown in Fig. 2. The drift velocity decreased continuously as we reduced the field gradient, obeying the relation (7) down to the zero velocity limit. The dipolar charge is zero in case (I), and indeed the Skyrmion remains stationary for sufficiently smooth E- field gradient. Even for this case, Skyrmions can move for field gradient modulations taking place on the length scale comparable to the Skyrmion radius, for the reason that the forces acting on the positive dipolar charge den- sity blobs (red in Fig. 1(a)) are not completely canceled by those on the negative dipolar charge density blobs (blue in Fig. 1(a)) for sufficiently rapid variations of thefield strength gradient. A small but non-zero drift veloc- ityensues, asshowninFig. 2. Longitudinalmotionalong the field gradient begins to develop with finite Gilbert damping, driving the Skyrmion center to the position of lowest “potential energy” E(r). For the Skyrmion lat- tice, imposing a uniform field gradient across the whole lattice may be too demanding experimentally, unless the magnetic crystal is cut in the form of a narrow strip the width of which is comparable to a few Skyrmion radii. In this case we indeed observe the constant drift of the Skyrmions along the length of the strip in response to the field gradient across it. The drift speed is still pro- portional to the field gradient, but about an order of magnitude less than that of an isolated Skyrmion under the same field gradient. We observed the excitation of breathing modes of Skyrmions when subject to a field gradient, and speculate that such breathing mode may interfere with the drift motion as the Skyrmions become closed-packed. 0 500 1000 1500 2000−30−25−20−15−10−505 ty(ab. unit) v(I) Hall=−7×10−4 v(II) Hall=−1.32×10−2 v(III) Hall=−1.49×10−2 v(II) Hall v(III) Hall≈Q(II) D Q(III) D=√ 3 2case (I) : B||[001] case (II) : B||[110] case (III): B||[111] FIG. 2: (color online) Skyrmion position versus time for cas es (I) through (III) for sinusoidal electric field modulation ( see text) with the Skyrmion center placed at the maximum field gradient position. The average Hall velocities (in arbitra ry units) in cases (II) and (III) indicated in the figure are ap- proximately proportional to the respective dipolar charge s, in agreement with Eq. (7). A small velocity remains in case (I) due to imperfect cancelation of forces across the dipola r charge profile. Several movie files are included in the supplementary information. II.gif and III.gif give Skyrmion motion for Ei=E0sin(2πxi/Lx) onLx×Ly= 66×66 lattice for magneto-electric couplings (II) and (III) in Eq. (3). III- Gilbert.gif gives the same E-field as III.gif, with finite Gilbert damping α= 0.2. I.gif describes the case (I) where the average dipolar charge is zero, with a rapidly varying electric field Ei=E0sin(2πxi/λx) andλxcom- parable to the Skyrmion radius. The case of a narrow strip with the field gradient across is shown in strip.gif. Mochizuki’s recent simulation[27] revealed that inter-4 nal motion of Skyrmions can be excited with the uniform a.c. magnetic field. Some of his predictions were con- firmed by the recent microwave measurement[29]. Here we show that uniform a.c. electric field can also ex- cite several internal modes due to the magneto-electric coupling. Time-localized electric field pulse was applied in the LLG simulation and the temporal response χ(t) was Fourier analyzed, where the response function χ(t) refersto(1 /2)/summationtext i([Sy i(t)]2−[Sx i(t)]2), (1/2)/summationtext i([Sz i(t)]2− [Sx i(t)]2), and (√ 3/2)/summationtext i[Sz i(t)]2for cases (I) through (III), respectively. (In Mochizuki’s work, the response function was the component of total spin along the a.c. magnetic field direction.) In case (I), the uniform electric field perturbs the ini- tial cylindrical symmetry ofSkyrmion spin profileso that/summationtext i([Sx i(t)]2−[Sy i(t)]2) becomes non-zero and the over- all shape becomes elliptical. The axes of the ellipse then rotates counter-clockwise about the Skyrmion center of mass as illustrated in supplementary figure, E-mode.gif. There are two additional modes of higher energies with broken cylindrical symmetry in case (I), labeled X1 and X2 in Fig. 3 and included as X1-mode.gif and X2- mode.gif in the supplementary. The rotational direction of the X1-mode is the same as in E-mode, while it is the opposite for X2-mode. As in Ref. [27], we find sharply defined breathing modesin cases(II) and(III) atthe appropriateresonance frequency ω, in fact the same frequency at which the a.c. magnetic field excites the breathing mode. The verti- cal dashed line in Fig. 3 indicates the common breath- ing mode frequency. Movie file B1-mode.gif shows the breathing mode in case (III). Additional, higher energy B2-mode (B2-mode.gif) was found in cases (II) and (III), which is the radial mode with one node, whereas the B1 mode is nodeless. In addition to the two breathing modes, E-mode and the two X-modes are excited in case (II) as well due to the partly in-plane nature of the spin perturbation, −(λE(t)/2)/summationtext i([Sz i(t)]2−[Sx i(t)]2). In contrast, case (III), where the perturbation −(√ 3λE(t)/2)/summationtext i[Sz i(t)]2 is purely out-of-plane, one only finds the B-modes. As a result, case (I) and (III) have no common resonance modes or peaks, while case (II) has all the peaks (though X1 and X2 peaks are small). Compared to the magnetic field-induced resonances, a richer variety of modes are excited by a.c. electric field. In particular, the E-mode has lower excitation energy than the B-mode and has a sharp resonance feature, which should make its detection a relatively straightforward task. Full analytic solution of the excited modes[28] will be given later. In summary, motivated by the recent discov- ery of magneto-electric material Cu 2OSeO3exhibiting Skyrmion lattice phase, we have outlined the theory of Skyrmion dynamics in such materials. Electric field gra- dient is identified as the source of Skyrmion Hall motion.00.20.40.60.81Imχ(ω) (ab. unit) B1 R1 R2(a)B||z,Bω||x,y B||z,Bω||z 00.05 0.10.15 0.20.25 0.30.3500.20.40.60.81 ωImχ(ω) (ab. unit) E X1X2B1 B2(b)case (I) : B||[001],Eω||[001] case (II) : B||[110],Eω||[001] case (III): B||[111],Eω||[111] FIG. 3: (color online) (a) Absorption spectra for a.c. unifo rm magnetic field as in Mochizuki’s work (reproduced here for comparison). (b) Absorption spectra for a.c. uniform elect ric field in cases (I) through (III). In case (I) where there is no net dipolar charge we find three low-energy modes E, X1, and X2. For case (III) where the dipolar charge is finite we find B1 and B2 radial modes excited. Case (II) exhibits all five modes. Detailed description of each mode is given in the text . 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1810.07384v2.Perpendicularly_magnetized_YIG_films_with_small_Gilbert_damping_constant_and_anomalous_spin_transport_properties.pdf

Perpendicularly magnetized YIG films with small Gilbert damping constant and anomalous spin transport properties Qianbiao Liu1, 2, Kangkang Meng1*, Zedong Xu3, Tao Zhu4, Xiao guang Xu1, Jun Miao1 & Yong Jiang1* 1. Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and Technology Beijing, Beijing 100083, China 2. Applied and Engineering physics, Cornell University, Ithaca, NY 14853, USA 3. Department of Physics, South University of Science and Technology of China , Shenzhen 518055, China 4. Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Email: kkmeng@ustb.edu.cn ; yjiang@ustb.edu.cn Abstract: The Y 3Fe5O12 (YIG) films with perpendicular magnetic anisotropy (PMA) have recently attracted a great deal of attention for spintronics applications. Here, w e report the induced PMA in the ultrathin YIG films grown on (Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12 (SGGG) substrate s by epitaxial strain without preprocessing. Reciprocal space mapping shows that the film s are lattice -matched to the substrate s without strain relaxation. Through ferromagnetic resonance and polarized neutron reflectometry measurements, we find that these YIG films have ultra-low Gilbert damping constant (α < 1×10-5) with a magnetic dead layer as thin as about 0.3 nm at the YIG/SGGG interfaces. Moreover, the transport behavior of the Pt/YIG/SGGG films reveals an enhancement of spin mixing conductance and a large non-monotonic magnetic field dependence of anomalous Hall effect as compared with the Pt/YIG/Gd 3Ga5O12 (GGG) films. The non- monotonic anomalous Hall signal is extracted in the temperature range from 150 to 350 K, which has been ascribed to the possible non -collinear magnetic order at the Pt/YIG interface induced by uniaxial strain. The spin transport in ferrim agnetic insulator (FMI) based devices has received considerable interest due to its free of current -induced Joule heating and beneficial for low-power spintronic s applications [1, 2]. Especially, the high-quality Y3Fe5O12 (YIG) film as a widely studied FMI has low damping constant, low magnetostriction and small magnetocrystalline anisotropy, making it a key material for magnonics and spin caloritronics . Though the magnon s can carr y information over distances as long as millimeters in YIG film , there remain s a challenge to control its magnetic anisotropy while maintaining the low damping constant [3] , especially for the thin film with perpendicular magnetic anisotropy (PMA) , which is very useful for spin polarizers, spin-torque oscillators, magneto -optical d evices and m agnon valve s [4-7]. In addition, the spin- orbit torque (SOT) induced magnetization switching with low current densities has been realized in non -magnetic heavy metal (HM)/FMI heterostructures , paving the road towards ultralow -dissipation SOT de vices based on FMI s [8-10]. Furthermore, p revious theoretical studies have pointed that the current density will become much smaller if the domain structures were topologically protected (chiral) [11]. However, most FMI films favor in-plane easy axis dominated by shape anisotropy , and the investigation is eclipsed as compared with ferromagnetic materials which show abundant and interesting domain structures such as chiral domain walls and magnetic skyrmions et al. [12-17]. Recently, the interface- induc ed chiral domain walls have been observed in centrosymmetric oxides Tm 3Fe5O12 (TmIG) thin films, and the domain walls can be propelled by spin current from an adjacent platinum layer [18]. Similar with the TmIG films, the possible chiral magnetic structures are also expected in the YIG films with lower damping constan t, which would further improve the chiral domain walls’ motion speed. Recently, several ways have been reported to attain the perpendicular ly magnetized YIG films , one of which is utiliz ing the lattice distortion and magnetoelastic effect induced by epitaxial strain [1 9-22]. It is noted that the strain control can not only enable the field -free magnetization switching but also assist the stabilization of the non- collinear magnetic textures in a broad range of magnetic field and temperature. Therefore, abundant and interesting physical phenomena would emerge in epitaxial grown YIG films with PMA. However, either varying the buffer layer or doping would increase the Gilbert damping constant of YIG, which will affect the efficiency of the SOT induced magnetization switching [20, 21]. On the other hand, these preprocessing would lead to a more complicate magnetic structures and impede the further discussion of spin transport properties such as possible topological Hall effect (THE). In this work, we realized the PMA of ultrathin YIG films deposited on SGGG substrates due to epitaxial strain . Through ferromagnetic resonance (FMR) and polarized neutron reflectometry (PNR) measurements, we have found that the YIG films had small Gilbert damping constant with a magnetic dead layer as thin as about 0.3 nm at the YIG/SGGG interfaces. Moreover, we have carried out the transport measurements of the Pt/YIG/SGGG films and observed a large non -monotonic magnet ic field dependence of the anomalous Hall resistivity, which did not exis t in the compared Pt/YIG/GGG films. The non -monotonic anomalous Hall signal was extracted in the temperature range from 150 to 350 K, and we ascribed it to the possible non -collinear magnetic order at the Pt/YIG interfaces induced by uniaxial strain. Results Structural and magnetic characterization. The epitaxial YIG films with varying thickness from 3 to 90 nm were grown on the [111] -oriented GGG substrate s (lattice parameter a = 1.237 nm) and SGGG substrates (lattice parameter a = 1.248 nm) respectively by pulsed laser deposition technique (see methods). After the deposition, we have investigated the surface morphology of the two kinds of films using atomic force microscopy (AFM) as shown in Fig. 1 ( a), and the two films have a similar and small surface roughness ~0.1 nm. Fig. 1 ( b) shows the enlarged XRD ω-2θ scan spectra of the YIG (40 nm) thin film s grow n on the two different substrates (more details are shown in the Supplementary Note 1 ), and they all show predominant (444) diffraction peaks without any other diffraction peaks, excluding impurity phases or other crystallographic orientation s and indicat ing the single -phase nature. According to the (444) diffraction pe ak position and the reciprocal space map of the (642) reflection of a 40 -nm-thick YIG film grown on SGGG as shown in Fig. 1(c), we have found that the lattice constant of SGGG (~1.248 nm) substrate was larger than the YIG layer (~1.236 nm). We quantify thi s biaxial strain as ξ = (aOP - aIP)/aIP, where a OP and aIP represent the pseudo cubic lattice constant calculated from the ou t-of-plane lattice constant d(4 4 4) OP and in-plane lattice constant d(1 1 0) IP, respectively, following the equation of 2 2 2lkhad ++= , with h, k, and l standing for the Miller indices of the crystal planes . It indicates that the SGGG substrate provides a tensile stress ( ξ ~ 0.84%) [21]. At the same time, the magnetic properties of the YIG films grown on the two different substrates were measured via VSM magnetometry at room temperature. According to the magnetic field ( H) dependence of the magnetization (M) as shown in Fig. 1 (d), the magnetic anisotropy of the YIG film grown on SGGG substrate has been modulated by strain, while the two films have similar in -plane M-H curves. To further investigate the quality of the YIG films grown on SGGG substrates and exclude the possibility of the strain induced large stoichiometry and lattice mismatch, compositional analyse s were carried out using x -ray photoelectron spectroscopy (XPS) and PNR. As shown in Fig. 2 (a), the difference of binding energy between the 2p 3/2 peak and the satellite peak is about 8.0 eV, and the Fe ions are determined to be in the 3+ valence state. It is found that there is no obvious difference for Fe elements in the YIG films grown on GGG and SGGG substrates. The Y 3 d spectrums show a small energy shift as shown in Fig. 2 (b) and the binding energy shift may be related to the lattice strain and the variation of bond length [21]. Therefore, the stoichiometry of the YIG surface has not been dramatically modified with the strain control. Furthermore, we have performed the PNR meas urement to probe the depth dependent struc ture and magnetic information of YIG films grown on SGGG substrates. The PNR signals and scattering length density (SLD) profiles for YIG (12.8 nm)/SGGG films by applying an in- plane magnetic field of 900 mT at room temperature are shown in Fig. 2 ( c) and ( d), respectively. In Fig. 2(c), R++ and R-- are the nonspin -flip reflectivities, where the spin polarizations are the same for the incoming and reflected neutrons. The inset of Fig. 2(c) shows the experimental and simulated spin -asymmetry (SA), defined as SA = ( R++ – R--)/(R++ + R--), as a function of scattering vector Q. A reasonable fitting was obtained with a three- layer model for the single YIG film, containing the interface layer , main YIG layer and surface layer. The nuclear SLD and magnetic SLD are directly proportional to the nuclear scattering potential and the magnetization , respectively . Then, the depth- resolved structural and magnetic SLD profiles delivered by fitting are s hown in Fig. 2(d) . The Z -axis represents the distance for the vertical direction of the film, where Z = 0 indicates the position at the YIG/SGGG interface. It is obvious that there is few Gd diffusion into the YIG film, and the dead layer (0.3 nm ) is much thinner than the reported values (5-10 nm) between YIG (or T mIG) and substrates [23 -25]. The net magnetization of YIG is 3.36 μB (～140 emu/cm3), which is similar with that of bulk YIG [2 6]. The PNR results also showed that besides the YIG/ SGGG interface region, there is also 1.51- nm-thick nonmagnetic surface layer, which may be Y2O3 and is likely to be extremely important in magnetic proximity effect [ 23]. Dynamical characterization and spin transport properties. To quantitatively determine the magnetic anisotropy and dynamic properties of the YIG films, the FMR spectra were measured at room temperature using an electron paramagnetic resonance spectrometer with rotating the films. Fig. 3(a) shows the geometric configuration of the angle reso lved FMR measurements. We use the FMR absorption line shape to extract the resonance field (H res) and peak -to-peak linewidth ( ΔHpp) at different θ for the 40 -nm-thick YIG fil ms grown on GGG and SGGG substrates, respectively. The details for 3 -nm-thick YIG film are show n in the Supp lementary Note 2 . According to the angle dependence of H res as shown in Fig. 3(b), one can find that as compared with the YIG films grown on GGG substrate s, the minimum Hres of the 40- nm-thick YIG film grown on SGGG substrate increases with varying θ from 0° to 90° .On the other hand, according to the frequency dependence of Hres for the YIG (40 nm) films with applying H in the XY plane as shown in Fig. 3(c), in contrast to the YIG/GGG films, the H res in YIG/SGGG films could not be fitted by the in-plane magnetic anisotropy Kittel formula 21)] 4 ( )[2(/ eff res res πM H Hπγ/ f + = . All these results indicate that the easy axis of YIG (40 nm) /SGGG films lies out -of-plane. The angle dependent ΔHpp for the two films are also compared as shown in Fig. 3(d) , the 40-nm-thick YIG film grown on SGGG substrate has an optimal value of Δ Hpp as low as 0.4 mT at θ =64°, and the corresponding FMR absorption line and Lorentz fitting curve are shown in Fig. 3(e). Generally , the ΔHpp is expected to be minimum (maximum) along magnetic easy (hard) axis, which is basically coincident with the angle dependent ΔHpp for the YIG films grown on GGG substrates. However, as shown in Fig. 3(d), the ΔHpp for the YIG/SGGG films shows an anomalous variation. The lowest ΔHpp at θ=64° could be ascribed to the high YIG film quality and ultrathin magnetic dead layer at the YIG/SGGG interface. It should be noted that , as compared with YIG/GGG films , the Δ Hpp is independent on the frequency from 5 GHz to 14 GHz as shown in Fig. 3(f). Then, w e have calculate d the Gilbert damping constant α of the YIG (40 nm)/SGGG films by extracting the Δ Hpp at each frequency as shown in Fig. 3(f). The obtained α is smaller tha n 1 × 10−5, which is one order of magnitude lower than t he report in Ref. [20] and would open new perspectives for the magnetization dynamics. According to the theor etical theme, the ΔHpp consists of three parts: Gilbert damping, two magnons scattering relaxation process and inhomogeneities, in which both the Gilbert damping and the two magnons scattering relaxation process depend on frequency. Therefore, the large Δ Hpp in the YIG/SGGG films mainly stems from the inhomogeneities, w hich will be discussed next with the help of the transport measurements. All of the above results have proven that the ultrathin YIG films grown on SGGG substrate s have not only evident PMA but also ultra-low Gilbert damping constant. Furthermore, we have also investigated the spin transport properties for the high quality YIG film s grown on SGGG substrate s, which are basically sensitive to the magnetic details of YIG. The magnetoresistance (MR) has been proved as a powerful tool to effectively explore magnetic information originating from the interfaces [ 27]. The temperature dependent spin Hall magnetoresistance (SMR) of the Pt (5 nm)/YIG (3 nm) films grown on the two different substrates were measured using a small and non-perturbative current densit y (~ 106 A/cm2), and the s ketches of the measurement is shown in Fig. 4 (a). The β scan of the longitudinal MR, which is defined as MR=ΔρXX/ρXX(0)=[ρXX(β) -ρXX(0)]/ρXX(0) in the YZ plane for the two films under a 3 T field (enough to saturate the magnetization of YIG ), shows cos2β behavior s with varying temperature for the Pt/YIG/GGG and Pt/YIG/SGGG films as shown in Fig. 4 (b) and (c), respectively. T he SMR of the Pt/YIG /SGGG films is larger than that of the Pt/YIG /GGG films with the same thickness of YIG at room temperature, indica ting an enhanced spin mixing conductance ( G↑↓) in the Pt/YIG /SGGG films. Here, it should be noted that the spin transport properties for the Pt layers ar e expected to be the same because of the similar resistivity and film s quality . Therefore, the SGGG substrate not only induces the PMA but also enhances G ↑↓ at the Pt/YIG interface. Then, we have also investigated the field dependent Hall resistivities in the Pt/YIG/SGGG films at the temperature range from 260 to 350 K as shown in Fig. 4(d). Though the conduction electrons cannot penetrate into the FMI layer, the possible anomalous Hall effect (AHE) at the HM/FMI interface is proposed to emerge, and the total Hall resistivity can usually be expressed as the sum of various contributions [28, 29]: S-A S H ρ ρ H R ρ + + =0 , (1) where R0 is the normal Hall coefficient, ρ S the transverse manifestation of SMR, and ρS-A the spin Hall anomalous Hall effect (SAHE) resistivity. Notably, the external field is applied out -of-plane, and ρs (~Δρ1mxmy) can be neglected [ 29]. Interestingly, the film grown on SGGG substrate shows a bump and dip feature during the hysteretic measurements in the temperature range from 260 to 350 K. In the following discussion, we term the part of extra anomalous signals as the anomalous SAHE resistivity ( ρ A-S-A). The ρ A-S-A signals clearly coexist with the large background of normal Hall effect. Notably, the broken (space) inversion symmetry with strong spin-orbit coupling (SOC) will induce the Dzyaloshinskii -Moriya interaction (DMI) . If the DMI could be compared with the Heisenberg exchange interaction and the magnetic anisotropy that were controlled by st rain, it c ould stabilize non-collinear magnetic textures such as skyrmions, producing a fictitious magnetic field and the THE . The ρA-S-A signals indicate that a chiral spin texture may exist, which is similar with B20-type compounds Mn 3Si and Mn 3Ge [ 30,31]. To more clearly demonstrate the origin of the anomalous signals, we have subtracted the normal Hall term , and the temperature dependence of ( ρS-A + ρ A-S-A) has been shown in Fig. 4 (e). Then, we can further discern the peak and hump structure s in the temperature range from 260 to 350 K. The SAHE contribution ρS-A can be expressed as 𝜌𝑆−𝐴=𝛥𝜌2𝑚𝑍 [32, 33], where 𝛥𝜌2 is the coefficient depending on the imaginary part of G ↑↓, and mz is the unit vector of the magnetization orientation along the Z direction . The extracted Hall resist ivity ρA-S-A has been shown in Fig. 4 (f), and the temperature dependence of the largest ρA-S-A (𝜌𝐴−𝑆−𝐴Max) in all the films have been shown in Fig. 4 (g). Finite values of 𝜌𝐴−𝑆−𝐴Max exist in the temperature range from 150 to 350 K , which is much d ifferen t from that in B20 -type bulk chiral magnets which are subjected to low temperature and large magnetic field [34]. The large non -monotonic magnetic field dependence of anomalous Hall resistivity could not stem from the We yl points, and the more detailed discussion was shown in the Supplementary Note 3. To further discuss the origin of the anomalous transport signals, we have investigated the small field dependence of the Hall resistances for Pt (5 nm) /YIG (40 nm)/SGGG films as shown in Fig. 5(a). The out-of-plane hysteresis loop of Pt/YIG/SGGG is not central symmetry, which indicates the existence of an internal field leading to opposite velocities of up to down and down to domain walls in the presence of current along the +X direction. The large field dependences of the Hall resistances are shown in Fig. 5(b), which could not be described by Equation (1). There are large variations for the Hall signals when the external magnetic field is lower than the saturation field ( Bs) of YIG film (~50 mT at 300 K and ~150 mT at 50 K). More interestingly, we have firstly applied a large out -of-plane external magnetic field of +0.8 T ( -0.8 T) above Bs to saturate the out -of-plane magnetization comp onent MZ > 0 ( MZ < 0), then decreased the field to zero, finally the Hall resistances were measured in the small field range ( ± 400 Oe), from which we could find that the shape was reversed as shown in Fig. 5(c). Here, we infer that the magnetic structures at the Pt/YIG interface grown on SGGG substrate could not be a simple linear magnetic order. Theoretically , an additional chirality -driven Hall effect might be present in the ferromagnetic regime due to spin canting [3 5-38]. It has been found that the str ain from an insulating substrate could produce a tetragonal distortion, which would drive an orbital selection, modifying the electronic properties and the magnetic ordering of manganites. For A 1-xBxMnO 3 perovskites, a compressive strain makes the ferromagnetic configuration relatively more stable than the antiferromagnetic state [3 9]. On the other hand, the strain would induce the spin canting [ 40]. A variety of experiments and theories have reported that the ion substitute, defect and magnetoelast ic interaction would cant the magnetization of YIG [41-43]. Therefore, if we could modify the magnetic order by epitaxial strain, the non-collinear magnetic structure is expected to emerge in the YIG film. For YIG crystalline structure, the two Fe sites ar e located on the octahedrally coordinated 16(a) site and the tetrahedrally coordinated 24(d) site, align ing antiparallel with each other [44]. According to the XRD and RSM results, the tensile strain due to SGGG substrate would result in the distortion ang le of the facets of the YIG unit cell smaller than 90 ° [45]. Therefore, the magneti zations of Fe at two sublattice s should be discussed separately rather than as a whole. Then, t he anomalous signals of Pt/YIG/SGGG films could be ascribed to the emergence o f four different Fe3+ magnetic orientation s in strained Pt/YIG films, which are shown in Fig. 5(d). For better to understand our results, w e assume that, in analogy with ρ S, the ρA-S-A is larger than ρA-S and scales linearly with m ymz and mxmz. With applying a large external field H along Z axis, the uncompensated magnetic moment at the tetrahedrally coordinated 24(d) is along with the external fields H direction for |H | > Bs, and the magnetic moment tends to be along A (-A) axis when the external fields is swept from 0.8 T (-0.8 T) to 0 T. Then, if the Hall resistance was measured at small out -of-plane field , the uncompensated magnetic moment would switch from A (-A) axis to B (-B) axis. In this case, the ρ A-S-A that scales with Δ ρ3(mymz+mxmz) would change the sign because the mz is switched from the Z axis to - Z axis as shown in Fig. 5(c). However, there is still some problem that needs to be further clarified. There are no anomalous signals in Pt/YIG/GGG films that could be ascribed to the weak strength of Δρ3 or the strong magnetic anisotropy . It is still valued for further discussion of the origin of Δ ρ3 that whether it could stem from the skrymions et al ., but until now we have not observed any chiral domain structures in Pt/YIG/SGGG films through the Lorentz transmission electron microscopy. Therefore, we hope that future work would involve more detailed magnetic microscopy imaging and microstructure analysis, which can further elucidate the real microscopic origin of the large non -monotonic magnetic field dependence of anomalous Hall resistivity. Conclusion In conclusion, the YIG film with PMA could be realized using both epitaxial strain and growth -induced anisotropies. These YIG films grown on SGGG substrates had low G ilbert damping constants (<1 ×10 -5) with a magnetic dead layer as thin as about 0.3 nm at the YIG/SGGG interface. Moreover, we observe d a large non -monotonic magnetic field dependence of anomalous Hall resistivity in Pt/YIG/SGGG films, which did not exist in Pt/YIG/GGG films. The non -monotonic anomalous portion of the Hall signal was extracted in the temperature range from 150 to 350 K and w e ascribed it to the possible non -collinear magnetic order at the Pt/YIG interface induced by uniaxial strain. The present work not only demonstrate that the strain control can effectively tune the electromagnetic properties of FMI but also open up the exp loration of non -collinear spin texture for fundamental physics and magnetic storage technologies based on FMI. Methods Sample preparation. The epitaxial YIG films with varying thickness from 3 to 90 nm were grown on the [111] -oriented GGG substrate s (lattice parameter a =1.237 nm) and SGGG substrates (lattice parameter a =1.248 nm) respectively by pulsed laser deposition technique . The growth temperature was TS =780 ℃ and the oxyg pressure was varied from 10 to 50 Pa . Then, the films were annealed at 780℃ for 30 min at the oxygen pressure of 200 Pa . The Pt (5nm) layer was deposited on the top of YIG films at room temperature by magnetron sputtering. After the deposition, the electron beam lithography and Ar ion milling were used to pattern Hall bars, and a lift-off process was used to form contact electrodes . The size of all the Hall bars is 20 μm×120 μm. Structural and magnetic characterization. The s urface morphology was measured by AFM (Bruke Dimension Icon). Magnetization measurements were carried out using a Physical Property Measurement System (PPMS) VSM. A detailed investigation of the magnetic information of Y IG was investigated by PNR at the Spallation Neutron Source of China. Ferromagnetic resonance measurements. The measurement setup is depicted in Fig. 3(a). For FMR measurements, the DC magnetic field was modulated with an AC field. The transmitted signal was detected by a lock -in amplifier. We observed the FMR spectrum of the sample by sweeping the external magnetic field. The data obtained were then fitted to a sum of symmetric and antisymmetric Lorentzian functions to extract the linewidth. Spin transport measurements . The measurements were carried out using PPMS DynaCool. Acknowledgments The authors thanks Prof. L. Q. Yan and Y. Sun for the technical assistant in ferromagnetic resonance measurement . This work was partially supported by the National Science Foundation of China (Grant Nos. 51971027, 51927802, 51971023 , 51731003, 51671019, 51602022, 61674013, 51602025), and the Fundamental Research Funds for the Central Universities (FRF- TP-19-001A3). References [1] Wu, M.-Z. & Hoffmann , A. Recent advances in magnetic insulators from spintronics to microwave applications. Academic Press , New York, 64 , 408 (2013) . [2] Maekawa, S. Concepts in spin electronics. Oxford Univ., ( 2006) . [3] Neusser, S. & Grundler, D. Magnonics: spin waves on the nanoscale. Adv. Mater., 21, 2927- 2932 ( 2009) . [4] Kajiwara , Y. et al. Transmission of electrical signals by spin -wave interconversion in a magnetic insulator. Nature 464, 262- 266 (2010). [5] Wu, H. et al. Magnon valve effect between two magnetic insulators. Phys. Rev. Lett. 120, 097205 ( 2018). [6] Dai, Y. et al. Observation of giant interfacial spin Hall angle in Y 3Fe5O12/Pt heterostructures. Phys. Rev. B . 100, 064404 ( 2019) . 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[14] Nii, Y. et al. Uniaxial stress control of skyrmion phase. Nat. Commun. 6, 1 -7 (2015) . [15] Meng, K. K. et al. Anomalous Hall effect and spin -orbit torques in MnGa/IrMn films: Modification from strong spin Hall effect of the antiferromagnet. Phys. Rev. B 94, 214413 (2016) . [16] Ado, I. A., Oleg A. T. & Titov, M. Microscopic theory of spin -orbit torques in two dimensions. Physical Review B 95, 094401 (2017) . [17] Liu, Q. B. et al. Modulated spin orbit torque in a Pt/Co/Pt /YIG multilayer by nonequilibrium proximity effect. A ppl. Phys. Lett. 112, 022402 (2018) . [18] Avci, C. O. et al. Interface -driven chiral magnetism and current -driven domain walls in insulating magnetic garnets. Nat. Nanotechnol. 14, 561-566 (2019). [19] Fu, J. et a l. Epitaxial growth of Y 3Fe5O12 thin films with perpendicular magnetic anisotropy. Appl. Phys. Lett. 110 , 202403 (2017) . [20] Soumah, L. et al. Ultra -low damping insulating magnetic thin films get perpendicular. Nat. Commun. 9, 1-6 (2018). [21] Wang C. T. et al. Con trolling the magnetic anisotropy in epitaxial Y 3Fe5O12 films by manganese doping. Phys. Rev. B 96, 224403 (2017). [22] Bauer J. J., Rosenberg E. R. & Ross C. A. Perpendicular magnetic anisotropy and spin mixing conductance in polycrystalline europium iron garnet thin films. Phys. Rev. Mater. 2 , 094405 (2018). [23] Cooper J. F. et al. Unexpected structural and magnetic depth dependence of YIG thin films Phys . Rev. B 96, 104404 (2017) . [24] Suturin S. M. et al. Role of gallium diffusion in the formation of a magnetically dead layer at the Y 3Fe5O12/Gd 3Ga5O12 epitaxial interface. 104404. Phys. Rev. Mater. 2, 104404 (2018) . [25] Shao Q. et al. Exploring interfacial exchang e coupling and sublattice effect in heavy metal/ferrimagnetic insulator heterostructures using Hall measurements, x-ray magnetic circular dichroism, and neutron reflectometry. Phys. Rev. B 99, 104401 (2019) . [26] Hansen P., Röschmann P & Tolksdorf W. Saturation magnetization of gallium ‐ substituted yttrium iron garnet. J. Appl. Phys. 45, 2728- 2732 (1974) . [27] Vélez S. et al. Competing effects at Pt/YIG interfaces: Spin Hall magnetoresistance, magnon excitations, and magnetic frustration. Phys. Rev. B 94, 174405 (2016) . [28] Chen Y. T. et al. Theory of spin Hall magnetoresistance. Phys. Rev. B 87 , 144411(2013). [29] Meyer S. et al. Anomalous Hall effect in YIG/Pt bilayers. Appl. Phys. Lett. 106, 132402 (2015). [30] Liang D. et al. Current -driven dynamics of skyrmions stabilized in M nSi nanowires revealed by topological Hall effect. Nat. Commun. 6, 1-8 (2015) . [31] Shibata K. et al. Towards control of the size and helicity of skyrmions in helimagnetic alloys by spin- orbit coupling. Nat. Nanotech. 8, 723- 728 (2013) . [32] Vlietstra N. et al. Spin -Hall magnetoresistance in platinum on yttrium iron garnet: Dependence on platinum thickness and in- plane/out -of-plane magnetization. Phys. Rev. B 87, 184421 (2013). [33] Xiao D., Chang M. C. & Niu Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010). [34] Neubauer A. et al. Topological Hall effect in the A phase of MnSi. Phys. Rev. Lett. 102, 186602 (2009). [35] Kimata M. et al. Magnetic and magnetic inverse spin Hall effects in a non-collinear antiferromagnet. Nature 565, 627-630 (2019). [36] Hou Z. et al. Observation of various and spontaneous magnetic skyrmionic bubbles at room temperature in a frustrated kagome magnet with uniaxial magnetic anisotropy. Adv. Mater. 29, 1701144 (2017) . [37] Leonov A. O. & Mostovoy M. Multiply periodic s tates and isolated skyrmions in an anisotropic frustrated magnet. Nat. Commun. 6, 1-8 (2015) . [38] Nakatsuji S., Kiyohara N. & Higo T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212-215 (2015). [39] Quindeau A. et al. Tm 3Fe5O12/Pt heterostructures with perpendicular magnetic anisotropy for spintronic applications. Adv. Electron. Mater. 3, 1600376 (2017). [40] Singh G. et al. Strain induced magnetic domain evolution and spin reorientation transition in epitaxial manganit e films. Appl. Phys. Lett. 101 , 022411 (2012). [41] Parker G. N. & Saslow W. M. Defect interactions and canting in ferromagnets. Phys. Rev. B 38, 11718 (1988). [42] Rosencwaig A. Localized canting model for substituted ferrimagnets. I. Singly substituted YIG systems. Can. J. Phys. 48, 2857- 2867(1970). [43] AULD B. A. Nonlinear magnetoelastic interactions. Proceedings of the IEEE, 53, 1517- 1533 (1965). [44] Ching W. Y., Gu Z. & Xu Y N. Th eoretical calculation of the optical properties of Y 3Fe5O12. J. Appl. Phys. 89, 6883- 6885 (2001). [45] Baena A., Brey L. & Calder ón M. J. Effect of strain on the orbital and magnetic ordering of manganite thin films and their interface with an insulator. Phys. Rev. B 83, 064424 (2011). Figure Captions Fig. 1 Structural and magnetic properties of YIG films. (a) AFM images of the YIG films grown on the two substrates (scale bar, 1 μ m). (b) XRD ω-2θ scans of the two different YIG films grown on the two substrates . (c) High -resolution XRD reciprocal space map of t he YIG film deposited on the SGGG substrate. (d) Field dependence of the normalized magnetization of the YIG films grown on the two different substrates . Fig. 2 Structural and magnetic properties of YIG films. Room temperature XPS spectra of (a) Fe 2p and (b) Y 3d for YIG films grown on the two substrates . (c) P NR signals (with a 900 mT in -plane field) for the spin -polarized R++ and R-- channels. Inset: The experimental and simulated SA as a function of scattering vector Q. (d) SLD profiles of the YIG/SGGG films. The nuclear SLD and magnetic SLD is directly proportional to the nuclear scattering potential and the magnetization , respectively. Fig. 3 Dynamical properties of YIG films . (a) The geometric configuration of the angle dependent FMR measurement. (b) The angle dependence of the H res for the YIG films on GGG and SGGG substrates. (c) The frequency dependence of the H res for YIG films grown on GGG and S GGG substrates. (d) The ang le dependence of Δ Hpp for the YIG films on GGG and SGGG substrates. (e) FMR spectrum of the 40-nm-thick YIG film grown on SGGG substrate with 9.46 GHz at θ =64°. (f) The frequency dependence of Δ Hpp for the 40 -nm-thick YIG films grown on GGG and SGGG substr ates. Fig. 4 Spin transport properties of Pt/YIG (3nm) films . (a) The definition of the angle, the axes and the measurement configurations. ( b) and ( c) Longitudinal MR at different temperatures in Pt/YIG/GGG and Pt/YIG/SGGG films respectively (The applied magnetic field is 3 T). (d) Total Hall resistivities vs H for Pt/YIG/SGGG films in the temperature range from 260 to 300 K. (e) (ρS-A+ρA-S-A) vs H for two films in the temperature range from 260 to 300 K. (f) ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. Inset: ρS-A and ρS-A + ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. (g) Temperature dependence of the 𝜌𝐴−𝑆−𝐴𝑀𝑎𝑥. Figure 5 S pin transport properties of Pt/YIG ( 40 nm) films . (a) and (b) The Hall resistances vs H for the Pt/YIG/SGGG films in the temperature range from 50 to 300 K in small and large magnetic field range, respectively. (c) The Hall resistances vs H at small magnetic field range after sweeping a large out -of-plane magnetic field +0.8 T (black line) and - 0.8 T (red line) to zero. (d) An illustration of the orientations of the magnetizations Fe ( a) and Fe ( d) in YIG films with the normal in -plane magnetic anisotropy (IMA), the ideal strain induced PMA and the actual magnetic anisotropy grown on SGGG in our work.

1909.05881v2.Spin_Transport_in_Thick_Insulating_Antiferromagnetic_Films.pdf

Spin Transport in Thick Insulating Antiferromagnetic Films Roberto E. Troncoso1, Scott A. Bender2, Arne Brataas1, and Rembert A. Duine1;2;3 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands and 3Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Spin transport of magnonic excitations in uniaxial insulating antiferromagnets (AFs) is investi- gated. In linear response to spin biasing and a temperature gradient, the spin transport properties of normal-metal{insulating antiferromagnet{normal-metal heterostructures are calculated. We fo- cus on the thick-lm regime, where the AF is thicker than the magnon equilibration length. This regime allows the use of a drift-diusion approach, which is opposed to the thin-lm limit considered by Bender et al. 2017, where a stochastic approach is justied. We obtain the temperature- and thickness-dependence of the structural spin Seebeck coecient Sand magnon conductance G. In their evaluation we incorporate eects from eld- and temperature-dependent spin conserving inter- magnon scattering processes. Furthermore, the interfacial spin transport is studied by evaluating the contact magnon conductances in a microscopic model that accounts for the sub-lattice sym- metry breaking at the interface. We nd that while inter-magnon scattering does slightly suppress the spin Seebeck eect, transport is generally unaected, with the relevant spin decay length being determined by non-magnon-conserving processes such as Gilbert damping. In addition, we nd that while the structural spin conductance may be enhanced near the spin ip transition, it does not diverge due to spin impedance at the normal metal|magnet interfaces. I. INTRODUCTION Spin-wave excitations in magnetic materials are a cor- nerstone in spintronics for the transport of spin-angular momentum1,2. The usage of antiferromagnetic materials has gained a renewed interest due to their high potential for practical applications. The most attractive proper- ties of antiferromagnets (AFs) are the lack of stray elds and the fast dynamics that can operate in the THz fre- quency range3. Those attributes have the potential to tackle current technological bottlenecks, like the absence of practical solutions to generate and detect electromag- netic waves in the spectrum ranging from 0.3 THz to 30 THz (the terahertz gap)2. Nevertheless, the control and access to the high-frequency response of AFs is challeng- ing. New proposals circumvent one of these obstacles by manipulating metallic AFs with charge currents, through the so-called spin-orbit torques4{6. Antiferromagnetic in- sulators , however, oer a compelling alternative since the Joule heating caused by moving electrons is absent. In such systems, the study of transport instead focuses on their magnetic excitations. In insulating AFs the spin-angular momentum is trans- ferred by their quantized low-energy excitations, i.e., magnons. Since the AF in its groundstate is composed of two collinear magnetic sublattices, magnons carry oppo- site spin angular momentum. The transport of magnons has been experimentally achieved through the longitu- dinal spin Seebeck eect in AF jNM7{13(NM, normal metal) and FMjAFjNM14{18(FM, ferromagnets) het- erostructures, in which magnons were driven by a ther- mal gradient across the AF. Alternatively, thermal injec- tion of magnons in AFs has been studied19,20by a spin accumulation at the contact with adjacent metals. Inaddition, it was shown that thermal magnon transport takes place at zero spin bias when the sublattice symme- try is broken at the interfaces, e.g., induced by interfacial magnetically uncompensated AF order21. Complemen- tarily, coherent spin transport induced by spin accumu- lation has been earlier considered and predicted to result in spin super uidity22,23or Bose-Einstein condensates of magnons24. Recently, it has been shown via non-local spin transport measurements that magnons remarkably propagate at long distances in insulating AFs; -Fe2O325, Cr2O326, MnPS 327and also in NiO via spin-pumping ex- periments with YIG14. Their exceptional transport prop- erties, as well as those reported in Refs.7{11, are governed by the spin conductance and spin Seebeck coecients. Rezendeetal:28discussed theoretically the spin Seebeck eect in AFs in contact with a normal metal. They ob- tain the Seebeck coecient in terms of temperature and magnetic eld, nding a good qualitative agreement with measurements in MnF 2/Pt7. In addition, it was found that magnon scattering processes aect signicantly the spin Seebeck coecient. Hitherto there has been no com- plete studies on the underlying mechanism for spin trans- port coecients, e.g., their thickness-, temperature- and eld-dependence, eects derived from magnon-magnon interactions or when the sublattice symmetry is broken at the interfaces. In this work, we describe spin transport though a left normal-metal{insulating antiferromagnet{right normal- metal (LNMjAFjRNM) heterostructure. As depicted in Fig. 1, magnon transport is driven by either a temper- ature gradient or spin biasing. We focus on the thick- lm limit, where the thickness dof the AF is greater than the internal equilibration length lfor the magnon gas. This limit implies a diusive regime where magnonsarXiv:1909.05881v2 [cond-mat.mes-hall] 4 Feb 20202 are in a local equilibrium described by a local tempera- ture and chemical potential. This is in contrast to our earlier, stochastic treatment of thin-lms ( dl) where spin waves do not establish a local equilibrium19. Specif- ically, we study the spin transport by evaluating, via a phenomenological theory, the structural spin Seebeck co- ecientSand magnon conductance G. Furthermore, we investigate their temperature- and magnetic-eld de- pendence by computing the interfacial conductance coef- cients in a microscopic model for the NM jAF interface and evaluating the various coecients using a Boltzmann approach. AF LNM RNM µ nT(x)d ˆxˆzjH FIG. 1. A normal-metal{insulating antiferromagnet{normal- metal heterostructure. An external eld His applied along thezdirection. A spatially dependent temperature T(x) and a spin biasis considered. As a result, a magnon spin current j ows through the AF of thickness d. The paper is outlined as follows. In Sec. II, we intro- duce the microscopic Hamiltonian for the bulk AF and its interaction with the metallic contacts. In Sec. III, we formulate the phenomenological spin diusion model, including scattering between magnon branches, and ob- tain expressions for the structural Seebeck coecient and magnon conductance. In Sec. IV, we compute the coef- cients for interfacial magnon transport from the micro- scopic model for the contacts. Based on this result, we estimate bulk transport coecients assuming the interac- tion parameters are eld and temperature independent. We conclude in Sec. VI with a discussion of our results. In the appendices, we detail various technical aspects of the calculations. II. MODEL We begin by dening the microscopic model for the LNMjAFjRNM heterostructure. The total Hamiltonian is^H=^HAF+^HI+^He, where ^HAFdescribes the AF spin system while ^HIrepresents their interfacial contact with the normal metals. The Hamiltonian ^Hedescribes the electronic states at the left- and right-lead. The coupling with LNM and RNM is modeled by a simple interfacialexchange Hamiltonian, ^HI=Z dxX iJii(x)^si
^S(x); (1) whereJiis the exchange coupling between the electronic spin density ^S(x) and the localized spin operator ^siat siteithat labels the lattice along the interface. Here i(x) is the density of the localized AF electron orbital representing eective spin densities at the interface. We will return to the study of ^HIin Sec. IV to determine the contact spin conductance. The AF spin Hamiltonian is introduced by labelling each square sublattice site by the position i. The nearest- neighbour Hamiltonian is ^HAF=JX hiji^si
^sjHX i^siz+ 2sX i^s2 ix+^s2 iy
;(2) with ^sithe spin operator at site i,J > 0 the antiferro- magnetic exchange biasing, Hthe magnetic eld, and  the uniaxial easy-axis anisotropy. We are interested in small spin uctuations (magnons) around the collinear bipartite ground state. The latter is the relevant ground state to expand around for magnetic eds below the spin- op eldHsf. Magnons are introduced by the Holstein- Primako transformation29, ^siz=s^ay i^ai;^si= ^ay iq 2s^ay i^ai; (3a) ^siz=s+^by i^bi;^si=q 2s^by i^bi^bi; (3b) and^si+=^sy i, when ibelongs to sublattice aandb, re- spectively. We expand the spin Hamiltonian, Eq. (2), in powers of magnon operators that includes magnon- magnon interactions, up to the fourth order, ^HAF= ^H(2) AF+^H(4) AF. To lowest order in s, excitations of ^H(2) AF are diagonalized through the Bogoliubov transformation (see Appendix A for denition), by the operators ^ qand ^qthat carry spin angular momentum + ~^zand~^zre- spectively, ^H(2) AF=X qh (q)^y q^q+(q)^y q^qi : (4) We refer to the magnons described by the operator  () as-()magnons, respectively. The dispersion re- lation is;(q) =H+q (6Js)2 1 2q
+H2cin a 3-dimensional lattice, where stands for the - and- magnon branch, respectively. Here, H2 c2+ 26Jsis the critical eld corresponding to the spin- op transition, while q= (1=3)P3 n=1cos(qna), where ais the lattice spacing. Magnon-magnon interactions are represented by the interacting Hamiltonian ^H(4) AF. In the diagonal basis, the interacting Hamiltonian becomes a lengthy ex- pression that is detailed in Eq. (A7) (Appendix A). It consists of nine dierent scattering processes among - and-magnons. Some of these processes allow for the exchange of population of - and-magnons, see Fig. 6.3 III. SPIN TRANSPORT: PHENOMENOLOGICAL THEORY We now outline the phenomenological spin trans- port theory for magnons across the LNM jAFjRNM het- erostructure. In the subsections that follow, we esti- mate the structural spin Seebeck coecient and struc- tural magnon conductances. The basic assumption is that the equilibration length for magnon-magnon inter- actions is much shorter than the system length d, so that the two magnon gases are parametrized by local chemical potentialsandand temperatures TandT. In keeping with our treatment of ferromagnets30, we assume that strong, inelastic spin-preserving processes x the lo- cal magnon temperatures to that of the local phonon temperature. This assumption is reasonable since the rate at which magnon temperature equilibrates with the phonon bath is dominated by both magnon-conserving and magnon-nonconserving scattering processes31. Thus, the magnon temperature reaches its equilibrium faster than the magnon chemical potential. The local phonon temperature, in turn, is assumed to be linear across the AF, and to be equal to the electronic temperatures in each of the metallic leads. Only the magnon chemical potentialsandare then left to be determined. We then express phenomenologically the spin conser- vation laws in terms of the chemical potentials. Its mi- croscopic derivation can be established from the Boltz- mann equation as is explained in Appendix B. Dening the magnon densities nandn, these read _n+r
j=rgg; (5a) _n+r
j=rgg: (5b) Here,ridescribes relaxation of spin into the lattice re- sulting from inelastic magnon-phonon interactions that do not conserve magnon number. In addition, gijde- scribes inelastic spin-conserving processes that accounts for, e.g., magnon-magnon and magnon-phonon scatter- ing, where the total number of magnons n+nmay change but the spin nnis constant. In what fol- lows, the coecients gij, by assumption, have their ori- gin in the coupling between magnons. The currents of - and-magnons, denoted as jandj, are given by j= r&rTandj=r&rT, where; and&;are the bulk magnon spin conductivities and Seebeck coecients, respectively. In writing the particle currents in the form above, we have neglected magnon- magnon drag, which stems from magnon-magnon inter- actions that transfer linear momentum from one magnon band to another in such a way that the total spin cur- rent is conserved. Such drag gives rise to cross-terms like j/r. We shall simply limit the discussion to the regime in which such momentum scattering in subdomi- nant to e.g. elastic disorder scattering. The bulk conti- nuity equations, Eqs. (5a) and (5b), are complemented by the boundary conditions at the NM jAF interfaces onthe spin currents j(s) =~jandj(s) =~j, x
j(s) (x=d=2) =G[L(d=2)]; (6a) x
j(s) (x=d=2) =G[L+(d=2)]; (6b) x
j(s) (x=d=2) =G[R(d=2)]; (6c) x
j(s) (x=d=2) =G[R+(d=2)]; (6d) withxthe unit vector along x-axis and where we have chosen the left and right interfaces to correspond to the planesx=d=2 andx=d=2. Inside the left and right normal metals the respective spin accumulations, the dif- ference between spin-up and spin-down chemical poten- tial, areLandR. HereG;are the contact magnon spin conductances of each interface. The contact Seebeck coecient does not appear, as we are assuming a contin- uous temperature prole across the structure, i.e., there is no temperature dierence between magnons at the in- terface and normal metal leads. For xed spin accumu- lationsL=R, Eqs. (5a-6d) form a closed set of equations with the parameters g,r;,&;,;andG;to be estimated from microscopic calculations (see Sec. IV). The inelastic spin-conserving terms gijcan be signif- icantly simplied by additional considerations. Impos- ing spin conservation one nds that g=gand g=g. This result is obtained from Eqs. (5a-6d) by equating _n_n= 0 in the absence of magnon currents and disregarding the relaxation term ri. In addition, we can estimate the eld- and temperature-dependence of the coecients gandg, in particular near the spin- op transition. For this purpose, we use Fermi's golden rule to calculate the transition rate of -magnons ( - magnons), dened as (), that represents the in- stantaneous leakage of magnons due to the conversion between- and-magnons. Among the dierent scatter- ing processes displayed in Fig. 6, few of them conserve the number of - or-magnons and thus do not con- tribute to the transition rate. As detailed in Appendix B, we sum over all the scattering rates and nd that = , which derives as a consequence of conser- vation of spin-angular momentum. Moreover, and more importantly, up to linear order in the chemical potential =g(+). Therefore, g=gg, mean- ing that a single scattering rate describes the inelastic spin-conserving process. The coecient gis expressed in terms of a complex integral, given in Eq. (B7), that can be estimated in certain limits. In the high temperature regime, where the thermal energy is much higher than the magnon gap, we obtain g= 2N =~s2
(kBT=Jsz )3 with a dimensionless integral dened in Appendix B. In the steady state limit the magnon chemical poten- tials are described by Eqs. (5a) and (5b), and are of the general form ex=. The collective spin decay lengthadmits two solutions, 22 1= 2 + 2 q 42 2+ (22 )2(7) 22 2= 2 + 2 +q 422 + (22 )2(8)4 where2 =g=,2 =g=, 2 = (g+r)=and 2 = (g+r)=. In the absence of magnetic eld, the magnon-bands are degenerate and therefore and have equal properties. Thus the collective spin diusion lengths become 2 1=r=and2 2= (2g+r)=that dif- fer due to the inelastic spin-conserving scattering ( g). In the following sections we evaluate the structural spin Seebeck coecient and structural magnon conductance. We will consider separately two scenarios, a tempera- ture gradient and spin bias across the LNM jAFjRNM heterostructure in Sec. III A and III B, respectively. A. Spin Seebeck Eect Let us assume a linear temperature gradient, with no spin accumulation in the normal metals. We solve for the spin current at the right interface, js=x
j(s) (d=2)+ x
j(s) (d=2) in presence of the temperature gradient
T. Then, the spin current owing through the right interface is related to the thermal gradient by js=S
T, where Sis the structural spin Seebeck coecient. The general solution forSis found in Appendix C (Eq. C4). In what follows we examine several regimes of interest. First, we consider the zero applied magnetic eld case, but allow for sublattice symmetry breaking at the nor- mal metal|magnet interfaces. Here, we have that the dispersion relations for the - and-magnons are identi- cal. Furthermore, the bulk transport properties becomes independent of the magnon band, i.e., ==and &=&=&. In this limit one nds S=2&(GG) (G1G2+G2G1)d1Cothd 21 ; (9) withGinthe eective conductances dened by Gin Gi+ (i=n) Coth [d=2n] fori=;andn= 1;2. We see thatSis proportional to the bulk spin Seebeck con- ductivity&. In the absence of symmetry breaking at the interfaces,G=G, the spin Seebeck eect vanishes as expected. When there is no magnetic eld, it is thus es- sential to have systems with uncompensated interfaces to get a nite Seebeck eect. In order to understand the dependence of Eq. (9) on the lm thickness d, it is useful to distinguish two thick- ness regimes. Let us rst introduce a \thin" lm regime, ddinnCoth1(Gin=) forn= 1;2 andi=;. In this limitGni(=n) Coth [d=2n]. The spin See- beck coecient becomes, S2(GG) dCoth [d=22]& ; (10) which in the extreme thin lm limit ( d2), be- comes independent of d,S!(GG)2&=. This can be understood as the sum of two independent par- allel channels, each with eective conductances renor- malized by the bulk transport parameters. When Gi>=n, we may also dene a \thick" regime ( ddin nCoth1(Gin=) for alli;n) in which the contact re- sistance dominates, i.e., GinGi(thick lm). In this case, one obtains, SCothd 21& d1 G1 G1  ; (11) andS  (&=d 1)G1 Tat long distances d1. In this case, the net interfacial conductance behaves as the sum of a series spin-channels, each with conductance G andG. Note that as the Seebeck coecient is dened through the relation js=S
T, the1=d-dependence means that js/@xTis independent of d; a Seebeck eect can thus originate for a thick AF due to a dierence between the impedances of the magnon-bands just at the interface where the signal is measured. Second, we consider eects of a nite applied magnetic eld. In addition, we assume no symmetry-breaking at interfaces,G=G=G. In the \thick" lm regime, we obtainsS (&&)=d, which is simply the bulk value of the Seebeck coecient. However, allowing symmetry breaking at the interfaces we can obtain Sin the weak coupling regime, i.e., gr;r, corresponding to slow scattering between the magnon branches (compared to Gilbert damping). Expanding the collective spin decay length, Eqs. (7) and (8), to linear order in g=r;, we get 1p =r(1g=r) and2p =r(1g=r). This expansion lead to corrections in the structural See- beck coecient,SS(0)+O(g=r), where S(0)=S(0) +S(0) =G& dG(0) 1G& dG(0) 2; (12) withG(0) nithe lowest order of the eective conductances. It is interesting to note that Eq. (12) consists of two completely decoupled parallel channels. In the partic- ular thick lm regime ( ddin), it reduces to S(0) (&&)=d, which is consistent with the result obtained at nite eld in the \thick" lm regime and G=G. Although we allow for symmetry-breaking at the inter- face here, all of the interfacial properties are washed out in the thick lm regime. Last, consider the regime in which interactions are strong:gr;randd2. Naively, one might expect interactions to greatly reduce the spin Seebeck ef- fect. In fact, one nds that all dependence on gdrops out: S=G+G d"( )2&& ( )2&+&# ; (13) Thus, even with strong interactions between magnon bands, the spin Seebeck eect becomes independent of gand nonzero. The eects of interband interactions are shown in Fig. (4a); while there is a slight suppression of the signal, the spin Seebeck eect is qualitatively un- changed by large scattering.5 B. Spin biasing Aside from a temperature gradient, a spin current may be generated by means of an electrically driven spin bi- asing across the spin (usually via the spin Hall eect in a normal metal contact)25. To model this, we consider the temperature constant throughout the structure, but a spin accumulation =^zis applied at the left inter- face, giving rise to a spin current j=G owing out of the opposite interface, parametrized by the structural conductance coecient G. The full steady-state solution to Eqs. (5a) and (5b) is given by Eq. (C7) in Appendix C. In order to nd simple relations for the spin conductance, we focus on three regimes. First, we consider the case of sublattice symmetry and zero magnetic eld At the interfaces, this entails G= G=G. In the bulk, this implies that bulk magnon spin conductivities and Seebeck coecients are identical for each magnon branch. Here we nd that only one collective spin decay length, r=p =r, plays a role in transport. One obtains, G=2G2=r [2=2r+G2] sinh d r + 2(=r)Gcosh d r: (14) (Note that as the eld - or symmetry-breaking at the in- terfaces - is turned on, the magnon-magnon interactions start to play a role). In the thin lm regime ( dr), GG, which is just the series conductance of two paral- lel channels, each with interfacial conductance G=2 (due to the two interfaces through which the spin current must pass). In the opposite limit, dr, we nd G4(=r)G2 (=r+G)2ed=r; (15) exhibiting an exponential decay over the distance r. Second, we consider the strongly interacting case where gr;randd2. Here, one nds that while the conductance generally depends on g, in this regime the conductance is nite and independent of g: G=GS+GB (16) where GS= G2 +2 G2 (+)=r sinh d rQ = G() r2 +2G() r (17) reduces to Eq. (14) at zero eld, while GB=1 2 2 2 
X =G() rGG() rG G() r2 +2G() r(18) is nonzero only when the magnetic eld is applied; here G() irGi+ (i=r) Tanh [d=2r] whileG(+) irGi+(i=r) Coth [d=2r]; the decay length ris given by the limit of 1in the large glimit, yielding 2 r= (r=+r=)=2. Thus, we nd that strong interactions do not radically alter the structural spin conductance in the sense that the spin conductance neither vanishes or diverges in this regime. When dr, Eq. (16) simplies to: G= 2(+) r G2 +2 G2  Gr2 +2Gr2ed=r: (19) Thus we nd that for large inter-band scattering, the nonlocal signal does not depend on gbut only on the decay processes (e.g. Gilbert damping) via ri. Third, we consider a nite magnetic eld and the limit when magnons are non-interacting. In the zero coupling regime,g= 0, one nds that the structural conductance is the sum of the parallel channels, G=G+G. Here, Gi=(i=ir)G2 i [2 i=2 ir+G2 i] sinh d ir + 2(i=ir)Gicosh d ir: (20) where2 ir=ri=iis determined by decay processes. For dir, we nd that Gi=2(i=ir)G2 i ((i=ir) +Gi)2ed=ir(21) which shows an exponential decay over distance. When - and-magnons are identical at the bulk and inter- faces, both Eqs. (21) and (19) reduce to Eq. (14). In the following sections we calculate and estimate the various parameters that enter into the phenomenological theory above. IV. TRANSPORT COEFFICIENTS: MICROSCOPIC THEORY In this section, we compute the interfacial spin con- ductances from a microscopic model for the interface. In addition, the bulk magnon conductance, as well as the bulk Seebeck coecient, are obtained in linear response from transport kinetic theory. Based on these results the structural Seebeck coecient is evaluated and plotted in Figs. 4. A. Contact magnon spin conductance In this section, we compute interface transport co- ecients appropriate to our bulk drift-diusion theory above, allowing for the boundary conditions, Eqs (6a) to be computed. Let us suppose that the spin degrees of freedom of the AF are coupled to those of the normal metals by the exchange Hamiltonian (1). Here it is understood that i6 labels the lattice along the interface (see Fig. 2). Speci- cally, the lattice is the set of vectors R2=fna^z+ma^yg. The integers nandmare such that icorresponds to a- andb-atoms when n+mare even and odd, respec- tively. In this model, we assume that aandbatoms are evenly spaced, which is not essential in what fol- lows. Besides, the itinerant electronic density, corre- sponding to evanescent modes in the x-direction, de- cays over an atomistic distance inside the AF. The spin density of itinerant electrons in the normal metal is ^S(x) = ( ~=2)P 0^ y (x)0^ 0(x), where ^ (x) is the electron operator and the Pauli matrix vector. The exchange coupling Ji=Ja, ifi2a, andJi=Jb, ifi2b, while the local spin density at each lattice site iis mod- ulated by the function i(x) =ji(x)j2, withithe lo- calized orbital at position i. Note that in general the orbitals for the aandbsublattices may be dierent. Printed by Mathematica for Students Printed by Mathematica for Students aJa⇢a(x?)Jb⇢b(x?)(uncompensated) (compensated) FIG. 2. Eective spin densities of AF jNM interface as experi- enced by normal metal electrons scattering o of the interface, for the compensated and uncompensated cases. Based on the model represented by the contact Hamil- tonian (1), we wish to obtain the magnonic spin current across the interface using Fermi's Golden Rule. We ex- pand ^HIin terms of magnon operators up to order ni=s, obtaining ^HI=^H(k) I+^H(sf) I. The rst term is the coher- ent Hamiltonian ^H(k) I=P kk0Ukk0 ^cy k"^ck0"^cy k#^ck0# , with ^ck ^cy k the fermionic operator that annihilate (create) and electron with spin- and momentum k. The term ^H(k) Igives rise to coherent spin torques, and magnonic corrections to it, n. Since we are assum- ing a xed order parameter nand focus only on thermal magnon spin currents, we need not consider this term. The second contribution, ^H(sf) I, is the spin- ip Hamilto- nian that describes processes in which both branches of magnons are annihilated and created at the interface by spin- ip scattering of electrons. This term reads, ^H(sf) I=X qkk0 V qkk0^y q^cy k#^ck0"+V qkk0^y q^cy k"^ck0# +h:c:; (22)where the matrix elements are V qkk0r 8S NZ dx  k(x) k0(x) ( a(q;x)Jacoshq b(q;x)Jbsinhq) (23) and V qkk0r 8S NZ dx  k(x) k0(x) (b(q;x)Jbcoshqa(q;x)Jasinhq):(24) Here, the function k(x) represent the eigenstates of the nonmagnetic Hamiltonian. Specically, in the yz direc- tions, the wavefunction is a delocalized Bloch state of the interfacial lattice, which we assume here for simplic- ity to be common to the both the metal and insulators (as is common in such heterostuctures); in the x direc- tion, the state is an evanescent mode on the insulator side of the interface, and a Bloch-like state of the metal- lic lattice on the other, which reduces to the usual 3D metallic Bloch state far inside the metal. The quantities aandbare dened by a(q;x) =P i2ai(x)eiq
iand b(q;x) =P i2bi(x)eiq
i, withi(x) =ji(x)j2as the density of the localized AF electron orbital at site i. The quantities cosh qand sinhqoriginate from the Bogoli- ubov transformation that diagonalizes the noninteracting magnon Hamiltonian32. It is instructive to consider the simplest case of in- terfacial spin transport. This occurs when the inter- face is fully compensated, i.e., i2a(x) =i2b(x+i)) andJa=Jb, see right side of Fig. 2. Because the normal metal electronic states k(x) are Bloch states of the interfacial nonmagnetic Hamiltonian, then trans- lation by the lattice spacing ain they(orz) direc- tion transform, k!eikya k. Usinga(b)(q;x) = eiqyab(a)(q;x+a^y) =ei2qyaa(b)(q;x+ 2a^y), it follows thatV qkk0=ei2a(q+kk0)yV qkk0. Applying translational invariance on the full Hamiltonian, under x!x+a^y, one has that this is independent of q+kk0, and it follows that V qkk0=ei V qk0k , with the phase factor dened by =a qy+kyk0 y
. Since all of the inter- facial transport coecients are proportional to jVj2, we establish that they become identical for both magnonic branches, at zero eld, for the case of fully compensated interface. It is also interesting to note the role played by Umk- lapp scattering processes at the interface. Suppose again a fully compensated interface. Then, in the small q limit, one nds cosh qsinhq, and the matrix ele- mentsV qkk0andV qkk0vanish when ( qk+k0)?= 0 (specular scattering of electrons), where the subscript \?" designates the in-plane components. However, when (qk+k0)?=Gmn, where Gmn=n(=a)^y+m(=a)^zis the reciprocal lattice vector, the matrix elements do not vanish for odd values of m+n, and transport for each7 species becomes possible. We therefore expect Umklapp scattering processes to play a crucial role in the low tem- perature behavior of the magnon conductance, as well as other interfacial linear transport coecients. This is con- sistent with Takei etal:21,22, where Umklapp scattering is found to be responsible for a nite spin-mixing conduc- tance, describing coherent spin torques, at an AF jNM interface. Umklapp processes, however, may only hap- pen when part of the yzcross section of the normal metal jkj= 2kFsurface lies outside the magnetic Brillioun zone of the lattice interface, for instance in a spherical Fermi surface this conditions becomes 2 kF>= a. We return to the general case in order to obtain the contact magnon conductances G;. This can be done by a straightforward application of Fermi's Golden rule to calculate the magnonic spin current owing across the interface. The magnon current is expressed as30 ji= 2D2 FZ dg(i) jVi()j2(0 i) [fim()fie()];(25) and therefore, the magnon spin current through the in- terface becomes js=~(jj). Here we have dened fori=;, jVi()j2=dAF D2 FX qkk01 g(i) Vi qkk02(kF) (k0F)(q);(26) withDFas the normal metal density of states and g(i)  thei-magnon density of states. In Eq. (25) 0 ;=, where we recall that is the spin accumulation. The Bose-Einstein distribution for the i-magnons is fim() = 1=[e(i)=kBTi1], andfie() = 1=[e(0 i)=kBTe1] cor- responds to the eective electron-hole-excitation density experienced by the i-magnons. In a simple model, we may take the atomic densities for both sublattices as i(x) =(x ri) and the normal metal wavefunctions k(x) = eik?
xFk(x)=pVNM. Here the function Fk(x) de- scribes the decay within the AF and VNM the nor- mal metal volume. Then, dening the spin-mixing con- ductanceg"# i16NdAFVAF(sDFJi=VNM)2, where =R dxFk0(x)F k(x), which we take to be momentum independent for simplicity, one may write the i-magnon spin current ~jiin Eq. (25) as ~ji=1 16 g"# a(i) aa+g"# b(i) bb+ 2q g"# ag"# b(i) ab ;(27) where the functions (i) ll0, which carry units of energy, are given by (i) ll0=1 D2 F(sVAF)X q;k;k0X m;nF(ll0) mnq(qi) (nimnie) (Fk)(Fk0)k0kq;Gmn;(28) withF(aa) mnq= cosh2q,F(bb) mnq= sinh2qandF(ab) mnq= F(ba) mnq= (1)m+n+1coshqsinhq. The motivation for 1246810J_bJ_a51015GIl2MG_bêG_a Printed by Mathematica for StudentsG↵/G10 15 5 1 2 4 6 8 10 24681002004006008001000 Printed by Mathematica for StudentsTHc1 with Umklappwithout UmklappJa/JbG↵/g"#↵FIG. 3. Ratio of interfacial conductances of the two magnon branchesandat zero eld for dierent ratios of interfacial sublattice exchange constants. Sublattice symmetry breaking (Ja6=Jb) is necessary to obtain a structural spin Seebeck eect in the absence of magnetic elds (see Eq. (9)). Inset: temperature dependence of Gincluding and excluding Umk- lapp scattering ( m6= 0 and/or n6= 0 in Eq. (28)). All curves are obtained for kF= 4=a, and 6Js2= 2Hc. expressing the spin current in the form of Eq. (27) is that in the case of particle-hole symmetry at the inter- face, ~ji= (g"#=4)(g"#=4)(~!i)(ni=s). In particular, the m=n= 0 term in Eq. (28) gives spec- tral scattering processes, while all others ( m6=n6= 0) correspond to Umklapp scattering. The contact spin conductances GandGare ob- tained by the linearization of the i-magnon current given by Eq. (27), i.e., Gi= (@ji=@i)ji=0. The ratio of in- terfacial conductances of the two magnon branches and is shown in Fig. 3 at zero eld and as a function of ratios of interfacial sublattice exchange constants Ja=Jb. The ratioG=Greaches a maximum value to later sat- urates when Ja=Jbis increased. In particular, we note thatG=Gwhen the interfacial exchange constants are equal. Thus, the breaking of sublattice symmetry (Ja6=Jb) is necessary in realizing a structural spin See- beck eect in the absence of a eld, as is seen from Eq. (9). In the inset of Fig. 3 we display the temperature dependence of G. In this plot we have included (solid line) and excluded (dashed line) Umklapp scattering. B. Bulk magnon conductances and spin Seebeck coecients In this section, we evaluate the bulk magnon con- ductances;and bulk spin Seebeck coecients &;. These are obtained from standard kinetic theory of trans- port. Unlike previous works28,33,34, here we consider the magnonic transport driven, in addition to thermal gra- dients, by spin biasing. The generic expressions for the8 FIG. 4. (a) Structural Seebeck coecient Sand (b) struc- tural spin conductance Gas functions of temperature T=Hc for a eldH= 0:2Hc. The temperature dependence of the inter-magnon scattering is given by g= (T=Tc)3(see Ap- pendix B). Shown for both plots are = 0 ;1;10;103;105, corresponding to a shift from blue to red coloring. While increased scattering slightly diminishes the SSE, it has no discernible eect on the spin conductance for these particular parameters. For these plots, the parameters g"# a= 1=100a2 (which for a= 1A corresponds to g"#1=nm2),kF= 1=a, 6Js2= 2Hc,= 103andd= 100 awere used. 0.20.40.60.81.00.00.51.01.52.0 Printed by Mathematica for StudentsH/HcH/HcG 0.20.40.60.81.02468101214 Printed by Mathematica for Studentsr H/Hc FIG. 5. Main gure: behavior of conductance Gnear spin op. While the spin diusion length rdiverges asjHj!Hc, the conductanceG, though sharply increasing, does not actually diverge because of bottlenecking by the interface impedances; for noninteracting magnons it has a maximium value of max(G=2;G=2) (see Eq. (34)). The colors and parameters are identical to those shown in Fig. 4. magnon current in the bulk are, ji=Zdq (2)3iv2 i@fi @x; (29) where the integration is over the Brillouin zone and i= ;. The magnon relaxation time is iand the magnon group velocity along the xdirection is vi=@i=~@kx. The number of i-magnons with momentum qis denoted byfiand given by the Bose-Einstein distribution func-tion. This yields the transport coecients, i=4~J4H2 c 9~2Zdq (2)3isin2(aqx) 2 q 1 +~J2 1 2q
ei (ei1)2;(30) &i=4~J4H2 c 9~2Zdq (2)3isin2(aqx) 2 q 1 +~J2 1 2q
i2ei (ei1)2;(31) where ~J6Js2=Hcis roughly the N eel temperature in units ofHcand= 1=kBT. Similarly, we may ob- tain expressions for the damping rates rifrom _nij= (2)3R dqni=igwithigthe Gilbert damping lifetime. From the above relation riis extracted and obeys ri=2 ~Zdq (2)32 iei (ei1)2; (32) andis being the Gilbert damping constant. The momentum relaxation rate, entering in the trans- port coecients obtained in Eqs. (30) and (31), has contributions from dierent sources; Gilbert damping, disorder scattering, magnon-phonon scattering and - magnon{-magnon scattering. For simplicity, we con- sider the regime in which Gilbert damping dominates transport: ~1 i~1 ig; (33) where ~1 ig= 2i=Hc. Note that tilde represents units ofHc. In Figs. 4 (a) and (b) we show the temperature- dependence of the spin Seebeck coecient and struc- tural spin condutance, respectively using the interface and bulk transport coecients above. The interaction parameter ggrows with temperature (see Appendix B and C). As shown in Fig. 4 (a), however, the eects of g are minimal, suppressing the spin Seebeck signal slightly and negligibly aecting the structural conductance. V. SPIN-FLOP TRANSITION The spin- op transition occurs as jHj!Hcfrom be- low. Here, the magnon spectrum becomes gapless and quadratic at low energies for one of the two magnon bands (say, the -band, for purposes of discussion). When Gilbert damping dominates the transport time (Eq. (33)), the bulk conductance in Eq. (30) demon- strates an infrared divergence, while &,randGare nite. It is straightforward to show that the Seebeck coef- cient, Eq. (C4), does not diverge in this case, consistent with19. In contrast to19, however, the structural conductance Gdoes not diverge in the diusive regime. Here, it is instructive to consider the noninteracting case, Eq. (20), which reduces to G=G2  (=r)(d=r) + 2G; (34)9 which shows an algebraic , rather than exponential, de- cay in lm thickness, due to a diverging decay length r (p). For a thin lm, this becomes G=G=2; while the AF bulk shows zero spin resistivity ( 1 = 0) due to the Bose-Einstein divergence at low energies, struc- tural transport is bottlenecked by the interface resistance G1 , which is only well dened in the diusive regime. (The eect is similar to a superconducting circuit, which with perfectly conducting components, shows a nite re- sistance due to normal metal contacts.) While the signal does not diverge, there is a clear enhancement due to the diminished spin resistivity, as well as long-distance transport (algebraic in d), manifesting as a peak in the signal near the spin- op transition25(see Fig. 5). A full calculation for nonlocal spin injection - including spin Hall/inverse spin Hall eects absent here - would show additional impedances to spin ow due to spin resistance in the normal metal injector and detector. VI. CONCLUSION AND DISCUSSION In summary, we have presented a study of spin trans- port of magnons in insulating AFs in contact with nor- mal metals. We focus on the thick-lm limit, wherein a diusive regime can be assumed and magnons are in a local thermodynamical equilibrium. The excitation of magnon currents is considered in linear response and driven by either a temperature gradient and/or spin bi- asing. The spin transport is studied by evaluating the structural magnon conductance and spin Seebeck eect within a phenomenological spin-diusion transport the- ory. These parameters were calculated in terms of bulk transport coecients as well as contact magnon conduc- tances. While the former were computed through kinetic theory of transport, the latter are obtained from a mi- croscopic model of the NM jAF interface. Furthermore, we allowed for the breaking of sublattice symmetry at the interface assuming an uncompensated magnetic or- der. In addition, the eld- and temperature-dependence of the inter-magnon scattering rates, which redistribute angular momentum between the magnon branches, were estimated. We nd that the eects of inter-magnon scat- tering, which lock the two magnon bands together, isnegligible. Furthermore, we show that even as the bulk spin resistivity vanishes near the spin op transition, nor- mal metal|magnet interface spin impedance ultimately bottleneck transport, irrespective of interactions, in con- trast to the stochastic theory19for thin lms. The phenomenological approach above ultimately breaks down for strong interactions (which occur near the spin- op transition), where the individual and clouds are no longer internally equilibrated with well- dened chemical potentials an interactions. Instead, a treatment of the interacting clouds (e.g. a kinetic theory approach) beyond the quasiequilibrium approach that is adopted here is needed. In addition, more sophisticated treatments of the transport time ihave been shown to more realistically reproduce experimental results33; such quasi-empirical transport times could be incorporated di- rectly into our phenomenology. Most importantly, our somewhat articial assumption that the magnetic eld is applied along the easy-axis in not necessarily realized in experiment. Instead, even simple bipartite AFs such as those modeled by our phenomenology above show com- plex paramagnetic behavior in response to a eld applied along dierent axes. In these scenarios, heterostuctures may manifest both antiferromagnetic and ferromagnetic transport behaviors25. Future work, such as the drift- diusion approach discussed above, is needed to fully understand such scenarios at a more fundamental level. ACKNOWLEDGMENTS This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under Grant DLV-737038 "TRANSPIRE", the Research Coun- cil of Norway through is Centres of Excellence funding scheme, Project No. 262633, "QuSpin" and European Research Council via Advanced Grant No. 669442 "Insu- latronics". Also we acknowledge funding from the Sticht- ing voor Fundamenteel Onderzoek der Materie (FOM) and the European Research Council via Consolidator Grant number 725509 SPINBEYOND. Note added{ During the submission of our work, we became aware of another related article35that considers magnon transport in AFs. Appendix A: Magnon-magnon interactions We start out by dening the AF Hamiltonian. Introducing a square lattice, labelling the sites in the lattice by i, on sub-latticesAandB, the nearest-neighbor Hamiltonian reads ^HAF=JX hi2A;j2Bi^si
^sjHX i2A;B^siz 2sX i2A;B^s2 iz; (A1) whereJ >0 is the exchange coupling, Hthe magnetic eld and >0 the uniaxial easy-axis anisotropy. Introducing the Holstein-Primako transformation, assuming a bipartite ground state, the spin operators in the limit of small spin10 uctuations reads ^sA iz=say iai; ^sB iz=s+by ibi; (A2a) ^sA i+=p 2sai1p 2say iaiai; ^sB i+=p 2sby i1p 2sby iby ibi; (A2b) ^sA i=p 2say i1p 2say iay iai ^sB i=p 2sbi1p 2sby ibibi: (A2c) The AF Hamiltonian Eq. (A1) is expanded up to fourth order in the magnon operators, Fourier transformed through the relations ai=1p NP ieik
iakandbi=1p NP ieik
jbkand expressed as HAF=E0+H(2) AF+H(4) AFwhere ^H(2) AF= (Jsz+)X qh (1 +h)ay qaq+ (1h)by qbq+ q(aqbq+ay qby q)i (A3) ^H(4) AF=Jz 2NX q1q2q3q4q1+q2q3q4h 2 q2q4ay q1by q4aq3bq2+ 2s ay q1ay q2aq3aq4+by q1by q2bq3bq4
+ q4 by q1bq2bq3aq4+by q3by q2bq1ay q4+ay q1aq2aq3bq4+ay q3ay q2aq1by q4i (A4) withh=H=(Jsz+),=Jsz= (Jsz+) and q=2 zP cos [q
] wherezis the coordination number. The quadratic part of the Hamiltonian, Eq. (A3), is diagonalized by the Bogoliubov transformation ^aq=lq^q+mq^y q (A5) ^by q=mq^q+lq^y q (A6) with the coecients lq= (Jsz+)+q 2q1=2 ,mq= (Jsz+)q 2q1=2 qlqandq= (Jsz+)q 12 2q, resulting in Eq. (4). In the diagonal basis, the interacting Hamiltonian Eq. (A4) nally becomes ^H(4) AF=X q1q2q3q4q1+q2q3q4h V(1) q1q2q3q4y q1y q2q3q4+V(2) q1q2q3q4y q1q2q3q4+V(3) q1q2q3q4y q1y q2q3y q4 +V(4) q1q2q3q4y q1q2q3y q4+V(5) q1q2q3q4q1q2q3y q4+V(6) q1q2q3q4y q1q2y q3y q4 +V(7) q1q2q3q4y q1y q2y q3y q4+V(8) q1q2q3q4q1q2q3q4+V(9) q1q2q3q4q1q2y q3y q4i (A7) where the scattering amplitudes are V(a) q1q2q3q4=Jz N
lq1lq2lq3lq4(a) 1234. The functions (a)are the following expressions11 (1) 1234 = q2q4q2q41 2( q2q2+ q4q4+ q2q1q3q4+ q4q1q2q3) + 2Jzs(1 +q1q2q3q4) (A8) (2) 1234 = q2q4q4 q1q4q1q2q4+ q4q1q3+ q4q2q4+1 2(q3q4( q1+ q2q1q2) + ( q2+ q1q1q2))  Jzs(q2+q1q3q4) (A9) (3) 1234 = q2q4q2 q2q3q2q3q4+ q2q1q3+ q2q2q4+1 2(q1q2( q3+ q4q3q4) + ( q4+ q3q3q4))  Jzs(q4+q1q2q3) (A10) (4) 1234 = q2q4+ q1q4q1q2+ q2q3q3q4+ q1q3q1q2q3q4+2 Jzs(q2q4+q1q3) (q1( q3+ q4q3q4) +q3( q1+ q2q1q4) +q4( q2+ q1q1q2) +q2( q4+ q3q3q4)) (A11) (5) 1234 = q2q4q1 q2q3q1q3q4+ q2q2q3+ q2q1q4+1 2(( q3+ q4q3q4) +q1q2( q4+ q3q3q4))  Jzs(q3+q1q2q4) (A12) (6) 1234 = q2q4q3 q1q4q1q2q3+ q4q1q4+ q4q2q3+1 2(( q1+ q2q1q2) +q3q4( q2+ q1q1q2))  Jzs(q1+q2q3q4) (A13) (7) 1234 = q2q4q2q31 2( q2q1+ q4q3+ q4q1q2q4+ q2q2q3q4) + 2Jzs(q3q4+q1q2) (A14) (8) 1234 = q2q4q1q41 2( q4q3+ q2q1+ q2q2q3q4+ q4q1q2q4) + 2Jzs(q1q2+q3q4) (A15) (9) 1234 = q2q4q1q31 2( q4q4+ q2q2+ q2q1q3q4+ q4q1q2q3) + 2Jzs(1 +q1q2q3q4) (A16) whereq= 1q 1+q1=2 . Note the symmetry relations among these coecients (3) 1234 = (2) 3412, (6) 1234 = (5) 3412 and (8) 1234 = (7) 3412. The form of these expressions dier from Ref.36, where a Dyson-Maleev transformation was considered. Appendix B: Scattering Lengths In this section, we compute the eld and temperature dependences of gandgthrough the Fermi's golden rule. To start with, we introduce the Boltzmann equation for the distribution of - and-magnons,f(x;q;t) and f(x;q;t) respectively, @f @t+@f @x
@! q @q= [q] + [q]; (B1) @f @t+@f @x
@! q @q= [q] + [q]; (B2) where q=~! q. The right-hand side are the total net rates of scattering into and out of a magnon state with wave vector q. The magnon spin diusion equations [Eqs. (5a) and (5b)] are obtained by linearizing the Boltzmann equations in terms of the small perturbations, e.g. the chemical potential. This is achieved, in addition, by integrating Eqs. (B1) and (B2) over all possible wave vectors q. The terms and originate from multiple eects such as, magnon-phonon collisions, elastic magnon scattering with defects, and magnon number and energy-conserving intraband magnon-magnon interaction. It is worth to mention that the estimation of each of those contribution, as was done in Ref.31for ferromagnets, is out of the scope of our work. However, we implement the basic assumption that the equilibration length for magnon-magnon interactions is much shorter than the system size, so that the two magnon gases are parametrized by local chemical potentialsandand temperatures TandT. Moreover, as was pointed out in Sec. IV, the magnon relaxation into the phonon bath is parametrized by the Gilbert damping.12 Now we focus on the magnon-magnon collisions described by and . These terms describe interband interaction between magnons that exchange the population of dierent magnon species. To calculate and  we consider the interacting Hamiltonian given by Eq. (A7) that represents all scattering processes among - and -magnons (depicted in Fig. 6). We emphasize that those processes represented in Fig. 6(b), (d) and (e), do not conserve the number of -magnons or -magnons, even though the dierence nnis constant due to conservation of spin-angular momentum. This inelastic spin-conserving processes contribute to the transfer of one magnon mode into the other, and thus determining the coecients gij. We quantify this eect evaluating perturbatively the rate of change of magnons using Fermi's golden rule. FIG. 6. Diagrammatic representation for the scattering processes experienced by - and-magnons. In (a), (c) and (f) are represented the processes with scattering amplitude V(1),V(4)andV(9), respectively. In (b), (d) and (e) is shown those inelastic processes that do not conserve the number of magnons. These are scattered by the interacting potential with amplitude V(3), V(6)andV(7), respectively. Those processes with amplitude V(2),V(5)andV(8)are the hermitian conjugate of the above and thus are omitted. Based on time-dependent perturbation theory, the transition rate between an initial jiiand a nal statejfiis given by Fermi's Golden Rule, which reads = (2 =~)P i;fWihfj^Vjii2 (fi). The sum runs over all possible initial and nal states, Wiis the Boltzmann weight that gives the probability of being in some initial state, ^Vis the matrix element of the Hamiltonian corresponding to the interactions and the delta function ensures energy conservation. We recognize that a nal state can be either any of those described in Eq. (A7). However, those processes that conserve the number of particles, i.e., -magnons and -magnons, have a null transition rate. Only the states described in Fig. 6(b), (d) and (e), will contribute to a nite transition rate. The net transition rate of scattering into and out of a magnon state with wave vector q, reads [q] =2 ~X q1q2q3V(3) qq1q2q32h 1 +f q
 1 +f q1 1 +f q2
f q3f qf q1f q2 1 +f q3
i qq1+q2q3  q+ q1+ q2 q3 +V(6) qq1q2q32h 1 +f q
 1 +f q1 1 +f q2 f q3f qf q1f q2 1 +f q3i qq1q2+q3  q+ q1+ q2 q3o (B3) and [q] =2 ~X q1q2q3V(3) qq1q2q32h 1 +f q 1 +f q1
1 +f q2
f q3f qf q1f q2 1 +f q3
i q+q1+q2q3  q+ q1+ q2 q3 +V(6) qq1q2q32h 1 +f q 1 +f q1
 1 +f q2 f q3f qf q1f q2 1 +f q3i q+q1q2+q3  q+ q1+ q2 q3o : (B4)13 We note that processes described by Fig. 6(e) do not contribute to the change of magnon density by invoking conservation of energy. The total rates and , obtained by summing up over all wave vectors q, are dened by =~X q[q]; (B5) =~X q[q]; (B6) which describes the net imbalance of the magnon densities nandnby the successive scatterings events between both magnon modes. Next, we will show that Eqs. (B5) and (B6) scale linearly with the magnon chemical potentials when the magnon distribution are close to the equilibrium. In order to calculate and we consider that magnons are near thermodynamic equilibrium. Thus, their distributions are parameterized by the Bose-Einstein distribution as f q= e( q)=kBT11 andf q=  e( q)=kBT11 , whereTis the temperature of the phonon bath. At equilibrium the rates obey = = 0, which is established when the chemical potentials satisfy += 0. This can be clearly seen when the distribution is expanded up to linear order on and. Using this expansion on Eqs. (B3) and (B4), is found that the total transition rates and become equals and proportional to the sum of the chemical potentials. Precisely, we obtain = =g(+) with the coecient ggiven by g=2 ~kBTX qq1q2q3qq1+q2q3V(3) qq1q2q32 f;0 qf;0 q1f;0 q2 1 +f;0 q3
  q+ q1+ q2 q3
+V(6) qq1q2q32 f;0 qf;0 q1 1 +f;0 q2
f;0 q3  q+ q1+ q3 q2
 ;(B7) wheref;0andf;0denote the equilibrium distribution evaluated at the chemical potential e =e . Comparing with the phenomenological Eqs. (5a) and (5b), we obtain g=g=g=g=g. Despite the complex expression for the factor g, it can be estimated in certain temperature regimes. For instance, at high temperatures the thermal energy is much higher than the magnon gap, therefore ;(q)=kBT(Jsz=kBT)ajqj, i.e., the exchange energy is the only magnetic coupling that becomes relevant. Thus, at large temperatures we obtain g=2 ~N s2kBT Jsz3 (B8) where is a dimensionless integral dened as =Zdp1 (2)3dp2 (2)3dp3 (2)3dp4 (2)3(p1+p2p3p4)v(3) p1p2p3p42 f;0 pf;0 p1f;0 p2 1 +f;0 p3 (p + p1+ p2p3) +v(6) p1p2p3p42 f;0 pf;0 p1 1 +f;0 p2 f;0 p3(p + p1+ p3p2) : (B9) To obtain Eq. (B8) the continuum limit was taken by the replacementP q!V((Jsz=kBT)a)3R dp=(2)3on Eq. (B7), where the dimensionless wavevector p = ( Jsz=kBT)ajqjwas introduced. We notice that in the limit of very large temperatures the Bose factors approach the Raleigh-Jeans distribution, i.e., f qf qkBT=(Jsz)ajqj, and becomes independent of temperature. The dimensionless scattering amplitudes v(i) q1q2q3q4=V(i) q1q2q3q4=v0, with v0= (Jsz)3=Ns(kBT)2, are evaluated and their asymptotic behaviour obeys v(3) p1p2p3p4=v(6) p1p2p3p4=2v(7) p1p2p3p4 with v(3) p1p2p3p421 p1p2p3p41=2 : (B10) Appendix C: Seebeck coecient and spin conductance To nd the structural spin Seebeck coecient and spin conductance we rst express the general solution for the magnon chemical potential, (x) = (Asinh [x= 1] +Bcosh [x= 1]) + (Dsinh [x= 2] +Ecosh [x= 2]) (C1) (x) =C(Asinh [x= 1] +Bcosh [x= 1]) + (Dsinh [x= 2] +Ecosh [x= 2]) (C2)14 whereCis a constant that is obtained from the eigenvalue problem that determines 1and2. From the boundary conditions, Eqs. (6a-6d), we nd the unknown coecients A,B,DandE. The net spin current crossing the right lead is, js=j(s) (d=2) +j(s) (d=2); (C3) wherej(s) =~jandj(s) =~j. In the absence of a spin accumulation, L=R= 0, we calculate the magnon currentsjandjto obtainS=js=d. Thus, the structural spin Seebeck coecient is: S=2(G2G&G1G&) +1(GG1&G2G&) + (G1G2)G&+12(G2G1)G& (G1G21+G2G12)d(C4) which is written in terms of the eective conductances GinGi+i n Cothd 2n (C5) forn= 1;2 andi=;, and 1= (~g~g+ ~r~r+)=2~g 2= (~g~g+ ~r~r+)=2~g (C6) where=p 4~g~g+ (~g~g+ ~r~r)2, and ~r=r=, and ~g=g=, with similar expressions for - parameters. The structural conductance is obtained by following the same procedure as before. However, in this case, we assume thatL6=RandrT= 0. In the general case, Gis given by: G=1 21(1+)G(+) 2G+2G(+) 2G 1G(+) 1G(+) 2+2G(+) 2G(+) 1Tanhd 21 +1 22(2)G(+) 1G1G(+) 1G 1G(+) 1G(+) 2+2G(+) 2G(+) 1Tanhd 22 +1 21(1+)G() 2G+2G() 2G 1G() 1G() 2+2G() 2G() 1Cothd 21 +1 22(2)G() 1G1G() 1G 1G() 1G() 2+2G() 2G() 1Cothd 22 (C7) whereG() inGi+ (i=n) Tanh [d=2n] whileG(+) inGi+ (i=n) Coth [d=2n]. 1X. W. P. W. J. Jungwirth, T. Marti, Nature Nanotechnol- ogy11, 231 (2016). 2V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018). 3T. Jungwirth, J. Sinova, A. Manchon, X. Marti, J. Wun- derlich, and C. Felser, Nature Physics 14, 200 (2018). 4J.Zelezn y, H. Gao, K. V yborn y, J. Zemen, J. Ma sek, A. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 113, 157201 (2014). 5P. Wadley et al., Science 351, 587 (2016). 6S. Y. Bodnar, L. Smejkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A. A. Sapozhnik, H. J. Elmers,M. Klaui, and M. 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1709.10365v1.Non_local_Gilbert_damping_tensor_within_the_torque_torque_correlation_model.pdf

Non-local Gilbert damping tensor within the torque-torque correlation model Danny Thonig,1,Yaroslav Kvashnin,1Olle Eriksson,1, 2and Manuel Pereiro1 1Department of Physics and Astronomy, Material Theory, Uppsala University, SE-75120 Uppsala, Sweden 2School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden (Dated: July 19, 2018) An essential property of magnetic devices is the relaxation rate in magnetic switching which depends strongly on the damping in the magnetisation dynamics. It was recently measured that damping depends on the magnetic texture and, consequently, is a non-local quantity. The damping enters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter , that does not, in a straight forward formulation, account for non-locality. Eorts were spent recently to obtain Gilbert damping from rst principles for magnons of wave vector q. However, to the best of our knowledge, there is no report about real space non-local Gilbert damping ij. Here, a torque-torque correlation model based on a tight binding approach is applied to the bulk elemental itinerant magnets and it predicts signicant o-site Gilbert damping contributions, that could be also negative. Supported by atomistic magnetisation dynamics simulations we reveal the importance of the non-local Gilbert damping in atomistic magnetisation dynamics. This study gives a deeper understanding of the dynamics of the magnetic moments and dissipation processes in real magnetic materials. Ways of manipulating non-local damping are explored, either by temperature, materials doping or strain. PACS numbers: 75.10.Hk,75.40.Mg,75.78.-n Ecient spintronics applications call for magnetic ma- terials with low energy dissipation when moving magnetic textures, e.g. in race track memories1, skyrmion logics2,3, spin logics4, spin-torque nano-oscillator for neural net- work applications5or, more recently, soliton devices6. In particular, the dynamics of such magnetic textures | magnetic domain walls, magnetic Skyrmions, or magnetic solitons | is well described in terms of precession and damping of the magnetic moment mias it is formulated in the atomistic Landau-Lifshitz-Gilbert (LLG) equation for sitei @mi @t=mi Beff i+ ms@mi @t ; (1) where andmsare the gyromagnetic ratio and the magnetic moment length, respectively. The precession eldBeff iis of quantum mechanical origin and is ob- tained either from eective spin-Hamilton models7or from rst-principles8. In turn, energy dissipation is dominated by the ad-hoc motivated viscous damping in the equation of motion scaled by the Gilbert damping tensor. Commonly, the Gilbert damping is used as a scalar parameter in magnetization dynamics simula- tions based on the LLG equation. Strong eorts were spend in the last decade to put the Gilbert damping to a rst-principles ground derived for collinear mag- netization congurations. Dierent methods were pro- posed: e.g. the breathing Fermi surface9{11, the torque- torque correlation12, spin-pumping13or a linear response model14,15. Within a certain accuracy, the theoretical models allow to interpret16and reproduce experimental trends17{20. Depending on the model, deep insight into the fun- damental electronic-structure mechanism of the Gilbertdampingis provided: Damping is a Fermi-surface ef- fect and depending on e.g. scattering rate, damping occurs due to spin- ip but also spin-conservative tran- sition within a degenerated (intraband, but also inter- band transitions) and between non-degenerated (inter- band transitions) electron bands. As a consequence of these considerations, the Gilbert damping is proportional to the density of states, but it also scales with spin-orbit coupling21,22. The scattering rate for the spin- ip tran- sitions is allocated to thermal, but also correlation ef- fects, making the Gilbert damping strongly temperature dependent which must be a consideration when applying a three-temperature model for the thermal baths, say phonon14, electron, and spin temperature23. In particu- lar, damping is often related to the dynamics of a collec- tive precession mode (macrospin approach) driven from an external perturbation eld, as it is used in ferromag- netic resonance experiments (FMR)24. It is also estab- lished that the Gilbert damping depends on the orien- tation of the macrospin25and is, in addition, frequency dependent26. More recently, the role of non-collective modes to the Gilbert damping has been debated. F ahnle et al.27 suggested to consider damping in a tensorial and non- isotropic form via ithat diers for dierent sites i and depends on the whole magnetic conguration of the system. As a result, the experimentally and theoret- ically assumed local Gilbert equation is replaced by a non-local equation via non-local Gilbert damping ijac- counting for the most general form of Rayleigh's dissi- pation function28. The proof of principles was given for magnetic domain walls29,30, linking explicitly the Gilbert damping to the gradients in the magnetic spin texture rm. Such spatial non-locality, in particular, for discrete atomistic models, allows further to motivate energy dis-arXiv:1709.10365v1 [cond-mat.mtrl-sci] 29 Sep 20172 ij αij q FIG. 1: Schematic illustration of non-local energy dissipation ijbetween site iandj(red balls) represented by a power cord in a system with spin wave (gray arrows) propagation q. sipation between two magnetic moments at sites iand j, and is represented by ij, as schematically illustrated in Fig. 1. An analytical expression for ijwas already proposed by various authors14,31,32, however, not much work has been done on a material specic, rst-principle description of the atomistic non-local Gilbert damping ij. An exception is the work by Gilmore et al.32who studied(q) in the reciprocal space as a function of the magnon wave vector qand concluded that the non-local damping is negligible. Yan et al.29and Hals et al.33, on the other hand, applied scattering theory according to Brataas et al.34to simulate non-collinearity in Gilbert damping, only in reciprocal space or continuous meso- scopic scale. Here we come up with a technical descrip- tion of non-locality of the damping parameter ij, in real space, and provide numerical examples for elemental, itinerant magnets, which might be of high importance in the context of ultrafast demagnetization35. The paper is organized as follows: In Section I, we introduce our rst-principles model formalism based on the torque-torque correlation model to study non-local damping. This is applied to bulk itinerant magnets bcc Fe, fcc Co, and fcc Ni in both reciprocal and real space and it is analysed in details in Section II. Here, we will also apply atomistic magnetisation dynamics to outline the importance in the evolution of magnetic systems. Fi- nally, in the last section, we conclude the paper by giving an outlook of our work. I. METHODS We consider the torque-torque correlation model in- troduced by Kambersk y10and further elaborated on by Gilmore et al.12. Here, nite magnetic moment rotations couple to the Bloch eigenenergies "n;kand eigenstates jnki, characterised by the band index nat wave vec-tork, due to spin-orbit coupling. This generates a non- equilibrium population state (a particle-hole pair), where the excited states relax towards the equilibrium distribu- tion (Fermi-Dirac statistics) within the time n;k=1=, which we assume is independent of nandk. In the adi- abatic limit, this perturbation is described by the Kubo- Greenwood perturbation theory and reads12,36in a non- local formulation (q) =g msZ X nmT nk;mk+q T nk;mk+q
Wnk;mk+qdk: (2) Here the integral runs over the whole Brillouin zone volume . A frozen magnon of wave vector qis consid- ered that is ascribed to the non-locality of . The scat- tering events depend on the spectral overlap Wnk;mk+q=R (")Ank(";)Amk+q(";) d"between two bands "n;k and"m;k+q, where the spectral width of the electronic bandsAnkis approximated by a Lorentzian of width . Note that is a parameter in our model and can be spin- dependent as proposed in Ref. [37]. In other studies, this parameter is allocated to the self-energy of the system and is obtained by introducing disorder, e.g., in an al- loy or alloy analogy model using the coherent potential approximation14(CPA) or via the inclusion of electron correlation38. Thus, a principle study of the non-local damping versus can be also seen as e.g. a temperature dependent study of the non-local damping. =@f=@"is the derivative of the Fermi-Dirac distribution fwith re- spect to the energy. T nk;mk+q=hnkj^Tjmk+qi, where =x;y;z , are the matrix elements of the torque oper- ator ^T= [;Hso] obtained from variation of the mag- netic moment around certain rotation axis e.and Hsoare the Pauli matrices and the spin-orbit hamilto- nian, respectively. In the collinear ferromagnetic limit, e=ezand variations occur in xandy, only, which al- lows to consider just one component of the torque, i.e. ^T=^Txi^Ty. Using Lehmann representation39, we rewrite the Bloch eigenstates by Green's function G, and dene the spectral function ^A= i GRGA
with the retarded (R) and advanced (A) Green's function, (q) =g mZ Z (")^T^Ak ^Ty^Ak+qdkd":(3) The Fourier transformation of the Green's function G nally is used to obtain the non-local Gilbert damping tensor23between site iat positionriand sitejat position rj,  ij=g mZ (")^T i^Aij ^T jy^Ajid": (4) Note that ^Aij= i GR ijGA ji
. This result is consis- tent with the formulation given in Ref. [31] and Ref. [14]. Hence, the denition of non-local damping in real space3 and reciprocal space translate into each other by a Fourier transformation, ij=Z (q) ei(rjri)
qdq: (5) Note the obvious advantage of using Eq. (4), since it allows for a direct calculation of ij, as opposed to tak- ing the inverse Fourier transform of Eq. (5). For rst- principles studies, the Green's function is obtained from a tight binding (TB) model based on the Slater-Koster parameterization40. The Hamiltonian consists of on-site potentials, hopping terms, Zeeman energy, and spin-orbit coupling (See Appendix A). The TB parameters, includ- ing the spin-orbit coupling strength, are obtained by t- ting the TB band structures to ab initio band structures as reported elsewhere23. Beyond our model study, we simulate material spe- cic non-local damping with the help of the full-potential linear mun-tin orbitals (FP-LMTO) code \RSPt"41,42. Further numerical details are provided in Appendix A. With the aim to emphasize the importance of non- local Gilbert damping in the evolution of atomistic magnetic moments, we performed atomistic magnetiza- tion dynamics by numerical solving the Landau-Lifshitz Gilbert (LLG) equation, explicitly incorporating non- local damping23,34,43 @mi @t=mi0 @ Beff i+X jij mj s@mj @t1 A:(6) Here, the eective eld Beff i =@^H=@miis allo- cated to the spin Hamiltonian entails Heisenberg-like ex- change couplingP ijJijmi
mjand uniaxial magneto- crystalline anisotropyP iKi(mi
ei)2with the easy axis alongei.JijandKiare the Heisenberg exchange cou- pling and the magneto-crystalline anisotropy constant, respectively, and were obtained from rst principles44,45. Further details are provided in Appendix A. II. RESULTS AND DISCUSSION This section is divided in three parts. In the rst part, we discuss non-local damping in reciprocal space q. The second part deals with the real space denition of the Gilbert damping ij. Atomistic magnetization dynam- ics including non-local Gilbert damping is studied in the third part. A. Non-local damping in reciprocal space The formalism derived by Kambersk y10and Gilmore12 in Eq. (2) represents the non-local contributions to the energy dissipation in the LLG equation by the magnonwave vector q. In particular, Gilmore et al.32con- cluded that for transition metals at room temperature the single-mode damping rate is essentially independent of the magnon wave vector for qbetween 0 and 1% of the Brillouin zone edge. However, for very small scat- tering rates , Gilmore and Stiles12observed for bcc Fe, hcp Co and fcc Ni a strong decay of withq, caused by the weighting function Wnm(k;k+q) without any sig- nicant changes of the torque matrix elements. Within our model systems, we observed the same trend for bcc Fe, fcc Co and fcc Ni. To understand the decay of the Gilbert damping with magnon-wave vector qin more de- tail, we study selected paths of both the magnon qand electron momentum kin the Brillouin zone at the Fermi energy"Ffor bcc Fe (q;k2!Handq;k2H!N), fcc Co and fcc Ni ( q;k2!Xandq;k2X!L) (see Fig. 2, where the integrand of Eq. (2) is plotted). For example, in Fe, a usually two-fold degenerated dband (approximately in the middle of H, marked by ( i)) gives a signicant contribution to the intraband damping for small scattering rates. There are two other contributions to the damping (marked by ( ii)), that are caused purely by interband transitions. With increasing, but small q the intensities of the peaks decrease and interband tran- sitions become more likely. With larger q, however, more and more interband transitions appear which leads to an increase of the peak intensity, signicantly in the peaks marked with ( ii). This increase could be the same or- der of magnitude as the pure intraband transition peak. Similar trends also occur in Co as well as Ni and are also observed for Fe along the path HN. Larger spectral width increases the interband spin- ip transitions even further (data not shown). Note that the torque-torque correlation model might fail for large values of q, since the magnetic moments change so rapidly in space that the adiababtic limit is violated46and electrons are not stationary equilibrated. The electrons do not align ac- cording the magnetic moment and the non-equilibrium electron distribution in Eq. (2) will not fully relax. In particular, the magnetic force theorem used to derive Eq. (3) may not be valid. The integration of the contributions in electron mo- mentum space kover the whole Brillouin zone is pre- sented in Fig. 3, where both `Loretzian' method given by Eq. (2) and Green's function method represented by Eq. (3) are applied. Both methods give the same trend, however, dier slightly in the intraband region, which was already observed previously by the authors of Ref. [23]. In the `Lorentzian' approach, Eq. (2), the electronic structure itself is unaected by the scattering rate , only the width of the Lorentian used to approx- imateAnkis aected. In the Green function approach, however, enters as the imaginary part of the energy at which the Green functions is evaluated and, conse- quently, broadens and shifts maxima in the spectral func- tion. This oset from the real energy axis provides a more accurate description with respect to the ab initio results than the Lorentzian approach.4 ΓHq(a−1 0) Γ H k(a−1 0) Fe ΓX Γ X k(a−1 0) Co ΓX Γ X k(a−1 0) Ni (i) (ii) (ii) FIG. 2: Electronic state resolved non-local Gilbert damping obtained from the integrand of Eq. (3) along selected paths in the Brillouin zone for bcc Fe, fcc Co and fcc Ni. The scattering rate used is = 0 :01 eV. The abscissa (both top and bottom in each panels) shows the momentum path of the electron k, where the ordinate (left and right in each panel) shows the magnon propagation vector q. The two `triangle' in each panel should be viewed separately where the magnon momentum changes accordingly (along the same path) to the electron momentum. Within the limits of our simplied electronic structure tight binding method, we obtained qualitatively similar trends as observed by Gilmore et al.32: a dramatic de- crease in the damping at low scattering rates (intra- band region). This trend is common for all here ob- served itinerant magnets typically in a narrow region 0<jqj<0:02a1 0, but also for dierent magnon propa- gation directions. For larger jqj>0:02a1 0the damping could again increase (not shown here). The decay of  is only observable below a certain threshold scattering rate , typically where intra- and interband contribu- tion equally contributing to the Gilbert damping. As already found by Gilmore et al.32and Thonig et al.23, this point is materials specic. In the interband regime, however, damping is independent of the magnon propa- gator, caused by already allowed transition between the electron bands due to band broadening. Marginal vari- ations in the decay with respect to the direction of q (Inset of Fig. 3) are revealed, which was not reported be- fore. Such behaviour is caused by the break of the space group symmetry due to spin-orbit coupling and a selected global spin-quantization axis along z-direction, but also due to the non-cubic symmetry of Gkfork6= 0. As a re- sult, e.g., in Ni the non-local damping decays faster along Kthan in X. This will be discussed more in detail in the next section. We also investigated the scaling of the non-local Gilbert damping with respect to the spin-orbit coupling strengthdof the d-states (see Appendix B). We observe an eect that previously has not been discussed, namely that the non-local damping has a dierent exponential scaling with respect to the spin-orbit coupling constant for dierentjqj. In the case where qis close to the Bril- louin zone center (in particular q= 0),/3 dwhereas for wave vectors jqj>0:02a1 0,/2 d. For largeq, typically interband transitions dominate the scatteringmechanism, as we show above and which is known to scale proportional to 2. Here in particular, the 2will be caused only by the torque operator in Eq. (2). On the other hand, this indicates that spin-mixing transitions become less important because there is not contribution infrom the spectral function entering to the damping (q). The validity of the Kambserk y model becomes ar- guable for3scaling, as it was already proved by Costa et al.47and Edwards48, since it causes the unphysical and strong diverging intraband contribution at very low temperature (small ). Note that there is no experi- mental evidence of such a trend, most likely due to that sample impurities also in uence . Furthermore, various other methods postulate that the Gilbert damping for q= 0 scales like 2 9,15,22. Hence, the current applied theory, Eq. (3), seems to be valid only in the long-wave limit, where we found 2-scaling. On the other hand, Edwards48proved that the long-wave length limit ( 2- scaling) hold also in the short-range limit if one account only for transition that conserve the spin (`pure' spin states), as we show for Co in Fig. 11 of Appendix C. The trendsversusjqjas described above changes drastically for the `corrected' Kambersk y formula: the interband re- gion is not aected by these corrections. In the intraband region, however, the divergent behaviour of disappears and the Gilbert damping monotonically increases with larger magnon wave vector and over the whole Brillouin zone. This trend is in good agreement with Ref. [29]. For the case, where q= 0, we even reproduced the re- sults reported in Ref. [21]; in the limit of small scattering rates the damping is constant, which was also reported before in experiment49,50. Furthermore, the anisotropy of(q) with respect to the direction of q(as discussed for the insets of Fig. 3) increases by accounting only for pure-spin states (not shown here). Both agreement with5 510−22Fe 0.000 0.025 0.050 0.075 0.100 q: Γ→H 2510−2α(q)Co q: Γ→X 510−225 10−310−210−110+0 Γ (eV)Ni q: Γ→X FIG. 3: (Color online) Non-local Gilbert damping as a func- tion of the spectral width for dierent reciprocal wave vector q(indicated by dierent colors and in units a1 0). Note that q provided here are in direct coordinates and only the direction diers between the dierent elementals, itinerant magnets. The non-local damping is shown for bcc Fe (top panel) along !H, for fcc Co (middle panel) along !X, and for fcc Ni (bottom panel) along !X. It is obtained from `Lorentzian' (Eq. (2), circles) and Green's function (Eq. (3), triangles) method. The directional dependence of for = 0:01 eV is shown in the inset. experiment and previous theory motivate to consider 2- scaling for all . B. Non-local damping in real space Atomistic spin-dynamics, as stated in Section I (see Eq. (6)), that includes non-local damping requires Gilbert damping in real-space, e.g. in the form ij. This point is addressed in this section. Such non-local con- tributions are not excluded in the Rayleigh dissipation functional, applied by Gilbert to derive the dissipation contribution in the equation of motion51(see Fig. 4). Dissipation is dominated by the on-site contribution -101 Fe αii= 3.552·10−3 ˜αii= 3.559·10−3 -101αij·10−4Co αii= 3.593·10−3 ˜αii= 3.662·10−3 -10 1 2 3 4 5 6 rij/a0Ni αii= 2.164·10−2 ˜αii= 2.319·10−2FIG. 4: (Color online) Real-space Gilbert damping ijas a function of the distance rijbetween two sites iandjfor bcc Fe, fcc Co, and fcc Ni. Both the `corrected' Kambersk y (red circles) and the Kambersk y (blue squares) approach is considered. The distance is normalised to the lattice constant a0. The on-site damping iiis shown in the gure label. The grey dotted line indicates the zero line. The spectral width is = 0:005 eV. iiin the itinerant magnets investigated here. For both Fe (ii= 3:55
103) and Co ( ii= 3:59
103) the on-site damping contribution is similar, whereas for Ni iiis one order of magnitude higher. O-site contri- butionsi6=jare one-order of magnitude smaller than the on-site part and can be even negative. Such neg- ative damping is discernible also in Ref. [52], however, it was not further addressed by the authors. Due to the presence of the spin-orbit coupling and a preferred global spin-quantization axis (in z-direction), the cubic symmetry of the considered itinerant magnets is broken and, thus, the Gilbert damping is anisotropic with re- spect to the sites j(see also Fig. 5 left panel). For ex- ample, in Co, four of the in-plane nearest neighbours (NN) areNN4:3
105, while the other eight are NN2:5
105. However, in Ni the trend is opposite: the out-of-plane damping ( NN1:6
103) is smaller than the in-plane damping ( NN 1:2
103). In- volving more neighbours, the magnitude of the non-local6 damping is found to decay as 1=r2and, consequently, it is dierent than the Heisenberg exchange parameter that asymptotically decays in RKKY-fashion as Jij/1=r353. For the Heisenberg exchange, the two Green's functions as well as the energy integration in the Lichtenstein- Katsnelson-Antropov-Gubanov formula54scales liker1 ij, G ij/ei(k
rij+) jrijj(7) whereas for simplicity we consider here a single-band model but the results can be generalized also to the multi- band case and where denotes a phase factor for spin =";#. For the non-local damping the energy integra- tion is omitted due to the properties of in Eq. (4) and, thus, ij/sin k"
rij+ " sin k#
rij+ # jrijj2:(8) This spatial dependency of ijsuperimposed with Ruderman-Kittel-Kasuya-Yosida (RKKY) oscillations was also found in Ref. [52] for a model system. For Ni, dissipation is very much short range, whereas in Fe and Co `damping peaks' also occur at larger distances (e.g. for Fe at rij= 5:1a0and for Co at rij= 3:4a0). The `long-rangeness' depends strongly on the parameter (not shown here). As it was already observed for the Heisenberg exchange interaction Jij44, stronger thermal eects represented by will reduce the correlation length between two magnetic moments at site iandj. The same trend is observed for damping: larger causes smaller dissipation correlation length and, thus, a faster decay of non-local damping in space rij. Dierent from the Heisenberg exchange, the absolute value of the non-local damping typically decreases with as it is demonstrated in Fig. 5. Note that the change of the magnetic moment length is not considered in the results discussed so far. The anisotropy with respect to the sites iandjof the non- local Gilbert damping continues in the whole range of the scattering rate and is controlled by it. For instance, the second nearest neighbours damping in Co and Ni become degenerated at = 0 :5 eV, where the anisotropy between rst-nearest neighbour sites increase. Our results show also that the sign of ijis aected by (as shown in Fig. 5 left panel). Controlling the broadening of Bloch spectral functions is in principal possible to evaluate from theory, but more importantly it is accessible from experimental probes such as angular resolved photoelec- tron spectroscopy and two-photon electron spectroscopy. The importance of non-locality in the Gilbert damping depend strongly on the material (as shown in Fig. 5 right panel). It is important to note that the total | dened as tot=P jijfor arbitrary i|, but also the local ( i=j) and the non-local ( i6=j) part of the Gilbert damping do not violate the thermodynamic principles by gaining an- gular momentum (negative total damping). For Fe, the -101 1. NN. 2. NN.Fe 34567αii αtot=/summationtext jαijαq=0.1a−1 0αq=0 -10αij·10−4Co 123456 αij·10−3 -15-10-50 10−210−1 Γ (eV)Ni 5101520 10−210−1 Γ (eV)FIG. 5: (Color online) First (circles) and second nearest neighbour (triangles) Gilbert damping (left panel) as well as on-site (circles) and total Gilbert (right panel) as a function of the spectral width for the itinerant magnets Fe, Co, and Ni. In particular for Co, the results obtained from tight binding are compared with rst-principles density functional theory results (gray open circles). Solid lines (right panel) shows the Gilbert damping obtained for the magnon wave vectors q= 0 (blue line) and q= 0:1a1 0(red line). Dotted lines are added to guide the eye. Note that since cubic symmetry is broken (see text), there are two sets of nearest neighbor parameters and two sets of next nearest neighbor parameters (left panel) for any choice of . local and total damping are of the same order for all , where in Co and Ni the local and non-local damp- ing are equally important. The trends coming from our tight binding electron structure were also reproduced by our all-electron rst-principles simulation, for both de- pendency on the spectral broadening (Fig. 5 gray open circles) but also site resolved non-local damping in the intraband region (see Appendix A), in particular for fcc Co. We compare also the non-local damping obtain from the real and reciprocal space. For this, we used Eq. (3) by simulating Nq= 151515 points in the rst magnon Brillouin zone qand Fourier-transformed it (Fig. 6). For7 -1.0-0.50.00.51.0αij·10−4 5 10 15 20 25 30 rij/a0FFT(α(q));αii= 0.003481 FFT(G(k));αii= 0.003855 FIG. 6: (Color online) Comparing non-local Gilbert damping obtained by Eq. (5) (red symbols) and Eq. (4) (blue symbols) in fcc Co for = 0 :005 eV. The dotted line indicates zero value. both approaches, we obtain good agreement, corroborat- ing our methodology and possible applications in both spaces. The non-local damping for the rst three nearest neighbour shells turn out to converge rapidly with Nq, while it does not converge so quickly for larger distances rij. The critical region around the -point in the Bril- louin zone is suppressed in the integration over q. On the other hand, the relation tot=P jij=(q= 0) for arbitrary ishould be valid, which is however violated in the intraband region as shown in Fig. 5 (compare tri- angles and blue line in Fig. 5): The real space damping is constant for small and follows the long-wavelength limit (compare triangles and red line in Fig. 5) rather than the divergent ferromagnetic mode ( q= 0). Two explanations are possible: i)convergence with respect to the real space summation and ii)a dierent scaling in both models with respect to the spin-orbit coupling. For i), we carefully checked the convergence with the summa- tion cut-o (see Appendix D) and found even a lowering of the total damping for larger cut-o. However, the non- local damping is very long-range and, consequently, con- vergence will be achieved only at a cut-o radius >>9a0. Forii), we checked the scaling of the real space Gilbert damping with the spin-orbit coupling of the d-states (see Appendix B). Opposite to the `non-corrected' Kam- bersk y formula in reciprocal space, which scales like 3 d, we nd2 dfor the real space damping. This indi- cates that the spin- ip scattering hosted in the real-space Green's function is suppressed. To corroborate this state- ment further, we applied the corrections proposed by Edwards48to our real space formula Eq. (4), which by default assumes 2(Fig. 4, red dots). Both methods, cor- rected and non-corrected Eq. (4), agree quite well. The small discrepancies are due to increased hybridisations and band inversion between p and d- states due to spin- orbit coupling in the `non-corrected' case. Finally, we address other ways than temperature (here represented by ), to manipulate the non-local damping. It is well established in literature already for Heisenberg exchange and the magneto crystalline anisotropy that -0.40.00.40.81.2αij·10−4 1 2 3 4 5 6 7 rij/a0αii= 3.49·10−3αii= 3.43·10−3FIG. 7: (Color online) Non-local Gilbert damping as a func- tion of the normalized distancerij=a0for a tetragonal dis- torted bcc Fe crystal structure. Here,c=a= 1:025 (red circles) andc=a= 1:05 (blue circles) is considered. is put to 0 :01 eV. The zero value is indicated by dotted lines. compressive or tensial strain can be used to tune the mag- netic phase stability and to design multiferroic materials. In an analogous way, also non-local damping depends on distortions in the crystal (see Fig. 7). Here, we applied non-volume conserved tetragonal strain along the caxis. The local damping iiis marginal biased. Relative to the values of the undistorted case, a stronger eect is observed for the non-local part, in particular for the rst few neighbours. Since we do a non-volume conserved distortion, the in-plane second NN component of the non-local damping is constant. The damping is in general decreasing with increasing distor- tion, however, a change in the sign of the damping can also occur (e.g. for the third NN). The rate of change in damping is not linear. In particular, the nearest- neighbour rate is about 0:4
105for 2:5% dis- tortion, and 2 :9
105for 5% from the undistorted case. For the second nearest neighbour, the rate is even big- ger (3:0
105for 2:5%, 6:9
105for 5%). For neigh- bours larger than rij= 3a0, the change is less signicant (0:6
105for 2:5%,0:7
105for 5%). The strongly strain dependent damping motivates even higher-order coupled damping contributions obtained from Taylor ex- panding the damping contribution around the equilib- rium position 0 ij:ij=0 ij+@ij=@uk
uk+:::. Note that this is in analogy to the magnetic exchange interaction55 (exchange striction) and a natural name for it would be `dissipation striction'. This opens new ways to dis- sipatively couple spin and lattice reservoir in combined dynamics55, to the best of our knowledge not considered in todays ab-initio modelling of atomistic magnetisation dynamics. C. Atomistic magnetisation dynamics The question about the importance of non-local damp- ing in atomistic magnetization dynamics (ASD) remains.8 0.40.50.60.70.80.91.0M 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t(ps)0.5 0.1 0.05 0.01αtot αij 0.5 1.0 1.5 2.0 2.5 3.0 t(ps)Fe Co FIG. 8: (Color online) Evolution of the average magnetic mo- mentMduring remagnetization in bcc Fe (left panel) and fcc Co (right panel) for dierent damping strength according to the spectral width (dierent colors) and both, full non- localij(solid line) and total, purely local tot(dashed line) Gilbert damping. For this purpose, we performed zero-temperature ASD for bcc Fe and fcc Co bulk and analysed changes in the average magnetization during relaxation from a totally random magnetic conguration, for which the total mo- ment was zero (Fig. 8) Related to the spectral width, the velocity for remag- netisation changes and is higher, the bigger the eective Gilbert damping is. For comparison, we performed also ASD simulations based on Eq. (2) with a scalar, purely local damping tot(dotted lines). For Fe, it turned out that accounting for the non-local damping causes a slight decrease in the remagnetization time, however, is overall not important for relaxation processes. This is under- standable by comparing the particular damping values in Fig. 5, right panel, in which the non-local part ap- pear negligible. On the other hand, for Co the eect on the relaxation process is much more signicant, since the non-local Gilbert damping reduces the local contribu- tion drastically (see Fig. 5, right panel). This `negative' non-local part ( i6=j) inijdecelerates the relaxation process and the relaxation time is drastically increased by a factor of 10. Note that a `positive' non-local part will accelerate the relaxation, which is of high interest for ultrafast switching processes. III. CONCLUDING REMARKS In conclusion, we have evaluated the non-locality of the Gilbert damping parameter in both reciprocal and real space for elemental, itinerant magnets bcc Fe, fcc Co and fcc Ni. In particular in the reciprocal space, our results are in good agreement with values given in the literature32. The here studied real space damping was considered on an atomistic level and it motivates to account for the full, non-local Gilbert damping in magnetization dynamic, e.g. at surfaces56or for nano- structures57. We revealed that non-local damping canbe negative, has a spatial anisotropy, quadratically scales with spin-orbit coupling, and decays in space as r2 ij. Detailed comparison between real and reciprocal states identied the importance of the corrections proposed by Edwards48and, consequently, overcome the limits of the Kambersk y formula showing an unphysical and experi- mental not proved divergent behaviour at low tempera- ture. We further promote ways of manipulating non-local Gilbert damping, either by temperature, materials dop- ing or strain, and motivating `dissipation striction' terms, that opens a fundamental new root in the coupling be- tween spin and lattice reservoirs. Our studies are the starting point for even further in- vestigations: Although we mimic temperature by the spectral broadening , a precise mapping of to spin and phonon temperature is still missing, according to Refs. [14,23]. Even at zero temperature, we revealed a signicant eect of the non-local Gilbert damping to the magnetization dynamics, but the in uence of non-local damping to nite temperature analysis or even to low- dimensional structures has to be demonstrated. IV. ACKNOWLEDGEMENTS The authors thank Lars Bergqvist, Lars Nordstr om, Justin Shaw, and Jonas Fransson for fruitful discus- sions. O.E. acknowledges the support from Swedish Re- search Council (VR), eSSENCE, and the KAW Founda- tion (Grants No. 2012.0031 and No. 2013.0020). Appendix A: Numerical details We performkintegration with up to 1 :25
106mesh points (500500500) in the rst Brillouin zone for bulk. The energy integration is evaluated at the Fermi level only. For our principles studies, we performed a Slater- Koster parameterised40tight binding (TB) calculations58 of the torque-torque correlation model as well as for the Green's function model. Here, the TB parameters have been obtained by tting the electronic structures to those of a rst-principles fully relativistic multiple scattering Korringa-Kohn-Rostoker (KKR) method using a genetic algorithm. The details of the tting and the tight binding parameters are listed elsewhere23,59. This puts our model on a rm, rst-principles ground. The tight binding Hamiltonian60H=H0+Hmag+ Hsoccontains on-site energies and hopping elements H0, the spin-orbit coupling Hsoc=S
Land the Zeeman termHmag=1=2B
. The Green's function is obtained byG= ("+ iH)1, allows in principle to consider disorder in terms of spin and phonon as well as alloys23. The bulk Greenian Gijin real space between site iandj is obtained by Fourier transformation. Despite the fact that the tight binding approach is limited in accuracy, it produces good agreement with rst principle band struc- ture calculations for energies smaller than "F+ 5 eV.9 -1.5-1.0-0.50.00.51.01.5 5 10 15 20 25 30 rij(Bohr radii)Γ≈0.01eVTB TBe DFT αDFT ii= 3.9846·10−3 αTB ii= 3.6018·10−3-1.5-1.0-0.50.00.51.01.5 Γ≈0.005eV αDFT ii= 3.965·10−3 αTB ii= 3.5469·10−3αij·10−4 FIG. 9: (Colour online) Comparison of non-local damping ob- tained from the Tight Binding method (TB) (red lled sym- bols), Tight Binding with Edwards correction (TBe) (blue lled symbols) and the linear mun tin orbital method (DFT) (open symbols) for fcc Co. Two dierent spectral broadenings are chosen. Equation (4) was also evaluated within the DFT and linear mun-tin orbital method (LMTO) based code RSPt. The calculations were done for a k-point mesh of 1283k-points. We used three types of basis func- tions, characterised by dierent kinetic energies with 2= 0:1;0:8;1:7 Ry to describe 4 s, 4pand 3dstates. The damping constants were calculated between the 3 d orbitals, obtained using using mun-tin head projection scheme61. Both the rst principles and tight binding im- plementation of the non-local Gilbert damping agree well (see Fig. 9). Note that due to numerical reasons, the values of used for the comparisons are slightly dierent in both electronic structure methods. Furthermore, in the LMTO method the orbitals are projected to d-orbitals only, which lead to small discrepancies in the damping. The atomistic magnetization dynamics is also per- formed within the Cahmd simulation package58. To reproduce bulk properties, periodic boundary condi- tions and a suciently large cluster (10 1010) are employed. The numerical time step is
t= 0:1 fs. The exchange coupling constants Jijare obtained from the Liechtenstein-Kastnelson-Antropov- Gubanovski (LKAG) formula implemented in the rst- principles fully relativistic multiple scattering Korringa- Kohn-Rostoker (KKR) method39. On the other hand, the magneto-crystalline anisotropy is used as a xed pa- rameter with K= 50eV. 012345678α·10−3 0.0 0.02 0.04 0.06 0.08 0.1 ξd(eV)2.02.22.42.62.83.03.2γ 0.0 0.1 0.2 0.3 0.4 q(a−1 0)-12-10-8-6-4-20αnn·10−5 01234567 αos·10−3 1.945 1.797 1.848 1.950 1.848 1.797 1.950FIG. 10: (Color online) Gilbert damping as a function of the spin-orbit coupling for the d-states in fcc Co. Lower panel shows the Gilbert damping in reciprocal space for dierent q=jqjvalues (dierent gray colours) along the !Xpath. The upper panel exhibits the on-site os(red dotes and lines) and nearest-neighbour nn(gray dots and lines) damping. The solid line is the exponential t of the data point. The inset shows the tted exponents with respect wave vector q. The colour of the dots is adjusted to the particular branch in the main gure. The spectral width is = 0 :005 eV. Appendix B: Spin-orbit coupling scaling in real and reciprocal space Kambersk y's formula is valid only for quadratic spin- orbit coupling scaling21,47, which implies only scattering between states that preserve the spin. This mechanism was explicitly accounted by Edwards48by neglecting the spin-orbit coupling contribution in the `host' Green's function. It is predicted for the coherent mode ( q= 0)21 that this overcomes the unphysical and not experimen- tally veried divergent Gilbert damping for low tem- perature. Thus, the methodology requires to prove the functional dependency of the (non-local) Gilbert damp- ing with respect to the spin-orbit coupling constant  (Fig. 10). Since damping is a Fermi-surface eects, it is sucient to consider only the spin-orbit coupling of the d-states. The real space Gilbert damping ij/ scales for both on-site and nearest-neighbour sites with 2. For the reciprocal space, however, the scaling is more complex and depends on the magnon wave vec- torq(inset in Fig. 10). In the long-wavelength limit, the Kambersk y formula is valid, where for the ferromag- netic magnon mode with 3 the Kambersk y formula is indenite according to Edwards48.10 10−32510−2α(q) 10−310−210−110+0 Γ (eV)0.000 0.025 0.050 0.075 0.100 q: Γ→XCo FIG. 11: (Colour online) Comparison of reciprocal non-local damping with (squares) or without (circles) corrections pro- posed by Costa et al.47and Edwards48for Co and dierent spectral broadening . Dierent colours represent dierent magnon propagation vectors q. Appendix C: Intraband corrections From the same reason as discussed in Section B, the role of the correction proposed by Edwards48for magnon propagations dierent than zero is unclear and need to be studied. Hence, we included the correction of Ed- ward also to Eq. (3) (Fig. 11). The exclusion of the spin- orbit coupling (SOC) in the `host' clearly makes a major qualitative and quantitative change: Although the in- terband transitions are unaected, interband transitions are mainly suppressed, as it was already discussed by Barati et al.21. However, the intraband contributions are not totally removed for small . For very small scat- tering rates, the damping is constant. Opposite to the `non-corrected' Kambersk y formula, the increase of the magnon wave number qgives an increase in the non- local damping which is in agreement to the observation made by Yuan et al.29, but also with the analytical modelproposed in Ref. [52] for small q. This behaviour was ob- served for all itinerant magnets studied here. Appendix D: Comparison real and reciprocal Gilbert damping The non-local damping scales like r2 ijwith the dis- tance between the sites iandj, and is, thus, very long range. In order to compare tot=P j2Rcutijfor arbi- traryiwith(q= 0), we have to specify the cut-o ra- dius of the summation in real space (Fig. 12). The inter- band transitions ( >0:05 eV) are already converged for small cut-o radii Rcut= 3a0. Intraband transitions, on the other hand, converge weakly with Rcutto the recipro- cal space value (q= 0). Note that (q= 0) is obtained from the corrected formalism. Even with Rcut= 9a0 which is proportional to 30000 atoms, we have not 0.81.21.62.0αtot·10−3 4 5 6 7 8 9 Rcut/a00.005 0.1 FIG. 12: Total Gilbert damping totfor fcc Co as a function of summation cut-o radius for two spectral width , one in intraband ( = 0 :005 eV, red dottes and lines) and one in the interband ( = 0 :1 eV, blue dottes and lines) region. The dotted and solid lines indicates the reciprocal value (q= 0) with and without SOC corrections, respectively. obtain convergence. Electronic address: danny.thonig@physics.uu.se 1S. S. P. Parkin, M. Hayashi, and L. 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2212.09164v1.Exponential_decay_of_solutions_of_damped_wave_equations_in_one_dimensional_space_in_the__L_p__framework_for_various_boundary_conditions.pdf

arXiv:2212.09164v1 [math.OC] 18 Dec 2022EXPONENTIAL DECAY OF SOLUTIONS OF DAMPED WAVE EQUATIONS IN ONE DIMENSIONAL SPACE IN THE LpFRAMEWORK FOR VARIOUS BOUNDARY CONDITIONS YACINE CHITOUR AND HOAI-MINH NGUYEN Abstract. We establish the decay of the solutions of the damped wave equ ations in one dimen- sional space for the Dirichlet, Neumann, and dynamic bounda ry conditions where the damping coefficient is a function of space and time. The analysis is bas ed on the study of the corresponding hyperbolic systems associated with the Riemann invariants . The key ingredient in the study of these systems is the use of the internal dissipation energy t o estimate the difference of solutions with their mean values in an average sense. Contents 1. Introduction 1 2. The well-posedness in Lp-setting 5 2.1. Proof of Proposition 2.1 7 2.2. Proof of Proposition 2.2 9 3. Some useful lemmas 9 4. Exponential decay in Lp-framework for the Dirichlet boundary condition 13 4.1. Proof of Theorem 1.2 14 4.2. Proof of Theorem 1.1 16 4.3. On the case anot being non-negative 16 5. Exponential decay in Lp-framework for the Neuman boundary condition 19 5.1. Proof of Theorem 5.2 19 5.2. Proof of Theorem 5.1 21 6. Exponential decay in Lp-framework for the dynamic boundary condition 21 6.1. Proof of Theorem 6.2 22 6.2. Proof of Theorem 6.1 23 References 23 1.Introduction This paper is devoted to the decay of solution of the damped wa ve equations in one dimensional space in the Lp-framework for 1 < p <+∞for various boundary conditions where the damping depends on space and time. More precisely, we consider the da mped wave equation (1.1)/braceleftBigg ∂ttu−∂xxu+a∂tu= 0 in R+×(0,1), u(0,·) =u0, ∂tu(0,·) =u1on (0,1), equipped with one of the following boundary conditions: (1.2) Dirichlet boundary condition: u(t,0) =u(t,1) = 0,fort≥0, 12 Y. CHITOUR AND H.-M. NGUYEN (1.3) Neumann boundary condition: ∂xu(t,0) =∂xu(t,1) = 0,fort≥0, and, for κ >0, (1.4) dynamic boundary condition: ∂xu(t,0)−κ∂tu(t,0) =∂xu(t,1)+κ∂tu(t,1) = 0,fort≥0. Hereu0∈W1,p(0,1) (with u0(0) =u0(1) = 0, i.e., u0∈W1,p 0(0,1), in the case where the Dirichlet boundarycondition is considered), and u1∈Lp(0,1) aretheinitial conditions. Moreover, a∈L∞/parenleftbig R+×(0,1)/parenrightbig is assumed to verify the following hypothesis: (1.5)a≥0,and∃λ,ε0>0,(x0−ε0,x0+ε0)⊂(0,1) such that a≥λonR+×(x0−ε0,x0+ε0), i.e,ais non-negative and a(t,x)≥λ >0 fort≥0 and for xin some open subset of (0 ,1). The region where a >0 represents the region in which the damping term is active. The decay of the solutions of ( 1.1) equipped with either ( 1.2), or (1.3), or (1.4) has been extensively investigated in the case where ais independent of t, i.e.,a(t,x) =a(x) and mainly in theL2-framework, i.e. within an Hilbertian setting. In this case , concerning the Dirichlet boundary condition, under the additional geometric multip lier condition on a, by the multiplier method, see e.g., [ 20,24], one can prove that the solution decays exponentially, i.e ., there exist positive constants Candγindependent of usuch that (1.6)/bardbl∂tu(t,·)/bardblL2(0,1)+/bardbl∂xu(t,·)/bardblL2(0,1)≤Ce−γt/parenleftBig /bardbl∂tu(0,·)/bardblL2(0,1)+/bardbl∂xu(0,·)/bardblL2(0,1)/parenrightBig , t≥0. Theassumption that asatisfies the geometric multiplier condition is equivalent to the requirement thata(x)≥λ >0 on some neighbourhood of 0 or 1. Based on more sophisticate a rguments in the seminal work of Bardos, Lebeau, and Rauch on the controll ability of the wave equation [ 3], Lebeau [23] showed that ( 1.6) also holds withoutthe geometric multiplier condition on a, see also the work of Rauch andTaylor [ 31]. When the dampingcoefficient ais also time-dependent, similar results have been obtained recently by Le Rousseau et al. in [ 22]. It is worth noticing that strong stabilization, i.e., the energy decay to zero for each traje ctory, has been established previously using LaSalle’s invariance argument [ 14,15]. The analysis of the nonlinear setting associated with (1.1) can be found in [ 6,17,26,27,34] and the references therein. Similar results holds for the Neumann boundary condition [ 3,22,26,34]. Concerning the dynamic boundary condition without interior damping effect, i.e., a≡0, the analysis for L2-framework was previously initiated by Quinn and Russell [ 30]. They proved that the energy exponentially decays in L2-framework in one dimensional space. The exponential decay for higher d imensional space was proved by Lagnese [ 21] using the multiplier technique (see also [ 30]). The decay hence was established for the geometric multiplier condition and this technique was l ater extended in [ 25], see also [ 1] for a nice account on these issues. Much less is known about the asymptotic stability of ( 1.1) equipped with either ( 1.2), or (1.3), or(1.4)inLp-framework. Thisisprobablyduetothefactthat forlinearw ave equationsconsidered in domains of Rdwithd≥2 is not a well defined bounded operator in general in Lpframework withp/ne}ationslash= 2, a result due to Peral [ 29]. As far as we know, the only work concerning exponential decay in the Lp-framework is due to Kafnemer et al. [ 19], where the Dirichlet boundary condition was considered. For the damping coefficient abeing time-independent, they showed that the decay holds under the additional geometric multiplier cond ition on afor 1< p <+∞. Their analysis is via the multiplier technique involving various non-linear test functions. In the case of zero damping and with a dynamic boundary condition, previ ous results have been obtained in [7] where the problem has been reduced to the study of a discrete time dynamical system over appropriate functional spaces.3 The goal of this paper is to give a unified approach to deal with all the boundary considered in (1.2), (1.3), and (1.4) in theLp-framework for 1 < p <+∞under the condition ( 1.5). Our results thus hold even in the case where ais a function of time and space. The analysis is based on the study of the corresponding hyperbolic systems associated w ith the Riemann invariants for which new insights are required. Concerning the Dirichlet boundary condition, we obtain the following result. Theorem 1.1. Let1< p <+∞,ε0>0,λ >0, and let a∈L∞/parenleftbig R+×(0,1)/parenrightbig be such that a≥0 anda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). Then there exist positive constants Candγdepending only on p,/bardbla/bardblL∞/parenleftbig R+×(0,1)/parenrightbig,ε0, andλsuch that for all u0∈W1,p 0(0,1) andu1∈Lp(0,1), the unique weak solution u∈C([0,+∞);W1,p 0(0,1))∩C1([0,+∞);Lp(0,1))of (1.1)and(1.2)satisfies (1.7) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤Ce−γt/parenleftBig /bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)/parenrightBig , t≥0. The meaning of the (weak) solutions given Theorem 1.1is given in Section 2(see Definition 2.1) and their well-posedness is also established there (see Pro position 2.1). Our analysis is via the study of the decay of solutions of hyperbolic systems which a re associated with ( 1.1) via the Riemann invariants. Such a decay for the hyperbolic system, even in the case p= 2, is new to our knowledge. The analysis of these systems has its own interes t and is motivated by recent analysis on the controllability of hyperbolic systems in one dimensi onal space [ 9–12]. As in [16,19], we set (1.8)ρ(t,x) =ux(t,x)+ut(t,x) and ξ(t,x) =ux(t,x)−ut(t,x) for (t,x)∈R+×(0,1). One can check that for a smooth solution uof (1.1) and (1.2), the pair of functions ( ρ,ξ) defined in (1.8) satisfies the system (1.9) ρt−ρx=−1 2a(ρ−ξ) in R+×(0,1), ξt+ξx=1 2a(ρ−ξ) in R+×(0,1), ρ(t,0)−ξ(t,0) =ρ(t,1)−ξ(t,1) = 0 in R+. Onecannothope the decay of a general solutions of ( 1.9) since any pair ( c,c) wherec∈Ris a constant is a solution of ( 1.9). Nevertheless, for ( ρ,ξ) being defined by ( 1.9) for a solution uof (1.1), one also has the following additional information (1.10)ˆ1 0ρ(t,x)+ξ(t,x)dx= 0 fort≥0. ConcerningSystem( 1.9) itself (i.e., withoutnecessarily assuming( 1.10)), weprovethefollowing result, which takes into account ( 1.10). Theorem 1.2. Let1< p <+∞,ε0>0,λ >0, anda∈L∞/parenleftbig R+×(0,1)/parenrightbig be such that a≥0 anda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). There exist a positive constant Cand a positive constant γdepending only on on p,/bardbla/bardblL∞/parenleftbig R+×(0,1)/parenrightbig,ε0, andλ such that the unique solution (ρ,ξ)of(1.9)with the initial condition ρ(0,·) =ρ0andξ(0,·) =ξ0 satisfies (1.11) /bardbl(ρ−c0,ξ−c0)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ(0,·)−c0,ξ(0,·)−c0)/bardblLp(0,1), t≥0,4 Y. CHITOUR AND H.-M. NGUYEN where (1.12) c0:=1 2ˆ1 0/parenleftbig ρ(0,x)+ξ(0,x)/parenrightbig dx, In Theorem 1.2, we consider the broad solutions. It is understood through t he broad solution in finite time: for T >0 and 1≤p <+∞, a broad solution uof the system (1.13) ρt−ρx=−1 2a(ρ−ξ) in (0 ,T)×(0,1), ξt+ξx=1 2a(ρ−ξ) in (0 ,T)×(0,1), ρ(t,0)−ξ(t,0) =ρ(t,1)−ξ(t,1) = 0 in (0 ,T), ρ(0,·) =ρ0, ξ(0,·) =ξ0 in (0,1), is a pair of functions ( ρ,ξ)∈C([0,T];/bracketleftbig Lp(0,1)/bracketrightbig2/parenrightbig ∩C([0,1];/bracketleftbig Lp(0,T)/bracketrightbig2/parenrightbig which obey the charac- teristic rules, see e.g., [ 10]. Thewell-posedness of ( 1.13) can befoundin [ 10] (see also theappendix of [13]). The analysis there is mainly for the case p= 2 but the arguments extend naturally for the case 1 ≤p <+∞. In theLp-framework, the Neumann boundary condition and its corresp onding hyperbolic sys- tems are discussed in Section 5and the dynamic boundary condition and its corresponding hy - perbolic systems are discussed in Section 6. Concerning the dynamic boundary condition, the decay holds even under the assumption a≥0. The analysis for the Neumann case shares a large part in common with the one of the Dirichlet boundary conditi on. The difference in their analysis comes from taking into account differently the boundary condi tion. The analysis of the dynamic condition is similar but much simpler. The study of the wave equation in one dimensional space via th e corresponding hyperbolic system is known. Thecontrollability and stability of hyper bolicsystems has been also investigated extensively. This goes back to the work of Russel [ 32,33] and Rauch and Taylor [ 31]. Many important progress has been obtained recently, see, e.g., [ 4] and the references therein. It is worth noting that many works have been devoted to the L2-framework. Less is studied in the Lp-scale. In this direction, we want to mention [ 9] where the exponential stability is studied for dissipativ e boundary condition. Concerning the wave equation in one dimensional space, the e xponential decay in L2-setting for the dynamic boundary condition is also established via its c orresponding hyperbolic systems [ 30]. However, to our knowledge, the exponential decay for the Dir ichlet and Neumann boundary conditions has not been established even in L2-framework via this approach. Our work is new and quite distinct from the one in [ 30] and has its own interest. First, the analysis in [ 30] uses essentially the fact that the boundary is strictly dissipat ive, i.e.,κ >0 in (1.4). Thus the analysis cannot be used for the Dirichlet and Neumann boundary condit ions. Moreover, it is not clear how to extend it to the Lp-framework. Concerning our analysis, the key observation i s that the information of the internal energy allows one to control the difference of the solutions and its mean value in the interval of time (0 ,T) in an average sense. This observation is implemented in two lemmas (Lemma 3.2and Lemma 3.3) after using a standard result (Lemma 3.1) presented in Section 3. These two lemmas are the main ingredients of our analysis fo r the Dirichlet and Neumann boundary conditions. The proof of the first lemma is m ainly based on the characteristic method while as the proof of the second lemma is inspired from the theory of functions with bounded mean oscillations due to John and Nirenberg [ 18]. As seen later that, the analysis for the dynamic boundary condition is much simpler for which the use of Lemma 3.1is sufficient.5 Aninterestingpointofouranalysisisthefactthatthesele mmasdonotdependontheboundary conditions used. In fact, one can apply it in a setting where a bound of the internal energy is accessible. This allows us to deal with all the boundary cond itions considered in this paper by the same way. Another point of our analysis which is helpful to be mentioned is that the asymptotic stability for hyperbolic systems in one dimensional space h as been mainly studied for general solutions. This is not the case in the setting of Theorem 1.2where the asymptotic stability holds under condition ( 1.10). It is also worth noting that the time-dependent coefficient s generally make the phenomena more complex, see [ 13] for a discussion on the optimal null-controllable time. The analysis in this paper cannot handle the cases p= 1 and p= +∞. Partial results in this direction for the Dirichlet boundary condition can be found in [19] whereais constant and in some range. These cases will be considered elsewhere by differ ent approaches. The paper is organized as follows. The well-posedness of ( 1.1) equipped with one of the bound- ary conditions ( 1.2) and (1.3) is discussed in Section 2, where a slightly more general context is considered (the boundary condition ( 1.4) is considered directly in Section 6; comments on this point is given in Remark 6.3). Section 4is devoted to the proof of Theorem 1.1and Theorem 1.2. We also relaxed slightly the non-negative assumption on ain Theorem 1.1and Theorem 1.2there (see Theorem 4.1and Theorem 4.2) using a standard perturbative argument. The Neumann boundary condition is studied in Section 5and the Dynamic boundary condition is considered in Section6. 2.The well-posedness in Lp-setting Inthissection, wegivethemeaningof thesolutions of theda mpedwave equation ( 1.1)equipped with either the Dirichlet boundary condition ( 1.2) or the Neumann boundary condition ( 1.3) and establish their well-posedness in the Lp-framework with 1 ≤p≤+∞. We will consider a slightly more general context. More precisely, we consider the syste m (2.1)/braceleftBigg ∂ttu−∂xxu+a∂tu+b∂xu+cu=fin (0,T)×(0,1), u(0,·) =u0, ∂tu(0,·) =u1in (0,1), equipped with either (2.2) Dirichlet boundary condition: u(t,0) =u(t,1) = 0 for t∈(0,T), or (2.3) Neumann boundary condition: ∂xu(t,0) =∂xu(t,1) = 0 for t∈(0,T). Herea,b,c∈L∞((0,T)×(0,1)) andf∈Lp((0,T)×(0,1)). We begin with the Dirichlet boundary condition. Definition 2.1. LetT >0,1≤p <+∞,a,b,c∈L∞((0,T)×(0,1)),f∈Lp((0,T)×(0,1)), u0∈W1,p 0(0,1), andu1∈Lp(0,1). A function u∈C([0,T];W1,p 0(0,1))∩C1([0,T];Lp(0,1))is called a (weak) solution of (2.1)and(2.2)(up to time T) if (2.4) u(0,·) =u0, ∂tu(0,·) =u1in(0,1),6 Y. CHITOUR AND H.-M. NGUYEN and (2.5)d2 dt2ˆ1 0u(t,x)v(x)dx+ˆ1 0ux(t,x)vx(x)dx+ˆ1 0a(t,x)ut(t,x)v(x)dx +ˆ1 0b(t,x)ux(t,x)v(x)dx+ˆ1 0c(t,x)u(t,x)v(x)dx=ˆ1 0f(t,x)v(x)dx in the distributional sense in (0,T)for allv∈C1 c(0,1). Definition 2.1can be modified to deal with the case p= +∞as follows. Definition 2.2. LetT >0,a,b,c∈L∞((0,T)×(0,1)),f∈L∞((0,T)×(0,1)),u0∈W1,∞ 0(0,1), andu1∈L∞(0,1). A function u∈L∞([0,T];W1,∞ 0(0,1))∩W1,∞([0,T];L∞(0,1))is called a (weak) solution of (2.1)and(2.2)(up to time T) ifu∈C([0,T];W1,2 0(0,1))∩C1([0,T];L2(0,1)) 1and satisfies (2.4)and(2.5). Concerning the Neumann boundary condition, we have the foll owing definition. Definition 2.3. LetT >0,1≤p <+∞,a,b,c∈L∞((0,T)×(0,1)),f∈Lp((0,T)×(0,1)), u0∈W1,p(0,1), andu1∈Lp(0,1). A function u∈C([0,T];W1,p(0,1))∩C1([0,T];Lp(0,1))is called a (weak) solution of (2.1)and(2.3)(up to time T) if(2.4)is valid and (2.6)d2 dt2ˆ1 0u(t,x)v(x)dx+ˆ1 0ux(t,x)vx(x)dx +ˆ1 0b(t,x)ux(t,x)v(x)dx+ˆ1 0c(t,x)u(t,x)v(x)dx+ˆ1 0a(t,x)ut(t,x)v(x)dx=ˆ1 0f(t,x)v(x)dx holds in the distributional sense in (0,T)for allv∈C1([0,1]). Definition 2.1can be modified to deal with the case p= +∞as follows. Definition 2.4. LetT >0,a,b,c∈L∞((0,T)×(0,1)),f∈L∞((0,T)×(0,1)),u0∈W1,∞(0,1), andu1∈L∞(0,1). A function u∈L∞([0,T];W1,∞(0,1))∩W1,∞([0,T];L∞(0,1))is called a (weak) solution of (2.1)and(2.3)(up to time T) ifu∈C([0,T];W1,2(0,1))∩C1([0,T];L2(0,1)) 2,(2.4)is valid, and (2.5)holds in the distributional sense in (0,T)for allv∈C1([0,1]). Concerningthe well-posedness of the Dirichlet system ( 2.1) and (2.2), we establish the following result. Proposition 2.1. LetT >0,1≤p≤+∞, anda,b,c∈L∞((0,T)×(0,1)), and let u0∈ W1,p 0(0,1),u1∈Lp(0,1), andf∈Lp/parenleftbig (0,T)×(0,1)/parenrightbig . Then there exists a unique (weak) solution uof(2.1)and(2.2). Moreover, it holds (2.7) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤C/parenleftBig /bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)+/bardblf/bardblLp/parenleftbig (0,T)×(0,1)/parenrightbig/parenrightBig , t≥0 for some positive constant C=C(p,T,/bardbla/bardblL∞,/bardblb/bardblL∞,/bardblc/bardblL∞)which is independent of u0,u1, and f. 1By interpolation, one can use C([0,T];W1,2 0(0,1))∩C1([0,T];L2(0,1)) instead of C([0,T];W1,2 0(0,1))∩ C1([0,T];L2(0,1)) for any 1 ≤q <+∞. This condition is used to give the meaning of the initial con ditions. 2By interpolation, one can use C([0,T];W1,2 0(0,1))∩C1([0,T];L2(0,1)) instead of C([0,T];W1,2(0,1))∩ C1([0,T];L2(0,1)) for any 1 ≤q <+∞. This condition is used to give the meaning of the initial con ditions.7 Concerning the well-posedness of the Neumann system ( 2.1) and (2.3), we prove the following result. Proposition 2.2. LetT >0,1≤p≤+∞, anda,b,c∈L∞((0,T)×(0,1)), and let u0∈ W1,p(0,1),u1∈Lp(0,1), andf∈Lp/parenleftbig (0,T)×(0,1)/parenrightbig . Then there exists a unique (weak) solution uof(2.1)and(2.3)and (2.8) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤C/parenleftBig /bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)+/bardblf/bardblLp/parenleftbig (0,T)×(0,1)/parenrightbig/parenrightBig , t≥0 for some positive constant C=C(p,T,/bardbla/bardblL∞,/bardblb/bardblL∞,/bardblc/bardblL∞)which is independent of u0,u1, and f. Remark 2.1. The definition of weak solutions and the well-posedness are s tated for p= 1 and p= +∞as well. The existence and the well-posedness is well-known in the case p= 2. The standard analysis in the case p= 2 is via the Galerkin method. The rest of this section is devoted to the proof of Propositio n2.1and Proposition 2.2in Section2.1and Section 2.2, respectively. 2.1.Proof of Proposition 2.1.The proof is divided into two steps in which we prove the uniqueness and the existence. •Step 1: Proof of the uniqueness. Assume that uis a (weak) solution of ( 2.1) withf= 0 in (0,T)×(0,1) andu0=u1= 0 in (0 ,1). We will show that u= 0 in (0 ,T)×(0,1). Set (2.9) g(t,x) =−a(t,x)∂tu(t,x)−b(t,x)∂xu(t,x)−c(t,x)u(t,x). Thenuis a weak solution of the system (2.10) ∂ttu−∂xxu=g in (0,T)×(0,1), u(t,0) =u(t,1) = 0 for t∈(0,T), u(0,·) = 0, ∂tu(0,·) = 0 in (0 ,1). Extenduandgin (0,T)×Rby appropriate reflection in xfirst by odd extension in ( −1,0), i.e., u(t,x) =−u(t,−x) andg(t,x) =−g(t,−x) in (0,T)×(−1,0) and so on, and still denote the extension by uandg. Thenu∈C([0,T];W1,p(−k,k))∩C1([0,t];Lp(−k,k)) andg∈Lp/parenleftbig (0,T)× (−k,k)/parenrightbig fork≥1 and for 1 ≤p <+∞, and similar facts holds for p= +∞. We also obtain that u(0,·) = 0 and ∂tu(0,·) = 0 inR, and (2.11) ∂ttu−∂xxu=gin (0,T)×Rin the distributional sense . The d’Alembert formula gives, for t≥0, that (2.12) u(t,x) =1 2ˆt 0ˆx+t−τ x−t+τg(τ,y)dydτ. We then obtain for t≥0 (2.13) ∂tu(t,x) =1 2ˆt 0g(τ,x+t−τ)+g(τ,x−t+τ)dτ and (2.14) ∂xu(t,x) =1 2ˆt 0g(τ,x+t−τ)−g(τ,x−t+τ)dτ.8 Y. CHITOUR AND H.-M. NGUYEN Using (2.9), we derive from ( 2.12), (2.13) and (2.14) that, for 1 ≤p <+∞and fort≥0, (2.15)ˆ1 0|∂tu(t,x)|p+|∂tu(t,x)|p+|∂xu(t,x)|pdx ≤Cˆt 0ˆ1 0/parenleftBig |∂tu(s,y)|p+|∂xu(s,y)|p+|u(s,y)|p/parenrightBig dyds, and, for p= +∞, (2.16)/bardblu(t,·)/bardblL∞(0,1)+/bardbl∂tu(t,·)/bardblL∞(0,1)+/bardbl∂xu(t,·)/bardblL∞(0,1) ≤Ct/parenleftBig /bardbl∂tu(t,·)/bardblL∞/parenleftbig (0,t)×(0,1)/parenrightbig+/bardbl∂xu(t,·)/bardblL∞/parenleftbig (0,t)×(0,1)/parenrightbig+/bardblu(t,·)/bardblL∞/parenleftbig (0,t)×(0,1)/parenrightbig/parenrightBig , for positive constant Conly depending only on p,T,/bardbla/bardblL∞,/bardblb/bardblL∞,/bardblc/bardblL∞. In the sequel, such constants will again be denoted by C. It is immediate to deduce from the above equations that u= 0 on [0 ,1/2C]×(0,1) and then u= 0 in (0 ,T)×(0,1). The proof of the uniqueness is complete. •Step 2: Proof of the existence. Let ( an), (bn), and (cn) be smooth functions in [0 ,T]×[0,1] such that supp an,suppbn,suppcn∩0×[0,1] =∅, (an,bn,cn)⇀(a,b,c) weakly star in/parenleftBig L∞/parenleftbig (0,T)×(0,1)/parenrightbig/parenrightBig3 , and (an,bn,cn)→(a,b,c) in/parenleftBig Lq/parenleftbig (0,T)×(0,1)/parenrightbig/parenrightBig3 for 1≤q <+∞. Letu0,n∈C∞ c(0,1) andu1,n∈C∞ c(0,1) be such that, if 1 ≤p <+∞, u0,n→u0inW1,p 0(0,1) and u1,n→u1inLp(0,1), and, ifp= +∞then the following two facts hold u0,n⇀ u0weakly star in W1,∞ 0(0,1) and u1,n⇀ u1weakly star in L∞(0,1), and, for 1 ≤q <+∞, u0,n→u0inW1,q 0(0,1) and u1,n→u1inLq(0,1). The existence of ( an,bn,cn) and the existence of u0,nandu1,nfollows from the standard theory of Sobolev spaces, see, e.g., [ 5]. Letunbe the weak solution corresponding to ( an,bn,cn) with initial data ( u0,n,u1,n). Thenun is smooth in [0 ,T]×[0,1]. Set gn(t,x) =−an(t,x)∂tun(t,x)−bn(t,x)∂xun(t,x)−cn(t,x)un(t,x) in (0,T)×(0,1). Extendun,gn, andfin (0,T)×Rby first odd refection in ( −1,0) and so on, and still denote the extension by unandgn, andf. We then have (2.17) ∂ttun−∂xxun=gn+fin (0,T)×R,9 The d’Alembert formula gives u(t,x) =1 2ˆt 0ˆx+t−τ x−t+τgn(τ,y)+f(τ,y)dydτ +1 2/parenleftBig un(0,x−t)+un(0,x+t)/parenrightBig +1 2ˆx+t x−t∂tun(0,y)dy. As in the proof of the uniqueness, we then have, for 1 ≤p <+∞and 0< t < T, (2.18)ˆ1 0|un(t,x)|p+|∂tun(t,x)|p+|∂xun(t,x)|pdx ≤Cˆt 0ˆ1 0/parenleftBig |∂tu(s,y)|p+|∂xu(s,y)|p/parenrightBig dyds +C/parenleftbigg /bardblun(0,·)/bardblp W1,p+/bardbl∂tun(0,·)/bardblp Lp+ˆt 0ˆ1 0|f(s,y)|pdyds/parenrightbigg , and, for p= +∞, (2.19)/bardblu(t,·)/bardblL∞(0,1)+/bardbl∂tu(t,·)/bardblL∞(0,1)+/bardbl∂xu(t,·)/bardblL∞(0,1) ≤Ct/parenleftbigg /bardbl∂tu(t,·)/bardblL∞/parenleftbig (0,t)×(0,1)/parenrightbig+/bardbl∂xu(t,·)/bardblL∞/parenleftbig (0,t)×(0,1)/parenrightbig/parenrightbigg +C/parenleftbigg /bardblun(0,·)/bardblW1,∞+/bardbl∂tun(0,·)/bardblL∞+/bardblf/bardblL∞/parenleftbig (0,t)×(0,1)/parenrightbig/parenrightbigg . Lettingn→+∞, we derive ( 2.8) from (2.18) and (2.19). To derive that u∈C([0,T];W1,p 0(0,1))∩C1([0,T];Lp(0,1)) in the case 1 ≤p <+∞and u∈C([0,T];W1,2 0(0,1))∩C1([0,T];L2(0,1)) otherwise, one just notes that ( un) is a Cauchy sequence in these spaces correspondingly. The proof is complete. /square Remark 2.2. Our proof on the well-posedness is quite standard and is base d on the d’Alembert formula. This formula was also used previously in [ 19]. Remark 2.3. There are several ways to give the notion of weak solution eve n in the case p= 2, see, e.g., [ 2,8]. The definitions given here is a nature modification of the ca sep= 2 given in [ 2]. 2.2.Proof of Proposition 2.2.The proof of Proposition 2.2is similar to the one of Propo- sition2.1. To apply the d’Alembert formula, one just needs to extend va rious function appro- priately and differently. For example, in the proof of the uniq ueness, one extend uandgin (0,T)×Rby appropriate reflection in xfirst by even extension in ( −1,0), i.e.,u(t,x) =u(t,−x) andg(t,x) =g(t,−x) in (0,T)×(−1,0) and so on. The details are left to the reader. /square 3.Some useful lemmas In this section, we prove three lemmas which will be used thro ugh out the rest of the paper. The first one is quite standard and the last two ones are the mai n ingredients of our analysis for the Dirichlet and Neumann boundary condition. We begin with the following lemma.10 Y. CHITOUR AND H.-M. NGUYEN Lemma 3.1. Let1< p <+∞,0< T <ˆT0, anda∈L∞((0,T)×(0,1))be such that a≥0in (0,T)×(0,1). There exists a positive constant Cdepending only on p,ˆT0, and/bardbla/bardblL∞such that, for(ρ,ξ)∈/bracketleftbig Lp/parenleftbig (0,T)×(0,1)/parenrightbig/bracketrightbig2, (3.1)ˆT 0ˆ1 0a|ρ−ξ|p(t,x)dxdt≤/braceleftBiggCmpifp≥2, C(mp+m2/p p)if1< p <2, where (3.2) mp=ˆ1 0ˆT 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dtdx. Proof.The proof of Lemma 3.1is quite standard. For the convenience of the reader, we pres ent its proof. There exists a positive constant Cpdepending only on psuch that •for 2≤p <+∞, it holds, for α,β∈R, (α−β)(α|α|p−2−β|β|p−2)≥Cp|α−β|p; •for 1< p <2, it holds, for α,β∈R3 (α−β)(α|α|p−2−β|β|p−2)≥Cpmin/braceleftbig |α−β|p,|α−β|2/bracerightbig . Using this, we derive that ˆT 0ˆ1 0 |ρ−ξ|≥1a|ρ−ξ|pdxdt+ˆT 0ˆ1 0 |ρ−ξ|<1a|ρ−ξ|max{p,2}dxdt≤mp. This yields (3.3)ˆT 0ˆ1 0a|ρ−ξ|p(t,x)dxdt≤Cmpifp≥2, and, using H¨ older’s inequality, one gets (3.4)ˆT 0ˆ1 0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p p) if 1< p≤2, The conclusion follows from ( 3.3) and (3.4). /square Thefollowing lemmais oneof themain ingredients in theanal ysis of theDirichlet andNeumann boundary conditions. Lemma 3.2. Let1< p <+∞,0< T0< T <ˆT0,ε0>0,λ >0, anda∈L∞((0,T)×(0,1))be such that T > T 0+4ε0,a≥0anda≥λ >0in(0,T)×(x0−ε0,x0+ε0)⊂(0,T)×(0,1)for somex0∈(0,1). Let(ρ,ξ)be a broad solution of the system (3.5)/braceleftBigg ρt−ρx=−1 2a(ρ−ξ)in(0,T)×(0,1), ξt+ξx=1 2a(ρ−ξ)in(0,T)×(0,1). Set (3.6) mp=ˆ1 0ˆT 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dtdx. 3Using the symmetry between αandβ, one can assume |α| ≥ |β|and by considering β/|α|, it is enough to prove these inequalities for α= 1 and β∈(−1,1). One finally reduces the analysis for β∈(0,1) and even βclose to one. The conclusion follows by performing a Taylor expansion wit h respect to 1 −β.11 Then there exists z∈(x0−ε0/2,x0+ε0/2)such that (3.7)ˆε0/2 0ˆT 0|ρ(t+s,z)−ρ(t,z)|pdtds+ˆε0/2 0ˆT 0|ξ(t+s,z)−ξ(t,z)|pdtds +ˆT 0|ρ(t,z)−ξ(t,z)|pdt+ˆT 0ˆ1 0a|ρ−ξ|p(t,x)dxdt ≤/braceleftBiggCmp ifp≥2, C(mp+m2/p p)if1≤p <2. for some positive constant Cdepending only on ε0,λ,p,T0,ˆT0, and/bardbla/bardblL∞. Proof.Set T1=T−4ε0andT2=T−2ε0. ThenT > T2> T1> T0. We have, for s∈(−ε0/2,ε0/2) andy∈(x0−ε0/2,x0+ε0/2), (3.8)ρ(t,y+2s)−ρ(t,y) =/parenleftBig ρ(t+2s,y)−ρ(t+s,y+s)/parenrightBig +/parenleftBig ρ(t+s,y+s)−ξ(t+s,y+s)/parenrightBig +/parenleftBig ξ(t+s,y+s)−ξ(t,y)/parenrightBig +/parenleftBig ξ(t,y)−ρ(t,y)/parenrightBig . By the characteristics method, we obtain (3.9) ξ(t+s,y+s)−ξ(t,y) =1 2ˆs 0a(t+τ,y+τ)/parenleftBig ρ(t+τ,y+τ)−ξ(t+τ,y+τ)/parenrightBig dτ and (3.10)ρ(t+2s,y)−ρ(t+s,y+s) =1 2ˆ2s sa(t+τ,y+2s−τ)/parenleftBig ρ(t+τ,y+2s−τ)−ξ(t+τ,y+2s−τ)/parenrightBig dτ. Combining ( 3.8), (3.9), and (3.10), after integrating with respect to tfrom 0 to T1, we obtain, for 0≤s≤ε0/2, ˆT1 0|ρ(t+2s,y)−ρ(t,y)|pdt≤4p−1/parenleftbiggˆT2 0|ρ(t,y+s)−ξ(t,y+s)|pdt +2ˆT2 0ˆ1 0ap|ρ−ξ|p(t,x)dtdx+ˆT 0|ρ(t,y)−ξ(t,y)|pdt/parenrightbigg .12 Y. CHITOUR AND H.-M. NGUYEN Integrating the above inequality with respect to sfrom 0 to ε0/2, we obtain (3.11)ˆε0/2 0ˆT1 0|ρ(t+2s,y)−ρ(t,y)|pdtds ≤4p/parenleftbiggˆx0+ε0 x0−ε0ˆT 0|ρ(t,x)−ξ(t,x)|pdtdx +ε0ˆT 0|ρ(t,y)−ξ(t,y)|pdt+ε0ˆ1 0ˆT 0ap|ρ−ξ|p(t,x)dtdx/parenrightbigg . Similarly, we have (3.12)ˆε0/2 0ˆT1 0|ξ(t+2s,y)−ξ(t,y)|pdtds ≤4p/parenleftbiggˆx0+ε0 x0−ε0ˆT 0|ρ(t,x)−ξ(t,x)|pdtdx +ε0ˆT 0|ρ(t,y)−ξ(t,y)|pdt+ε0ˆ1 0ˆT 0ap|ρ−ξ|p(t,x)dtdx/parenrightbigg . Takey=z∈(x0−ε0/2,x0+ε0/2) such that (3.13)ˆT 0|ρ(t,z)−ξ(t,z)|pdt≤1 ε0ˆx0+ε0 x0−ε0ˆT 0|ρ−ξ|p(t,x)dxdt. By choosing y=zin (3.11) and (3.12), then by using ( 3.13) and the fact that (itself consequence of (1.5)) ˆx0+ε0 x0−ε0ˆT 0|ρ(t,x)−ξ(t,x)|pdtdx≤C(a,p)ˆ1 0ˆT 0ap|ρ−ξ|p(t,x)dtdx, for some positive constant C(a,p) only depending on a,p, one gets the conclusion. /square The next lemma is also a main ingredient of our analysis for th e Dirichlet and Neumann boundary conditions. Lemma 3.3. Let1≤p <+∞andL > l > 0, and let u∈Lp(0,L+l). Then there exists a positive constant Cdepending only on p,L, andlsuch that (3.14)ˆL 0|u(x)− L 0u(y)dy|pdx≤Cˆl 0ˆL 0|u(x+s)−u(x)|pdxds. Here and in what follows,fflb ameans1 b−a´b aforb > a. Proof.By scaling, one can assume that L= 1. Fix n≥2 such that 2 /n≤l≤2/(n−1). One first notes that, for x∈[0,1], (3.15) x+1/n x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(x)− x+1/n xu(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep dxJensen ≤ x+1/n x x+1/n x|u(x)−u(y)|pdxdy ≤n2ˆ2/n 0ˆ1 0|u(x+s)−u(x)|pdxds13 and (3.16)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x+1/n xu(s)ds− x+2/n x+1/nu(t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep Jensen ≤ x+1/n x x+2/n x+1/n|u(s)−u(t)|pdtds ≤n2ˆ2/n 0ˆ1 0|u(x+s)−u(x)|pdxds. For 0≤k≤n−1, set ak= k/n+1/n k/nu(s)ds. We then derive from ( 3.16) that, for 0 ≤i < j≤n−1, |aj−ai|p≤(|ai+1−ai|+···+|aj−aj−1|)p ≤np−1(|ai+1−ai|p+···+|aj−aj−1|p) ≤np+1ˆ2/n 0ˆ1 0|u(x+s)−u(x)|pdxds. This implies, for 0 ≤k≤n−1, (3.17)/vextendsingle/vextendsingle/vextendsingle/vextendsingleak−ˆ1 0u(t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 nn−1/summationdisplay i=0|ak−ai|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep ≤1 nn−1/summationdisplay i=0|ak−ai|p≤np+1ˆ2/n 0ˆ1 0|u(x+s)−u(x)|pdxds. We have (3.18)ˆ1 0/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(x)−ˆ1 0u(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep dx=n−1/summationdisplay k=0ˆk/n+1/n k/n/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(x)−ˆ1 0u(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep dx ≤2p−1n−1/summationdisplay k=0ˆk/n+1/n k/n|u(x)−ak|pdx+2p−1n−1/summationdisplay k=0/vextendsingle/vextendsingle/vextendsingle/vextendsingleak−ˆ1 0u(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep The conclusion with C= 2pnp+1now follows from ( 3.15), (3.17), and (3.18) after noting that L= 1 and 2 /n≤l. /square Remark 3.1. Related ideas used in the proof of Lemma 3.3was implemented in the proof of Caffarelli-Kohn-Nirenberg inequality for fractional Sobol ev spaces [ 28]. 4.Exponential decay in Lp-framework for the Dirichlet boundary condition In this section, we prove Theorem 1.1and Theorem 1.2. We begin with the proof Theorem 1.2 in the first section, and then use it to prove Theorem 1.1in the second section. We finally extend these results for awhich might be negative in some regions using a standard pert urbation argument in the third section.14 Y. CHITOUR AND H.-M. NGUYEN 4.1.Proof of Theorem 1.2.We will only consider smooth solutions ( ρ,ξ)4. The general case will follow by regularizing arguments. Moreover, replacin g (ρ,ξ) by (ρ−c0,ξ−c0), where the constant c0is defined in ( 1.12), we can assume that ˆ1 0(ρ0+ξ0)dx= 0. Multiplying the equation of ρwithρ|ρ|p−2, the equation of ξwithξ|ξ|p−2, and integrating the expressions with respect to x, after using the boundary conditions, we obtain, for t >0, (4.1)1 pd dtˆ1 0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1 2ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx= 0. This implies (4.2)1 p/bardbl(ρ,ξ)(t,·)/bardblp Lp(0,1)+1 2ˆt 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt=1 p/bardbl(ρ0,ξ0)/bardblp Lp(0,1). Integrating the equations of ρandξ, summing them up and using the boundary conditions, we obtain d dtˆ1 0/parenleftBig ρ(t,x)+ξ(t,x)/parenrightBig dx= 0 fort >0. It follows that (4.3)ˆ1 0/parenleftBig ρ(t,x)+ξ(t,x)/parenrightBig dx=ˆ1 0/parenleftBig ρ(0,x)+ξ(0,x)/parenrightBig dx= 0 fort≥0. By (4.2) and (4.3), toderive ( 1.11), it sufficesto prove that thereexists aconstant c >0depending only on/bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that for any T >2, there exists cT>0 only depending onp,T,aso that (4.4)ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt≥cT/bardbl(ρ0,ξ0)/bardblp Lp(0,1). By scaling, without loss of generality, one might assume tha t (4.5) /bardbl(ρ0,ξ0)/bardblLp(0,1)= 1 Set mp:=ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt. Applying Lemma 3.1, we have (4.6)ˆT 0ˆ1 0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p p). By Lemma 3.2there exists z∈(x0−ε0/2,x0+ε0/2) such that (4.7)ˆε0/2 0ˆT 0|ρ(t+s,z)−ρ(s,z)|pdtds+ˆε0/2 0ˆT 0|ξ(t+s,z)−ξ(s,z)|pdtds +ˆT 0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p p). 4We thus assume that ais smooth. Nevertheless, the constants in the estimates whi ch will be derived in the proof depend only on p,/bardbla/bardblL∞,λ, andε0.15 By Lemma 3.3, we have (4.8)ˆT 0|ρ(t,z)−Aρ|pdt≤Cˆε0/2 0ˆT 0|ρ(t+s,z)−ρ(s,z)|pdtds and (4.9)ˆT 0|ξ(t,z)−Aξ|pdt≤Cˆε0/2 0ˆT 0|ξ(t+s,z)−ξ(s,z)|pdtds. where we have set (4.10) Aρ:= T 0ρ(s,z)ds, A ξ:= T 0ξ(s,z)ds. Combining ( 4.7), (4.8), and (4.9) yields (4.11)ˆT 0|ρ(t,z)−Aρ|pdt+ˆT 0|ξ(t,z)−Aξ|pdt +ˆT 0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p p). We next prove the following estimates (4.12)ˆ1 0|ρ(0,x)−Aξ|pdx≤C(mp+m2/p p) and (4.13)ˆ1 0|ξ(0,x)−Aρ|pdx≤C(mp+m2/p p). The arguments being similar, we only provide that of ( 4.12). Forx∈(0,1), one has, by using the boundary condition at x= 0, i.e., ρ(·,0) =ξ(·,0), ρ(0,x) =/parenleftBig ρ(0,x)−ρ(x,0)/parenrightBig +ρ(x,0) =/parenleftBig ρ(0,x)−ρ(x,0)/parenrightBig +ξ(x,0) =/parenleftBig ρ(0,x)−ρ(x,0)/parenrightBig +/parenleftBig ξ(x,0)−ξ(x+z,z)/parenrightBig +ξ(x+z,z), which yields, after substracting Aξto both sides of the above equality, (4.14)ˆ1 0|ρ(0,x)−Aξ|pdx≤3p−1/parenleftbiggˆ1 0|ρ(0,x)−ρ(x,0)|pdx+ˆ1 0|ξ(x,0)−ξ(x+z,z)|pdx +ˆ1 0|ξ(x+z,z)−Aξ|pdx/parenrightbigg . We use the characteristics method and ( 3.9),(3.10) to upper bound the first two integrals in the right-hand side of ( 4.14) byC(mp+m2/p p). As for the third integral in the right-hand side of16 Y. CHITOUR AND H.-M. NGUYEN (4.14), we perform the change of variables t=x+zto obtain ˆ1 0|ξ(x+z,z)−Aξ|pdx=ˆz+1 z|ξ(t,z)−Aξ|pdt ≤ˆT 0|ξ(t,z)−Aξ|pdt, which is upper bounded by C(mp+m2/p p) according to ( 4.11). The proof of ( 4.12) is complete. We now resume the argument for ( 4.4). We start by noticing that, for every t∈(0,T) |Aρ−Aξ| ≤ |Aρ−ρ(t,z)|+|Aξ−ρ(t,z)|+|ρ(t,z)−ξ(t,z)|. Taking the p-th power, integrating over t∈(0,T) and using ( 4.11), one gets that (4.15) |Aρ−Aξ|p≤C(mp+m2/p p). Similarly, for every x∈(0,1), Aρ+Aξ=/parenleftbig Aρ−ξ(0,x)/parenrightbig +/parenleftbig Aξ−ρ(0,x)/parenrightbig +/parenleftbig ρ(0,x)+ξ(0,x)/parenrightbig . Integrating over x∈(0,1) and using ( 4.3), then taking the p-th power and using ( 4.12) and (4.13) yield (4.16) |Aρ+Aξ|p≤C(mp+m2/p p). Still, for x∈(0,1), it holds |ρ(0,x)|p+|ξ(0,x)|p≤2p−1/parenleftBig |Aρ−ξ(0,Aξ|p+|Aρ−ξ(0,x)|p/parenrightBig +|Aρ|p+|Aξ|p. Integrating over x∈(0,1) and using ( 4.5), one gets (4.17) 1 ≤ |Aρ|p+|Aξ|p+C(mp+m2/p p). Since it holds |a|p+|b|p≤ |a+b|p+|a−b|pfor every real numbers a,b, one deduces from ( 4.15), (4.16) and (4.17) that 1≤C(mp+m2/p p) and hence mp≥c3for some positive constant depending only on /bardbla/bardblL∞(R+×(0,1)),ε0,γ, andp (after fixing for instance T= 3). The proof of the theorem is complete. /square 4.2.Proof of Theorem 1.1.Using Theorem 1.2, we obtain the conclusion of Theorem 1.1 for smooth solutions. The proof in the general case follows f rom the smooth case by density arguments. /square 4.3.On the case anot being non-negative. In this section, we first consider the following perturbed system of ( 1.9): (4.18) ρt−ρx=−1 2a(ρ−ξ)−b(ρ−ξ) in R+×(0,1), ξt+ξx=1 2a(ρ−ξ)+b(ρ−ξ) in R+×(0,1), ρ(t,0)−ξ(t,0) =ρ(t,1)−ξ(t,1) = 0 in R+. We establish the following result.17 Theorem 4.1. Let1< p <+∞,ε0>0,λ >0, anda,b∈L∞/parenleftbig R+×(0,1)/parenrightbig be such that a≥0 anda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). There exists a positive constant αdepending only on p,/bardbla/bardblL∞,ε0, andλsuch that if (4.19) /bardblb/bardblL∞≤α, then there exist constants C,γ >0depending only on p,/bardbla/bardblL∞,ε0, andλsuch that, if´1 0ρ0+ ξ0dx= 0, then the solution (ρ,ξ)of(4.18)satisfies (4.20) /bardbl(ρ,ξ)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ0,ξ0)/bardblLp(0,1), t≥0. Proof.Multiplying the equation of ρwithρ|ρ|p−2, the equation of ξwithξ|ξ|p−2, and integrating the expressions with respect to x, after using the boundary conditions, we obtain 1 pd dtˆ1 0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1 2ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx +ˆ1 0b(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx= 0. This implies (4.21)1 p/bardbl(ρ,ξ)(t,·)/bardblp Lp(0,1)+1 2ˆt 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−1−ξ|ξ|p−1)(t,x)dxdt +ˆt 0ˆ1 0b(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx=1 p/bardbl(ρ0,ξ0)/bardblp Lp(0,1). Integrating the equation of ρandξand using the boundary condition, we obtain d dtˆ1 0/parenleftBig ρ(t,x)+ξ(t,x)/parenrightBig dx= 0,fort >0. It follows that (4.22)ˆ1 0/parenleftBig ρ(t,x)+ξ(t,x)/parenrightBig dx=ˆ1 0/parenleftBig ρ(0,x)+ξ(0,x)/parenrightBig dx= 0,fort >0. By (4.21) and (4.22), to derive ( 4.20), it suffices to prove that there exists a constant c >0 depending only on /bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that for T= 35, it holds (4.23)ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt≥c/bardbl(ρ0,ξ0)/bardblp Lp(0,1). Using the facts that a≥0 andbis bounded, a simple application of Gronwall’s lemma to ( 4.21) yields the existence of α >0 depending only on /bardblb/bardblL∞(R+×(0,1))so that (4.24) /bardbl(ρ,ξ)(t,·)/bardblp Lp(0,1)≤epαt/bardbl(ρ,ξ)(0,·)/bardblp Lp(0,1)fort∈[0,T]. 5It holds for T >2 withc=cT.18 Y. CHITOUR AND H.-M. NGUYEN Let (ρ1,ξ1) be the unique solution of the system (4.25) ρ1,t−ρ1,x=−1 2a(ρ1−ξ1)−b(ρ−ξ) inR+×(0,1), ξ1,t+ξ1,x=1 2a(ρ1−ξ1)+b(ρ−ξ) in R+×(0,1), ρ1(t,0)−ξ1(t,0) =ρ1(t,1)−ξ1(t,1) = 0 in R+, ρ1(0,·) =ξ1(0,·) = 0 in (0 ,1). Thus−b(ρ−ξ) andb(ρ−ξ) can be considered as source terms for the system of ( ρ1,ξ1). We then derive from ( 4.24) that (4.26) /bardbl(ρ1,ξ1)/bardblLp(T,·)≤Cα/bardbl(ρ,ξ)(0,·)/bardblp Lp(0,1). Set /tildewideρ=ρ−ρ1and/tildewideξ=ξ−ξ1. Then (4.27) /tildewideρt−/tildewideρx=−1 2a(t,x)(/tildewideρ−/tildewideξ) in R+×(0,1), /tildewideξt+/tildewideξx=1 2a(t,x)(/tildewideρ−/tildewideξ) in R+×(0,1), /tildewideρ(t,0)−/tildewideξ(t,0) =/tildewideρ(t,1)−/tildewideξ(t,1) = 0 in R+, /tildewideρ(0,·) =ρ0,/tildewideξ(0,·) =ξ0 in (0,1). Applying Theorem 1.2, we have (4.28) /bardbl(/tildewideρ,/tildewideξ)(T,·)/bardblLp≤c/bardbl(/tildewideρ,/tildewideξ)(0,·)/bardblLp for some positive constant cdepending only on /bardbla/bardblL∞,ε0, andλ. The conclusion now follows from (4.26) and (4.27). /square Regarding the wave equation, we have Theorem 4.2. Let1< p <+∞,ε0>0,λ >0, anda,b∈L∞/parenleftbig R+×(0,1)/parenrightbig be such that a≥0 anda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1). There exists a positive constant α depending only on p,/bardbla/bardblL∞,ε0, andλsuch that if (4.29) /bardblb/bardblL∞≤α, then there exist positive constants Candγdepending on p,/bardbla/bardblL∞/parenleftbig R+×(0,1)/parenrightbig,ε0, andλsuch that for allu0∈W1,p 0(0,1)andu1∈Lp(0,1), the unique weak solution u∈C([0,+∞);W1,p 0(0,1))∩ C1([0,+∞);Lp(0,1))of (4.30) ∂ttu−∂xxu+/parenleftBig a(t,x)+b(t,x)/parenrightBig ∂tu= 0inR+×(0,1), u(t,0) =u(t,1) = 0 inR+, u(0,·) =u0, ∂tu(0,·) =u1 in(0,1), satisfies (4.31) /bardbl∂tu(t,·)/bardblp Lp(0,1)+/bardbl∂xu(t,·)/bardblp Lp(0,1)≤Ce−γt/parenleftBig /bardblu1/bardblp Lp(0,1)+/bardbl∂xu0/bardblp Lp(0,1)/parenrightBig , t≥0. Proof.The proof of Theorem 4.2is similar to that of Theorem 1.1however instead of using Theorem 1.2one apply Theorem 4.1. The details are left to the reader. /square19 5.Exponential decay in Lp-framework for the Neuman boundary condition In this section, we study the decay of the solutions of the dam ped wave equation equipped the Neumann boundary condition and the solutions of the corresp onding hyperbolic systems. Here is the first main result of this section concerning the wave equa tion. Theorem 5.1. Let1< p <+∞,ε0>0,λ >0, and let a∈L∞/parenleftbig R+×(0,1)/parenrightbig be such that a≥0 anda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). There exist positive constants Candγdepending only on p,/bardbla/bardblL∞/parenleftbig R+×(0,1)/parenrightbig,ε0, andλsuch that for all u0∈W1,p(0,1) andu1∈Lp(0,1), the unique weak solution u∈C([0,+∞);W1,p(0,1))∩C1([0,+∞);Lp(0,1))of (1.1)and(1.3)satisfies (5.1) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤Ce−γt/parenleftBig /bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)/parenrightBig , t≥0. As in the case where the Dirichlet condition is considered, w e use the Riemann invariants to transform ( 1.1) with Neumann boundary condition into a hyperbolic system. Set (5.2)ρ(t,x) =ux(t,x)+ut(t,x) and ξ(t,x) =ux(t,x)−ut(t,x),for (t,x)∈R+×(0,1). One can check that for smooth solutions uof (1.1), the pair of functions ( ρ,ξ) defined in ( 1.8) satisfies the system (5.3) ρt−ρx=−1 2a(ρ−ξ) in R+×(0,1), ξt+ξx=1 2a(ρ−ξ) in R+×(0,1), ρ(t,0)+ξ(t,0) =ρ(t,1)+ξ(t,1) = 0 in R+. Concerning ( 5.3), we prove the following result. Theorem 5.2. Let1< p <+∞,ε0>0,λ >0, anda∈L∞/parenleftbig R+×(0,1)/parenrightbig be such that a≥0and a≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). Then there exist positive constants C,γdepending only on on p,/bardbla/bardblL∞/parenleftbig R+×(0,1)/parenrightbig,ε0, andλsuch that the unique solution uof(5.3)with the initial condition ρ(0,·) =ρ0andξ(0,·) =ξ0satisfies (5.4) /bardbl(ρ,ξ)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ0,ξ0)/bardblLp(0,1). The rest of this section is organized as follows. The first sub section is devoted to the proof of Theorem 5.2and the second subsection is devoted to the proof of Theorem 5.1. 5.1.Proof of Theorem 5.2.The argument is in the spirit of that of Theorem 1.2. As in there, we will only consider smooth solutions ( ρ,ξ). Multiplying the equation of ρwithρ|ρ|p−2, the equation of ξwithξ|ξ|p−2, and integrating the expressions with respect to x, after using the boundary conditions, we obtain, for t >0, (5.5)1 pd dtˆ1 0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1 2ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx= 0. This implies (5.6)1 p/bardbl(ρ,ξ)(t,·)/bardblp Lp(0,1)+1 2ˆt 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−1−ξ|ξ|p−1)(t,x)dxdt=1 p/bardbl(ρ0,ξ0)/bardblp Lp(0,1). By (5.6), to derive ( 5.4), it suffices to prove that there exists a constant c >0 depending only on/bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that for any T >2, there exists cT>0 only depending on20 Y. CHITOUR AND H.-M. NGUYEN p,T,aso that (5.7)ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt≥cT/bardbl(ρ0,ξ0)/bardblp Lp(0,1). By scaling, without loss of generality, one might assume tha t (5.8) /bardbl(ρ0,ξ0)/bardblLp(0,1)= 1 Set mp:=ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt. Applying Lemma 3.1, we have (5.9)ˆT 0ˆ1 0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p p). By Lemma 3.2there exists z∈(x0−ε0/2,x0+ε0/2) such that (5.10)ˆε0/2 0ˆT 0|ρ(t+s,z)−ρ(t,z)|pdtds+ˆε0/2 0ˆT 0|ξ(t+s,z)−ξ(t,z)|pdtds +ˆT 0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p p). Applying Lemma 3.3, we obtain (5.11)ˆT 0|ρ(t,z)− T 0ρ(s,z)ds|pdt≤Cˆε0/2 0ˆT 0|ρ(t+s,z)−ρ(t,z)|pdtds and (5.12)ˆT 0|ξ(t,z)− T 0ξ(s,z)ds|pdt≤Cˆε0/2 0ˆT 0|ξ(t+s,z)−ξ(t,z)|pdtds. Combining ( 5.10), (5.11), and (5.12) yields (5.13)ˆT 0|ρ(t,z)− T 0ρ(τ,z)dτ|pdt+ˆT 0|ξ(t,z)− T 0ξ(τ,z)dτ|pdt +ˆT 0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p p). Using the characteristics method to estimate ρ(τ,0) byρ(τ−z,z) andξ(τ,0) byξ(τ+z,z) after using the boundary condition at 0 and choosing appropr iatelyτ, we derive from ( 5.9) that (5.13) that (5.14)/vextendsingle/vextendsingle/vextendsingle/vextendsingle T 0ρ(t,z)dt+ T 0ξ(t,z)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep ≤C(mp+m2/p p). As done to obtain ( 4.12) and (4.13), we use the characteristic methods to estimate ρ(0,·) via ξ(t,z) andξ(0,·) viaρ(t,z) after taking into account the boundary conditions (at x= 0 forρ(0,·) and atx= 1 forξ(0,·)), we derive from ( 5.9) and (5.13) that (5.15)/vextendsingle/vextendsingle/vextendsingle/vextendsingle T 0ρ(t,z)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep +/vextendsingle/vextendsingle/vextendsingle/vextendsingle T 0ξ(t,z)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep ≥1−C(mp+m2/p p).21 Combining ( 5.14) and (5.15), we derive (after choosing T= 3) that there exists a postive constant c3only depending on /bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that mp≥c. The proof of the theorem is complete. /square 5.2.Proof of Theorem 5.1.The proof of Theorem 5.1is in the same spirit of Theorem 1.1. However, instead of using Theorem 1.2, we apply Theorem 5.2. In fact, as in the proof of Theo- rem1.1, we have ˆ1 0|∂tu(t,x)−∂xu(t,x)|p+|∂tu(t,x)+∂xu(t,x)|pdx ≤Ce−γtˆ1 0|∂tu(0,x)−∂xu(0,x)|p+|∂tu(0,x)+∂xu(0,x)|pdx. Assertion ( 5.1) follows with two different appropriate positive constants Candγ. /square Remark 5.1. We can also consider the setting similar to the one in Section 4.3and establish similar results. This allows one to deal with a class of afor which ais not necessary to be non-negative. The analysis for this is almost the same lines as in Section 4.3and is not pursued here. 6.Exponential decay in Lp-framework for the dynamic boundary condition In this section, we study the decay of the solution of the damp ed wave equation equipped the dynamic boundary condition and of the solutions of the corre sponding hyperbolic systems. Here is the first main result of this section concerning the wave eq uation. Theorem 6.1. Let1< p <+∞,κ >0, anda∈L∞/parenleftbig R+×(0,1)/parenrightbig non negative. Then there exist positive constants C,γdepending only on p,κ, and/bardbla/bardblL∞/parenleftbig R+×(0,1)/parenrightbigsuch that for all u0∈ W1,p(0,1)andu1∈Lp(0,1), there exists a unique weak solution u∈C([0,+∞);W1,p(0,1))∩ C1([0,+∞);Lp(0,1))such that ∂tu,∂xu∈C([0,1];Lp(0,T))for allT >0of (6.1) ∂ttu−∂xxu+a∂tu= 0 inR+×(0,1), ∂xu(t,0)−κ∂tu(t,0) =∂xu(t,1)+κ∂tu(t,1) = 0 inR+, u(0,·) =u0, ∂tu(0,·) =u1 in(0,1), satisfies (6.2) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤Ce−γt/parenleftBig /bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)/parenrightBig , t≥0. Remark 6.1. In Theorem 6.1, a weak considered solution of ( 6.1) means that ∂ttu(t,x)− ∂xxu(t,x) +a(t,x)∂tu= 0 holds in the distributional sense, and the boundary and th e initial conditions are understood as usual thanks to the regularity imposing condition on the solutions. As previously, we use the Riemann invariants to transform th e wave equation into a hyperbolic system. Set (6.3)ρ(t,x) =ux(t,x)+ut(t,x) and ξ(t,x) =ux(t,x)−ut(t,x) for (t,x)∈R+×(0,1).22 Y. CHITOUR AND H.-M. NGUYEN One can check that for smooth solutions uof (1.1), the pair of functions ( ρ,ξ) defined in ( 1.8) satisfies the system (6.4) ρt−ρx=−1 2a(t,x)(ρ−ξ) in R+×(0,1), ξt+ξx=1 2a(t,x)(ρ−ξ) in R+×(0,1), ξ(t,0) =c0ρ(t,0), ρ(t,1) =c1ξ(t,1) inR+, wherec0=c1= (κ−1)/(κ+1). Regarding System ( 6.4) withc0,c1not necessarily equal, we prove the following result. Theorem 6.2. Let1< p <+∞,c0,c1∈(−1,1), anda∈L∞/parenleftbig R+×(0,1)/parenrightbig non negative. Then there exist positive constants C,γdepending only on c0,c1, and/bardbla/bardblL∞/parenleftbig R+×(0,1)/parenrightbigsuch that the unique solution uof(6.4)with the initial condition ρ(0,·) =ρ0andξ(0,·) =ξ0satisfies (6.5) /bardbl(ρ,ξ)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ0,ξ0)/bardblLp(0,1), t≥0. The rest of this section is organized as follows. The proof of Theorem 6.2is given in the first section and the proof of Theorem 6.1is given in the second section. 6.1.Proof of Theorem 6.2.We will only consider smooth solutions ( ρ,ξ). Multiplying the equation of ρwithρ, the equation of ξwithξ, and integrating the expressions with respect to x, after using the boundary conditions, we obtain, for t >0, (6.6)1 pd dtˆ1 0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1 2ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx 1 p/parenleftBig (1−|c1|p)|ξ(t,1)|p+(1−|c0|p)|ρ(t,0)|p/parenrightBig = 0. This implies (6.7)1 p/bardbl(ρ,ξ)(t,·)/bardblp Lp(0,1)+1 2ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt +1 pˆT 0/parenleftBig (1−|c1|p)|ξ(t,1)|p+(1−|c0|p)|ρ(t,0)|p/parenrightBig dt=1 2/bardbl(ρ0,ξ0)/bardbl2 L2(0,1). To derive ( 6.5) from (6.7), it suffices to prove that there exists a constant c >0 depending only on/bardbla/bardblL∞(R+×(0,1)),c0,c1,ε0,γ, andpsuch that for for T= 36, it holds (6.8)ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt +ˆT 0/parenleftBig |ξ(t,1)|p+|ρ(t,0)|p/parenrightBig dt≥c/bardbl(ρ0,ξ0)/bardblp Lp(0,1). After scaling, one might assume without loss of generality t hat (6.9) /bardbl(ρ0,ξ0)/bardblLp(0,1)= 1 6It holds for T >2 withc=cT.23 Applying Lemma 3.1, we have (6.10)ˆT 0ˆ1 0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p p), where mp:=ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt. Using the characteristics method (in particular equations (3.9), (3.10)), we derive that (6.11) /bardbl(ρ,ξ)(T,·)/bardblp Lp(0,1)≤CˆT 0/parenleftBig |ξ(t,1)|p+|ρ(t,0)|p/parenrightBig dt+CˆT 0ˆ1 0ap|ρ−ξ|p(t,x)dxdt. As a consequence of ( 6.7), (6.9), (6.10), and (6.11), we have ˆT 0ˆ1 0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt+ˆT 0/parenleftBig |ξ(t,1)|p+|ρ(t,0)|p/parenrightBig dt≥c. The proof of the theorem is complete. /square Remark 6.2. In the case a≡0, one can show that the exponential stability for 1 ≤p≤+∞by noting that /bardbl/parenleftbig ρ(t+1,0),ρ(t+1,1)/parenrightbig /bardbl ≤max{|c0|,|c1|}/bardbl/parenleftbig ρ(t,0),ρ(t,1)/parenrightbig /bardbl. The conclusion then follows using the characteristics meth od. 6.2.Proof of Theorem 6.1.Wefirstdealwiththewell-posednessofthesystem. Theuniqu eness follows as in the proof of Proposition 2.1via the d’Alembert formula. The existence can be proved byapproximation arguments. Firstdeal withsmoothsolutio ns (withsmooth a) usingTheorem 6.2 and then pass to the limit. The details are omitted. The proof of ( 6.5) is in the same spirit of ( 1.7). However, instead of using Theorem 1.2, we apply Theorem 6.2. The details are left to the reader. /square Remark 6.3. One can prove the well-posedness of ( 1.1) and (1.4) directly in Lp-framework. 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1804.07080v2.Damping_of_magnetization_dynamics_by_phonon_pumping.pdf

Damping of magnetization dynamics by phonon pumping Simon Streib,1Hedyeh Keshtgar,2and Gerrit E. W. Bauer1, 3 1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 2Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran 3Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan (Dated: July 11, 2018) We theoretically investigate pumping of phonons by the dynamics of a magnetic film into a non- magnetic contact. The enhanced damping due to the loss of energy and angular momentum shows interferencepatternsasafunctionofresonancefrequencyandmagneticfilmthicknessthatcannotbe described by viscous (“Gilbert”) damping. The phonon pumping depends on magnetization direction as well as geometrical and material parameters and is observable, e.g., in thin films of yttrium iron garnet on a thick dielectric substrate. The dynamics of ferromagnetic heterostructures is at the root of devices for information and communication technologies [1–5]. When a normal metal contact is at- tached to a ferromagnet, the magnetization dynamics drives a spin current through the interface. This effect is known as spin pumping and can strongly enhance the (Gilbert) viscous damping in ultra-thin magnetic films [6–8]. Spin pumping and its (Onsager) reciprocal, the spin transfer torque [9, 10], are crucial in spintronics, as they allow electric control and detection of magnetiza- tion dynamics. When a magnet is connected to a non- magnetic insulator instead of a metal, angular momen- tum cannot leave the magnet in the form of electronic or magnonic spin currents, but they can do so in the form of phonons. Half a century ago it was reported [11, 12] and explained [13–16] that magnetization dynamics can generate phonons by magnetostriction. More recently, the inverse effect of magnetization dynamics excited by surface acoustic waves (SAWs) has been studied [17–20] and found to generate spin currents in proximity normal metals [21, 22]. The emission and detection of SAWs was combined in one and the same device [23, 24], and adia- batic transformation between magnons and phonons was observed in inhomogeneous magnetic fields [25]. The an- gular momentum of phonons [26, 27] has recently come into focus again in the context of the Einstein-de Haas effect [28] and spin-phonon interactions in general [29]. The interpretation of the phonon angular momentum in termsoforbitalandspincontributions[29]hasbeenchal- lenged [30], a discussion that bears similarities with the interpretation of the photon angular momentum [31]. In our opinion this distinction is rather semantic since not required to arrive at concrete results. A recent quantum theory of the dynamics of a magnetic impurity [32] pre- dicts a broadening of the electron spin resonance and a renormalized g-factor by coupling to an elastic contin- uum via the spin-orbit interaction, which appears to be related to the enhanced damping and effective gyromag- netic ratio discussed here. A phonon current generated by magnetization dynam- ics generates damping by carrying away angular momen- tum and energy from the ferromagnet. While the phonon phonon sinkzmagnet non-magnet0 phononsmHFigure 1. Magnetic film (shaded) with magnetization mat- tached to a semi-infinite elastic material, which serves as an ideal phonon sink. contribution to the bulk Gilbert damping has been stud- ied theoretically [33–38], the damping enhancement by interfaces to non-magnetic substrates or overlayers has to our knowledge not been addressed before. Here we present a theory of the coupled lattice and magnetiza- tion dynamics of a ferromagnetic film attached to a half- infinite non-magnet, which serves as an ideal phonon sink. We predict, for instance, significantly enhanced damping when an yttrium iron garnet (YIG) film is grown on a thick gadolinium gallium garnet (GGG) sub- strate. We consider an easy-axis magnetic film with static ex- ternal magnetic field and equilibrium magnetization ei- ther normal (see Fig. 1) or parallel to the plane. The magnet is connected to a semi-infinite elastic material. Magnetization and lattice are coupled by the magne- tocrystalline anisotropy and the magnetoelastic interac- tion, giving rise to coupled field equations of motion in the magnet [39–42]. By matching these with the lattice dynamics in the non-magnet by proper boundary con- ditions, we predict the dynamics of the heterostructure as a function of geometrical and constitutive parameters. We find that magnetization dynamics induced, e.g., by ferromagnetic resonance (FMR) excites the lattice in the attachednon-magnet. Inanalogywiththeelectroniccase wecallthiseffect“phononpumping” thataffectsthemag- netization dynamics. We consider only equilibrium mag- netizations that are normal or parallel to the interface, in which the pumped phonons are pure shear waves that carry angular momentum. We note that for general mag-arXiv:1804.07080v2 [cond-mat.mes-hall] 16 Jul 20182 netization directions both shear and pressure waves are emitted, however. We consider a magnetic film (metallic or insulating) that extends from z=dtoz= 0. It is subject to suffi- ciently high magnetic fields H0such that magnetization is uniform, i.e. M(r) =M:For in-plane magnetizations, H0> Ms, where the magnetization Msgoverns the de- magnetizing field [43]. The energy of the magnet|non- magnet bilayer can be written E=ET+Eel+EZ+ED+E0 K+Eme;(1) which are integrals over the energy densities "X(r). The different contributions are explained in the following. The kinetic energy density of the elastic motion reads "T(r) =( 1 2_u2(r); z> 0 1 2~_u2(r);d<z< 0; (2) and the elastic energy density [44] "el=( 1 2(P X(r))2+P X2 (r); z> 0 1 2~(P X(r))2+ ~P X2 (r);d<z< 0; (3) where;2fx;y;zg,andare the Lamé parameters andthe mass density of the non-magnet. The tilded parameters are those of the magnet. The strain tensor Xis defined in terms of the displacement fields u(r), X(r) =1 2@u(r) @r+@u(r) @r : (4) EZ=0VM
Hextis the Zeeman energy for Hext= H0+h(t), where h(t)is time-dependent. ED= 1 20VMTDMis the magnetostatic energy with shape- dependent demagnetization tensor DandVthe volume of the magnet. For a thin film with zaxis along the sur- face normal n0,Dzz= 1while the other components van- ish.E0 K=K1V(mn0)2is the uniaxial magnetocrys- talline anisotropy in the absence of lattice deformations, where m=M=MsandK1is the anisotropy constant. The magnetoelastic energy Emecouples the magnetiza- tion to the lattice, as discussed in the following. The magnetoelastic energy density can be expanded as "me(r) =1 M2sX ;M(r)M(r) [BX(r) +C (r)]:(5) For an isotropic medium the magnetoelastic constants Bread [45] B=Bk+ (1)B?: (6) Rotational deformations as expressed by the tensor (r) =1 2@u(r) @r@u(r) @r (7)are often disregarded [39–42, 46], but lead to a position dependence of the easy axis n(r)from the equilibrium value n0=ezand an anisotropy energy density [29, 47, 48] "K(r) =K1 M2s[Mn(r)]2: (8) To first order in the small deformation n(r) =n(r)n0=0 @ xz(r) yz(r) 01 A; (9) "K(r) ="0 K+ 2K1(n0mzm)
n(r):(10) From = it follows that (for non-chiral crystal structures) C=C. For the uniaxial anisotropy considered here Cxz=Cyz=K1. The magnetoelastic coupling due to the magnetocrystalline anisotropy thus contributes [47] "K me(r) =2K1 M2sMz(r) [Mx(r) xz(r) +My(r) yz(r)]: (11) PureYIGismagneticallyverysoft, sothemagnetoelastic constants are much larger than the anisotropy constant [49, 50] Bk= 3:48105Jm3; B?= 6:96105Jm3; K1=6:10102J m3; (12) but this ratio can be very different for other magnets. We find below that for the Kittel mode dynamics both coupling processes cannot be distinguished, even though they can characteristically affect the magnon-phonon coupling for finite wave numbers. The magnetization dynamics within the magnetic film is described by the Landau-Lifshitz-Gilbert (LLG) equa- tion [51, 52] _m= 0mHe+() m; (13) where is the gyromagnetic ratio, the effective mag- netic field which includes the magnetoelastic coupling He=rmE=(0VMs); (14) and the Gilbert damping torque [52] () m=m_m: (15) The equation of motion of the elastic continuum reads [44] u(r;t) =c2 t4u(r;t) + (c2 lc2 t)r[r
u(r;t)];(16) with longitudinal and transverse sound velocities cl=s + 2 ; ct=r ; (17)3 where elastic constants and mass density of non-magnet and magnet can differ. A uniform precession of the magnetization interacts with the lattice deformation at the surfaces of the mag- netic film [13, 14] and at defects in the bulk. The present theorythenholdswhenthethicknessofthemagneticfilm dp A, whereAis the cross section area. The Kittel mode induces lattice distortions that are uniform in the film planeu(r) =u(z)[14]. The elastic energy density is then affected by shear waves only: "el(z) =(  2 u02 x(z) +u02 y(z)
; z> 0 ~ 2 u02 x(z) +u02 y(z)
;d<z< 0;(18) whereu0 (z) =@u(z)=@z. The magnetic field Hext=hx(t); hy(t); H 0
Twith monochromatic drive hx;y(t) = Re hx;yei!t
and static component H0along thezaxis. At the FMR frequency !?=!H+!Awith !H= 0H0and!A= (2K1=Ms Ms). The equi- librium magnetization is perpendicular for !?>0. The magnetoelastic energy derived above then simplifies to Ez me=(B?K1)A MsX =x;yM[u(0)u(d)];(19) whichresultsinsurfaceshearforces F(0) =F(d) = (B?K1)Am, withF=FxiFy. These forces generate a stress or transverse momentum current in the zdirection (see Supplemental Material) j(z) =(z)u0 (z); (20) with(z) =forz >0and(z) = ~ford < z < 0, andu=uxiuy, which is related to the transverse mo- mentump(z) =( _ux(z)i_uy(z))by Newton’s equa- tion: _p(z) =@ @zj(z): (21) The boundary conditions require momentum conserva- tion and elastic continuity at the interfaces, j(d) = (B?K1)m;(22) j(0+)j(0) =(B?K1)m;(23) u(0+) =u(0): (24) We treat the magnetoelastic coupling as a small pertur- bation and therefore we approximate the magnetization mentering the above boundary conditions as indepen- dent of the lattice displacement u. The loss of angular momentum (see Supplemental Material) affects the mag- netization dynamics in the LLG equation in the form of a torque, which we derive from the magnetoelastic energy (19), _mjme=i!c d[u(0)u(d)] =i!cRe(v)m!cIm(v)m;(25)where!c= (B?K1)=Ms(for YIG:!c= 8:76 1011s1) andv= [u(0)u(d)]=(dm). We can distinguish an effective field Hme=!c 0Re(v)ez; (26) and a damping coefficient (?) me=!c !Imv: (27) The latter can be compared with the Gilbert damping constantthat enters the linearized equation of motion as _mj=i_m=!m: (28) With the ansatz u(z;t) =( Ceikzi!t; z> 0 Dei~kzi!t+Eei~kzi!t;d<z< 0; (29) we obtain v=Ms!c ! d~~ct2h cos(~kd)1i ict ~~ctsin(~kd) sin(~kd) +ict ~~ctcos(~kd);(30) and the damping coefficient for perpendicular magneti- zation (?) me=!c !2Ms d~~ctct ~~ct4 sin4~kd 2 sin2(~kd) + ct ~~ct2 cos2(~kd); (31) where!=ctk= ~ct~k. The oscillatory behavior of the damping(?) mecomes from the interference of the elastic waves that are generated at the top and bottom surfaces of the magnetic film. When they constructively (destruc- tively) interfere at the FMR frequency, the damping is enhanced(suppressed), becausethemagnon-phononcou- pling and phonon emission are large (small). Whenct~~ct(soft substrate) or when acoustic impedances are matched ( ct= ~~ct), damping at the resonance ~kd= (2n+ 1)withn2N0[14] simplifies to (?) me!!c !24Ms dct: (32) Whenct~~ct(hard substrate), the magnet is acousti- cally pinned at the interface and the acoustic resonances are at ~kd= (2n+ 1)=2[14] with (?) me!!c !2Ms d~~ctct ~~ct: (33) In contrast to Gilbert damping, (?) medepends on the frequency and vanishes in the limits !!0and!!1. Therefore, it does not obey the LLG phenomenology4 and in the non-linear regime does not simply enhance in Eq. (15). The magnetization damping 0in bulk magnetic insulators, on the other hand, is usually of the Gilbert type. It is caused by phonons as well, but not necessarily the magnetoelastic coupling. A theory of Gilbert damping [38] assumes a bottleneck process by sound wave attenuation, which appears realistic for magnets with high acoustic quality such as YIG. In the present phonon pumping model, energy and angular mo- mentum is lost by the emission of sound waves into an attached perfect phononwaveguide, so thepumpingpro- cess dominates. Such a scenario could also dominate the damping in magnets in which the magnetic quality is rel- atively higher than the acoustic one. When the field is rotated to Hext = hx(t); H 0; hz(t)
T, the equilibrium magnetization is in the in-plane ydirection and the magnetoelastic energy couples only to the strain uy, Ey me=(B?K1)A MsMz[uy(0)uy(d)]:(34) The FMR frequency for in-plane magnetization !k= !Hp 1!A=!Hwith!A< !H. The magnetoelastic coupling generates again only transverse sound waves. The linearized LLG equation including the phononic torques reads now _mx= (!H+!me)mz 0hz!Amz + (+me) _mz; (35) _mz=!Hmx+ 0hx_mx; (36) wheremeis given by Eq. (27) and !me= 0Hmewith effective field Hme=Hme
ezgiven by Eq. (26). Both Hmeandmecontribute only to _mx. The phonon pump- ing is always less efficient for the in-plane configuration: (k) me=1 1 + (!k=!H)2(?) me: (37) As an example, we insert parameters for a thin YIG film on a semi-infinite gadolinium gallium garnet (GGG) substrate at room temperature. We have chosen YIG because of its low intrinsic damping and high quality in- terface to the GGG substrate. Substantially larger mag- netoelastic coupling in other materials should be offset against generally larger bulk damping. For GGG, = 7:07103kg m3,cl= 6411 m s1, andct= 3568 m s1 [53]. For YIG, Ms= 1:4105A m1, = 1:76 1011s1T1,~= 5170 kg m3,~cl= 7209 m s1,~ct= 3843 m s1, and!c= 8:761011s1[49, 50]. The ratio of the acoustic impedances ~~ct=ct= 0:79. The damp- ing enhancement (?) meis shown in Fig. 2 over a range of FMR frequencies and film thicknesses. The FMR fre- quencies!?=!H+!Aand!k=!Hp 1!A=!Hfor the normal and in-plane configurations are tunable by the static magnetic field component H0via!H= 0H0. 0 200 400 600 800 1000 magnetic film thickness d[nm]0246810FMR frequency ν[GHz] <10−510−410−3 α(⊥) meFigure 2. Damping enhancement (?) meby phonon pumping in a YIG film on a semi-infinite GGG substrate, as given by Eq. (31). The damping enhancement peaks at acoustic resonance frequencies n~ct=(2d). The counter-intuitive result that the damping increases for thicker films can be un- derstood by the competition between the magnetoelastic effect that increases with thickness at the resonances and wins against the increase in total magnetization. How- ever, with increasing thickness the resonance frequencies decrease below a minimum value at which FMR can be excited. For a fixed FMR frequency me!0ford!1: For comparison, the Gilbert damping in nanometer thin YIG films is of the order 104[54] which is larger than corresponding values for single crystals. We con- cludethattheenhanceddampingisatleastpartlycaused by interaction with the substrate and not by a reduced crystal quality. The resonances in the figures are very broad because thect~~ctimplies very strong coupling of the discrete phonons in the thin magnetic layer with the phonon con- tinuum in the substrate. When an acoustic mismatch is introduced, the broad peaks increasingly sharpen, re- flecting the increased lifetime of the magnon polaron res- onances in the magnet. The frequency dependent effective magnetic field H(?) me is shown in Fig. 3. The frequency dependence of H(?) me implies a weak frequency dependence of the effective gy- romagnetic ratio (?) e= 1 + 0H(?) me !! : (38) In the limit of vanishing film thickness, 0H(?) me! (B?K1)2=(Ms~). We assumed that the non-magnet is an ideal phonon sink, which means that injected sound waves do not return. In the opposite limit in which the phonons cannot escape, i.e. when the substrate is a thin film with high acoustic quality, the additional damping van- ishes. This can be interpreted in terms of a phonon5 0 200 400 600 800 1000 magnetic film thickness d[nm]0246810FMR frequency ν[GHz] −60−45−30−15015304560 µ0H(⊥) me[µT] Figure3. Effectivefield H(?) megeneratedbythemagnetoelastic generation of phonons in a YIG film on a semi-infinite GGG substrate, as given by Eq. (26). accumulation that, when allowed to thermalize, gener- ates a phonon chemical potential and/or non-equilibrium temperature. The non-equilibrium thermodynamics of phonons in magnetic nanostructures is subject of our on- going research. The damping enhancement by phonons may be com- pared with that from electronic spin pumping [6–8], sp= ~ 4dMsh e2g; (39) which is inversely proportional to the thickness dof the magnetic film and does not depend on the FMR fre- quency, i.e. obeys the LLG phenomenology. Here, g is the spin mixing conductance per unit area at the in- terface. While phonons can be pumped into any elastic material, spin pumping requires an electrically conduct- ing contact. With a typical value of hg=e21018m2 the damping enhancement of YIG on platinum is sp 102nm=d:The physics is quite different, however, since sp;in contrast to me, does not require coherence over the interface. In conclusion, the pumping of phonons by magnetic anisotropy and magnetostriction causes a frequency- dependent contributions to the damping and effective field of the magnetization dynamics. The generation of phonons by magnetic precession can cause significant damping in a magnetic film when grown on an insulating, non-magnetic substrate and partly explains the increased dampinginvariablyobservedforthinnerfilms. Theimpli- cations of further reaching ramifications, such as phonon- induced dynamic exchange interactions, phonon accumu- lations and phonon spin Seebeck effect require additional research. This work is financially supported by the Nederlandse OrganisatievoorWetenschappelijkOnderzoek(NWO)as well as by the Grant-in-Aid for Scientific Research on In- novative Area, ”Nano Spin Conversion Science” (Grant No. 26103006). H. K. acknowledges support from theIran Science Elites Federation. We acknowledge use- ful discussions with Yaroslav Blanter, Rembert Duine, Akashdeep Kamra, Eiji Saitoh, and Sanchar Sharma. [1] S. Bader and S. Parkin, Annu. Rev. Condens. Matter Phys. 1, 71 (2010). [2] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. 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The angular mo- mentum density relative to the origin of an elastic body with displacement field u(r;t)and constant mass density reads l(t) =(r+u(r;t))_u(r;t): (S2) With uniaxial anisotropy axis along z, FMR generates the transverse elastic wave u(z;t) = Re2 40 @ux uy 01 Aeikzi!t3 5; (S3) with dispersion relation !=ctk. Defining the time av- erage hf(t)i= lim T!11 TZT 0dtf(t); (S4) hli= (0;0;hlzi)can be expressed as hlzi=hux_uy_uxuyi =! 4 ju+j2juj2 ; (S5) whereu=uxiuyand where we used Re(aei!t)Re(bei!t) =1 2Re (ab):(S6) The non-magnet harbors a constant phonon angular mo- mentum density in the z-direction, which implies pres- ence of a phonon angular momentum current Ahjz liat the interface to the magnet with area Athat is ab- sorbed at the phonon sink. In our model the angular momentum loss rate of the magnet by phonon pumping h_Szjmei=Ahjz liand D _Sz meE =hfSz;Ez megi=MsV(?) me! 4  jm+j2jmj2 (S7)wheref;gis the Poisson bracket and Ez methe magnetoe- lastic energy Eq. (19) in the main text. From Eq. (31) and u=Cm; (S8) with C=(B?K1)h cos(~kd)1i ikcos(~kd) +~k~sin(~kd); (S9) we obtain the relation hjz li=cthlzi; (S10) whichagreeswiththesimplephysicalpictureofanelastic wave carrying away its angular momentum density hlzi with transverse sound velocity ct. II. TRANSVERSE MOMENTUM For the transverse elastic wave (S3) in a magnet ex- tending from z=z0toz=z1(withz1> z0), the time derivative of the transverse momentum P=PxiPy reads _P=Z Vd3ru(z;t) =A u0 (z1;t)u0 (z0;t) :(S11) The change of momentum can be interpreted as a trans- verse momentum current density j(z0) =u0 (z0) flowing into the magnet at z0and a current j(z1) = u0 (z1)flowing out at z1. The momentum current is related to the transverse momentum density p(z) = _u(z)by _p(z) =@ @zj(z); (S12) which confirms that j(z;t) =u0 (z;t): (S13) The instantaneous conservation of transverse momentum isaboundaryconditionsattheinterface. Itstimeaverage hji= 0, but the associated angular momentum along z is finite, as shown above. III. SANDWICHED MAGNET When a non-magnetic material is attached at both sides of the magnet and elastic waves leave the magnet atz= 0andz=d, the boundary condition are j(d)j(d+) = (B?K1)m;(S14) j(0+)j(0) =(B?K1)m;(S15) u(0+) =u(0); (S16) u(d+) =u(d); (S17)2 withd=d0+. Since the Hamiltonian is piece-wise constant u(z;t) =8 >< >:Ceikzi!t; z> 0 Dei~kzi!t+Eei~kzi!t;d<z< 0 Feikzi!t: z< d; (S18) Using the boundary conditions v=u(0)u(d) dm=Ms!c ! d~~ct2 ict ~~ctcot(~kd 2);(S19) leading to the damping coefficient (?) me=!c !2Ms d~~ct2 ct ~~ct+~~ct ctcot2~kd 2:(S20) When ~~ct=ct, (?) me=!c !22Ms dctsin2 ~kd 2! ;(S21) which differs from the sin4 ~kd=2 dependence obtained for the magnet|non-magnet bilayer. This result can be explained by the phonon angular momentum leaking at two interfaces that should enhance the damping for thin magneticfilms. However, thephononpumpingisacoher- ent process that couples both interfaces, so the damping is not increased simply by a factor of 2 as in case of in- coherent spin pumping of a magnetic film sandwiched by metals. The position of the resonances, ~kd= (2n+1)=2 withn2N0, are independent of the ratio ct=~~ctwith me=!c !22Ms dct: (S22) (?) meand the effective magnetic field for YIG sandwiched between two infinitely thick GGG layers are shown in Figs. S1 and S2. IV. FLEXURAL (BENDING) WAVES IN THIN BEAMS In the main text we focus on the generation of trans- verseorlongitudinalsoundwaves. Infree-standingstruc- tured samples such as cantilevers, additional modes be- come important that can be excited by magnetization dynamics as well. This can be illustrated by a thin cylindrical elastic beam (see Fig. S3) with cross section areaA=r2, in which flexural waves are generated by the magnet of volume V=Adattached to the top of the beam. The elastic energy according to the Euler- Bernoulli beam theory [44] Eel=ZL 0dz1 2A_u2(z;t) +1 2EYI?u002(z;t) ;(S23) 0 200 400 600 800 1000 magnetic film thickness d[nm]0246810FMR frequency ν[GHz] <10−510−410−3 α(⊥) meFigure S1. Phonon pumping-enhanced (?) mein a YIG film sandwiched between two infinitely thick GGG layers. 0 200 400 600 800 1000 magnetic film thickness d[nm]0246810FMR frequency ν[GHz] −40−30−20−10010203040 µ0H(⊥) me[µT] Figure S2. Phonon pumping effective field H(?) mein a YIG film sandwiched between two infinitely thick GGG layers. leads to the equation of motion for the flexural waves [44] Au(z;t) +EYI?u(4) (z;t) = 0;(S24) whereI?=R dAx2=r4=4and elastic modulus EY= (3+ 2)=(+). The dispersion relation of the flex- ural waves is quadratic, !=s EYI? Ak2: (S25) When the dimensions of the magnet are much smaller than the wavelength of the elastic waves, the magne- toelastic coupling is suppressed and magnetization and lattice are coupled exclusively by the magnetocrystalline and, in contrast to the bulk magnet, also the shape anisotropies, EA=EK+ED; (S26) with EK=K1V(mn)2; (S27) ED=1 20VMTDM: (S28)3 m H z 0magnet non-magnet flexural wave Figure S3. Thin film magnet (shaded) with magnetization m attached to a thin semi-infinite elastic beam. For a thin magnetic film Dzz=D3= 1. When the magnet become very thick ( dr), i.e. a needle with its point forming the contact, and a coordinate system withzaxis along the surface normal n,Dxx=Dyy= D?=1=2. All otherDvanish. In contrast to the extended bilayer treated in the main text nis now a dynamic variable with n=u0 (0;t). The mechanical torque exerted by the magnet on the elastic beam reads =_L=VMs _m+_J; (S29) where _J=0VMsmHextis the torque exerted by the external magnetic field on the total angular momentum. For a magnet with equilibrium magnetization mkn0 =if(mn); (S30) wheref=VMs!A= and !A=( 2 K1=Ms 0Ms;thin film 2 K1=Ms+1 2 0Ms;needle:(S31) In order to compute the angular momentum current pumped into the attached non-magnet, _L=, we have to specify four boundary conditions. Two are pro- vided by the assumption that the beam is infinitely long so that the are no reflections. The absence of shear forces at the boundary is expressed by u000(0;t) = 0, while the bending by the torque follows from the principle of least action [55] u00 (0;t) =i EYI?: (S32)The general solution for the differential equation can be written u(z;t) =ei!t Aeikz+Bekz
;(S33) because there are no back-reflections in the semi-infinite beam. We find n=wm with w=2f EYI?k 1 +i2f EYI?k1 ;(S34) and the following source term in the LLG equation, _mjan=i!An =i!ARe(w)m!AIm(w)m;(S35) The first term on the right-hand-side is a field-like torque equivalent to the effective field 0Han=!A Re(w)ez; (S36) and the second one a damping-like torque with damping coefficient an=!A !Imw: (S37) Since for weak magnetoelastic coupling we expect an 1and thereforejwj  1, which is equivalent to 2f=(EYI?k)1, we may approximate wf(1i) EYI?k; (S38) anVMs!2 A !k I?EY; (S39) 0HanVMs!2 A 2EYI?kez: (S40) The damping enhancement scales as an/V A2!3 2: (S41) For a needle-shaped YIG magnet attached to GGG with EY= 2:51011Paand!A= 1:41010s1 an8:6106d=nm (=GHz)3 2(r=nm)2;(S42) 0jHanj3:1107d=nm (=GHz)1 2(r=nm)2T;(S43) which are very small numbers even at nanoscale dimen- sions.

2203.06312v1.Stability_for_nonlinear_wave_motions_damped_by_time_dependent_frictions.pdf

Stability for nonlinear wave motions damped by time-dependent frictions? Zhe Jiaoa, Yong Xua,b, Lijing Zhaoa, aDepartment of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China bMIIT key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China Abstract We are concerned with the dynamical behavior of solutions to semilinear wave systems with time- varying damping and nonconvex force potential. Our result shows that the dynamical behavior of solution is asymptotically stable without any bifurcation and chaos. And it is a sharp condition on the damping coecient for the solution to converge to some equilibrium. To illustrate our theoretical results, we provide some numerical simulations for dissipative sine-Gordon equation and dissipative Klein-Gordon equation. Keywords: dissipative wave system, nonautonomous damping, convergence to equilibria, decay rate 2010 MSC: 35L70, 35B35 1. Introduction In this paper, we consider the initial-boundary value problem for semilinear wave systems with time-varying dissipation: 8 < :utt
u+h(t)ut+f(u) = 0 ( t;x)2R+ ; u= 0 ( t;x)2R+@ ; (u(0;x);ut(0;x)) = (u0(x);u1(x))x2 ;(1) where is a bounded domain in Rd,d1, with smooth boundary @ , and the initial data u02H2( )\H1 0( ) andu12H1 0( ). The main purpose of the paper is to study the dynamical behavior of the solution of dissipative system (1) damped by time-dependent frictions. It is clear that the nonlinearity fand the coecient h(t) play signicant roles in the analysis. In particular, the set of equilibria associated to system (1) depends on the assumptions on the nonlinearity f. And whether h(t) is asymptotically vanishing or becoming larger too rapidly as time goes to innity, convergence to some equilibrium for the solution of system (1) may fail. Let us rst begin the introduction with a short survey in the literature. The dynamical behavior of solution to system (1) and its ODE version x(t) +h(t) _x(t) +r(x(t)) = 0; t0 with a potential  2W2;1 loc(Rd;R), which models the heavy ball with friction, have been studied by many authors under various assumptions on the damping and potential terms (cf., e.g., [1{6] and references therein). Recently, assuming the nonlinearity fis monotone and conservative, which is equivalent to the convexity of the potential, the authors in [7, 8] proved that the solution converges weakly to an equilibrium if the damping coecient behaves likeC tfor someC > 0 and2(0;1). However, in the case of nonconvex force potential, the question of convergence to an equilibrium for the solution of the system (1) is left open. The new estimates in this paper allow to solve ?This research was partially supported by the National Natural Science Foundation of China (11802236) and the Fundamental Research Funds for the Central Universities (310201911CX033). Corresponding author Email addresses: zjiao@nwpu.edu.cn (Zhe Jiao), hsux3@nwpu.edu.cn (Yong Xu), zhaolj@nwpu.edu.cn (Lijing Zhao) Preprint submitted to CNSNS March 15, 2022arXiv:2203.06312v1 [math.AP] 12 Mar 2022this question and to propose necessary and sucient conditions on the damping coecient for convergence, see main theorem and remarks below. Before giving the detailed statement of our main theorem, we make assumptions on the non- linearityfand the damping coecient h: (I-1)f2W1;1 loc(R) satises lim inf jsj!+1f(s) s1; (2) where the constant 1<  0, and0is the rst eigenvalue of
in with zero Dirichlet boundary condition; and that jf0(s)jC(1 +jsjp); s2R; (3) whereC > 0 andp0, with (d2)p<2, are constants. (I-2) Put G(v) :=
v+f(v); E 0(v) :=1 2krvk2 L2+Z F(v)dx; whereF(s) :=Rs 0f(t)dt, and S:=f 2H2( )\H1 0( ) :G( ) = 0g (the set of equilibria). There exists a number 2(0;1 2] such that for each 2S, kG(v)kH1c jE0(v)E0( )j1; (4) wheneverv2H1 0( ),kv kH1 ; here,c and are constants depending on . (I-3) The damping coecient h(t)2W1;1 loc(R+) is a nonnegative function, and there exist con- stantsc;C > 0, and 2[0;(1)1) (5) such that c (t+ 1)h(t)C (t+ 1);8t0; (6) or cth(t)Ct;8t0: (7) In [9], the authors proved the convergence result under the conditions (I-1), (I-2), (I-3) without (7), and the following assumption (I-4). (I-4) For any a>0, inf t>0Zt+a th(s)ds> 0: Condition (I-4) is a technical assumption, which is only used to show decay to zero of utinL2 and then the precompactness of the trajectories of system (1). However, condition (I-4) implies thathdoes not tend to 0 at innity. Stimulated by all the work above, the major contribution of this paper is to present an eective method to prove the convergence to equilibrium of solutions of system (1) without the assumption (I-4). More precisely, we will prove the main theorem as follows. Theorem 1. Assume Conditions (I-1), (I-2) and (I-3). Let u2W1;1 loc(R+;H1 0)\W2;1 loc(R+;L2) be a solution of (1). Then f(0; ) : 2Sgis the attracting set for system (1), that is, lim t!+1(kut(t;
)kL2+ku(t;
) kH1) = 0: Moreover, there exist positive constants c, andCsuch that ku kL2 c(1 +t); 2(0;1 2); Cexp(ct1); =1 2; where2(0;(1) 12). 2Remark 1. It follows from [10] that (I-2) holds true if fis analytic. For example, the analytic functionf(u) =bsinu,bsome positive constant, with Lojasiewicz exponent =1 2. Then equation (1) is a damped sine-Gordon equation. As we know in [11], (I-2) is also suitable for some non-analytic functions, for instance, f(u) =au+jujp1u; p> 1: And ifa>0, then (I-2) is satised with the Lojasiewicz exponent =1 2; otherwise, (I-2) holds with=1 p+1. Thus, equation (1) becomes a damped nonlinear Klein-Gordon equation. Remark 2. The condition (I-3) on the damping coecient is optimal, which prevent the damping term from being either too small, or too large as t!+1. From Theorem 1, we obtain the convergence results for equation (1) with a small damping coecienth(t)1 (t+1), asymptotically vanishing, or a large damping coecient h(t)t, 2[0;(1)1). Here, the notation means that the coecient grows like a polynomial function. As forh(t) = (t+ 1),jj>1, Theorem 1 do not apply, and the solution u(t;x) may oscillate or approach to some functions (not an equilibrium) as time goes to innity. Indeed, if > 1, there exist oscillating solution that do not approach zero as t!1 . For example, we consider the problem 8 < :utt
u+1 (t+1)ut+bu= 0 (t;x)2R+ ; u= 0 ( t;x)2R+@ ; (u(0;x);ut(0;x)) = (u0(x);u1(x))x2 ; whereis an eigenvalue of
+b, having the corresponding eigenfunction (x). Takingu(t;x) = !(t) (x), we have !tt+1 (t+ 1)!t+!= 0: When<1, solutions again do not in general approach zero, though their behavior is quite dierent from the case <1. Note that an interesting solution u(t;x) = (1(t+ 1)1+ 1 +) (x) solves utt
u+h(t)ut+bu= 0 (t;x)2R+ ; u= 0 ( t;x)2R+@ ; where the damping coecient is as follows h(t) =t 1 +tt1; and the initial data (u(0;x);ut(0;x)) = (1 + 1 + (x); 1 + (x)); x2 : But it is easy to see that u(t;x) approaches to (x), which is not an equilibrium, as t!1 . The plan of this paper is as follows. In Section 2, we make some estimates of solutions. A key lemma in this section is to prove a generalized type of Lojasiewicz-Simon inequality, which is rstly given in the literature. In Section 3, we prove our main results. And we will give some numerical analysis to illustrate our results in Section 4. Throughout the paper, c,c1,c2


,C,C1,C2


denote corresponding constants. 32. Preliminary By the semigroup theory (e.g., [12], section 6.1) and regard h(t)ut+f(u) as a perturbation term), we know that under Condition (3) and for u02H2( )\H1 0( ) andu12H1 0( ), Problem (1) has a unique solution u(t;x)2W1;1 loc(R+;H1 0( ))\W2;1 loc(R+;L2( )): The solution energy is dened by Eu(t) :=1 2(kutk2 L2+kruk2 L2) +Z F(u)dx: Notice that Eu(
) is non-increasing by E0 u(t) =h(t)kutk2 L20. Lemma 2. Letu2W1;1 loc(R+;H1 0)TW2;1 loc(R+;L2)be a solution to (1). There is a positive constantcdepending only on the norm of initial data in the energy space H1 0( )L2( ), such that kutkL2+kukH1+kf(u)kL2cfort0: (8) And the functional E0(u)has at least a minimizer v2H1( ). Proof. For every>0, it follows from (2) that there exists M()1 such that forjsj>M () f(s) s(1+); and so F(s)1+ 2s2: Then we haveZ F(u)dx=Z juj>M()F(u)dx+Z jujM()F(u)dx 1+ 2Z juj2dx+j jsup jujM()jF(s)j; which implies that Eu(t)1 2kutk2 L2+1 2 11+ 0
kruk2 L2+j jsup jujM()jF(s)j: (9) SinceEu(
) is non-increasing, then by using the Poincar e inequality, it implies from (9) that kutk2 L2+kuk2 H1c1; wherec1depends on the norm of initial data in the energy space. Accordingly, kf(u)kL2is also bounded by (I-1). Then (8) is proved. And also we know from (9) that E0(u) is bounded below. Thus, there exists a minimizing sequenceun2H1( ) such that E0(un) = infu2H1E0(u). By (3) H older inequality and Sobolev inequalities, we have kf(un)f(v)k2 L2c2Z (junj2p+jvj2p+ 1)junvj2dx c3k(junj2p+jvj2p+ 1)k L3p 1qkjunvj2k L6 1+2q c4(kunk2p H1q+kvk2p H1q+ 1)kunvk2 H1q withq=2p 2(1+p)2(0;1). From (8) and the Aubin's compactness theorem, we know that unis relatively compact in L1([0;T];H1q( )). Then there is a subsequence, denoted still by un, such thatun!vinH1q, andv2H1. It implies jZ F(un)F(v)dxj=jZ [Z1 0f(sun)unf(sv)vds]dxj =jZ [Z1 0f(sun)unf(sun)v+f(sun)vf(sv)vds]dxj c5fkunvkL2+kf(sun)f(sv)kL2g!0 4asngoes to innity. Because kukH1is weakly lower semicontinuous, we obtain E0(v) = infu2H1E0(u), that is,vis a minimizer of the functional E0(u).  Remark 3. From this Lemma, we know the following static problem associated to system (1) 
+f( ) = 0x2 ; = 0 x2@ admits at least a classical solution, which means that the set of equilibria is nonempty. The following inequality is a generalized type of Simon-Lojasiewicz inequality. Lemma 3. Assume that u2W1;1 loc(R+;H1 0)TW2;1 loc(R+;L2)is a solution to (1) and 2S. There exist constants c>0,T >0,2(0;1 2]and >0depending on such that for kG(u)kH1cjE0(u)E0( )j1 providedku kH1q( )<,q=2p 2(1+p). Proof. Ifku kH1( ), then forz=u , we have 
z=utt+h(t)ut+f(u)f( )x2 ; z= 0 x2@ From the regularity theory for elliptic problem we know kzkH1c1k
zkH1; (10) wherec1is a constant independent of u. From (3), we have kf(u)f( )kL2c2 kukp H1q+k kp H1q+ 1
ku kH1q: There exists ~ >0 depending on such that for any v,ku kH1q<~, kf(u)f( )kL2< 2c1: (11) Then it infers from (10), (11) and Lemma 2 that k
u+f(u)kH1=k
z+f(u)f( )kH1 k
zkH1kf(u)f( )kH1 1 c1kzkH1kf(u)f( )kH1 1 2c1kzkH1 c3jE0(u)E0( )j1: For the other case kv kH1( )<, it follows from (I-2) that the estimate in this Lemma holds. Thus, our proof is completed.  3. Main Result In this section, we give the proof of Theorem 1. Proof. The subsequent proof consists of several steps. Step 1. From (8) and the Aubin's compactness theorem, we know that uis relatively compact in L1([0;T];H1q( )). It follows that there exist a sequence tn!+1and 2H1qsuch that ku(tn;
) kH1q!0; n!+1: (12) Let2(0;1), we dene H(t) =Eu(t)E0( ) + (t+ 1)hG(u);uti; 5whereh
;
iisH1inner product. Then we have H0(t) =h(t)kutk2 L2 (t+ 1)+1hG(u);uti+ (t+ 1)hG0(u)ut;uti  (t+ 1)kG(u)kH1 (t+ 1)hG(u);h(t)uti: Ifh(t) satises (6), then we obtain H0(t)c (t+ 1)kutk2 L2 (t+ 1)kG(u)kH1+ (t+ 1)hG0(u)ut;uti + ( (t+ 1)+1+C (t+ 1)2)jhG(u);utij: And whenh(t) satises (7), then we obtain H0(t)ctkutk2 L2 (t+ 1)kG(u)kH1+ (t+ 1)hG0(u)ut;uti + ( (t+ 1)+1+Ct (t+ 1))jhG(u);utij ctkutk2 L2 (t+ 1)kG(u)kH1+ (t+ 1)hG0(u)ut;uti + ( (t+ 1)+1+C)jhG(u);utij: Since we know jhG(u);utijkG(u)kH1kutkH11 2kG(u)k2 H1+1 2kutk2 L2; or jhG(u);utijkG(u)kH1kutkH1C1 2(1 +t)kG(u)k2 H1+t 2C2kutk2 L2; moreover, k(
)1(f0(u))utkC3(1 +kukH1)kutkL2 by (3), and so jhG0(u)ut;utijC4kutk2 L2; then it implies from choosing small enough that H0(t)C5 (t+ 1)(kutk2 L2+kG(u)k2 H1) (13) regardless of whether h(t) satises (6) or (7). And (13) implies that H(t) is non-increasing. It follows that H(t) has a nite limit as time goes to + 1. Step 2. Due to the Lemma 2, uis weakly compact in H1. And then we note that kG(u(t;
))G(u(s;
))kH1 = sup 2ZfjhG(u(t;
))G(u(s;
));ijg sup 2Zfjh
u(t;
)
u(s;
);ijg+ sup 2Zfjhf(u(t;
))f(u(s;
));ijg !0; t!1: whereZ=f2H1 0:kkH1= 1g. Then we have kG(u)Zt+1 tG(u)dskH1Zt+1 tkG(u(t;
))G(u(s;
))kH1ds !0; t!+1:(14) BecausekutkH1is uniformly continuous, we deduce that kut(t+ 1;
)ut(t;
)kH1!0; t!+1: (15) 6Ifh(t) satises (6), we have kZt+1 th(s)usdskH1kZt+1 th(s)usdskL2Zt+1 th(s)kuskL2ds (Zt+1 th(s)ds)1 2(Zt+1 th(s)kusk2 L2ds)1 2 C6(Zt+1 th(s)kusk2 L2ds)1 2 C6(E(t)E(t+ 1))1 2!0; t!+1:(16) LetJ(t) =kutk2 H1 (t+1). We know d dtJ(t) = t+ 1J(t) +2 (t+ 1)hut;utti = t+ 1J(t)2h(t)J(t) +2 (t+ 1)hut;G(u)i (2h(t)1 + t+ 1)J(t) +1 (t+ 1)kG(u)k2 H1 (2h(t)1 + t+ 1)J(t)1 C5H0(t) by (13). Note that sup t0Zt+1 tH0(s)ds= 0; and fort >1, 2h(t)1 + t+1>0 if h(t) satises (7). Then it follows from a Grownwall type inequality (see Lemma 2.2 in [13]) that kutk2 H1C7(t+ 1)et; t> 1: Furthermore, h(t) satises (7), we have kZt+1 th(s)usdskH1kZt+1 th(s)usdskL2Zt+1 th(s)kuskL2ds !0; t!+1:(17) Now we examine the term kG(u)kH1. Since we know by (1) that G(u) =G(u)Zt+1 tG(u)dsZt+1 t(uss+h(s)us)ds =G(u)Zt+1 tG(u)ds(ut(t+ 1;
)ut(t;
))Zt+1 th(s)usds then we have kG(u)kH1kG(u)Zt+1 tG(u)dskH1+kut(t+ 1;
)ut(t;
)kH1 +kZt+1 th(s)usdskH1: Thanks to (14), (15), (16) and (17), we have lim t!1kG(u)kH1= 0 Therefore, we know 2H1and
+f( ) =G( ) = lim n!+1G(u(tn;
)) = 0; 7which means that is an equilibrium. Step 3. For any 0<< , there exists an integer Nsuch that for any nN [H(tn)](1 )[H(t)](1 ) 2; t>tn0; (18) whereis the constant in Lemma 3. Dene ^tn= supft>tn:ku(s;
) (
)kH1q<;8s2[tn;t]g: Due to 2(12)0 and the uniform boundedness in Lemma 2, we deduce from Lemma 3 that fort2[tn;^tn),n>N , kutk2 L2+kG(u)k2 H1 kutk2 L2+1 2kG(u)k2 H1+1 2jE0(u)E0( )j2(1) kutk2(12) L2 C8kutk2 L2+kG(u)k2(12) H1 C9kG(u)k2 H1d+1 2jE0(u)E0( )j2(1) C10fkutk4(1) L2 +kG(u)k4(1) H1+jE0(u)E0( )j2(1)g C11fkutk4(1) L2 +kutk2(1) L2kG(u)k2(1) H1+jE0(u)E0( )j2(1)g C12H(t)2(1):(19) Then we deduce from (13) that for t2[tn;^tn),n>N , H0(t)C13 (t+ 1)H(t)2(1); so that H(t) C14(1 +t)1 12; 2(0;1 2); C15exp(C11t1)=1 2:(20) Taking 2(0;1), from (13) and (19) we also have d dt[H(t)(1 )] =(1 )H0(t)[H(t)](1 )1 C16(1 )(kutkL2+kG(u)kH1)2H(t)(1 )1 C17 (t+ 1)H(t) (kutkL2+kG(u)kH1) fort2[tn;^tn),n>N . Note that (1)(12)1 > . By (20), we obtain C17 (t+ 1)H(t) !+1; t!+1: Therefore, kutkL2C18d dt[H(t)(1 )] (21) fort2[tn;^tn),nlarge enough. Then we see that sup t2[tn;^tn)ku(t;
) kL2ku(tn;
) kL2+C19jZ^tn tnd dt[H(t)(1 )]dtj; which implies that ^tn= +1whennis large enough. Then we assert that u(t;
) converges to in L2astgoes to +1. Sinceuis relatively compact in H1q( ) (see in Lemma 2), then we have ku(t;
) kH1q!0; t!+1: (22) 8From (22), we can deduce jZ F(u)F( )dxj!0; t!+1: Due to lim t!1H(t) = lim t!1fkut(t;
)kL2+E0(v)E0( )g= 0; then we knowf(0; ) : 2Sgis the attracting set for system (1), that is, lim t!+1(kut(t;
)kL2+ku(t;
) kH1) = 0: Step 4. From (21) and (22), it infers that kv(t;
) kL2Z1 tkvtkL2dsC20[H(t)](1 ); which implies the required speed estimates from (20).  4. Numerical simulation In this section, we present several examples to illustrate the evolution of the solutions to the system (1). We will show how the role of damping coecient h(t) aects the dynamical behaviors of the solution. We consider the following one-dimensional equations 8 < :utt+h(t)utuxx+G(u) = 0 (t;x)2(0;T)(L;L) u(t;L) =u(t;L) = 0 t2(0;T) (u(0;x);ut(0;x) = (f(x);g(x))x2(L;L)(23) with the initial condition f(x) = 0; g(x) =4p 1c2 cosh(xp 1c2): Takeh(t) =hi(t),i= 0, 1, ..., 6 with h0= 0; h 1= 1; h 2=1pt+ 1; h 3=1 t+ 1; h4=p t; h 5=t; h 6=t3 2; also, we consider two cases of the function G(u): one is G(u) =G1(u) =au+jujp1u; p1; and the other is G(u) =G2(u) =bsinu; b> 0: The system 23 with the nonlinear term G1(u) is so-called dissipative Klein-Gordon equation, and withG2(u) is dissipative sine-Gordon equation. Here, the Lojasiewicz exponent for sine-Gordon equation is1 2. And ifa>2 L2, which is the rst eigenvalue for one-dimensional Laplace operator with Dirichlet boundary condition, then the Lojasiewicz exponent for Klein-Gordon equation is1 2. Ifa <2 L2, the Lojasiewicz exponent is1 p+1. We will use central dierences for both time and space derivatives and the following parameters set: Space interval L= 20 Space discretization
x= 0:1 Time discretization
t= 0:05 Amount of time steps T= 200 Velocity of initial wave c= 0:2 9As for the non-damping case h(t) =h0= 0, the numerical results can be seen in gure 1 (a) and (b). As can be seen easily, The solutions to these two equations does not decay as time goes to innity. This is because the systems are conservative, that is, the energy of these two systems Eu(t) keep to be some constant Eu(0) depended fully on the initial data. If the systems are damped by time-dependent damping, the numerical results can be seen in gure 2 (a) and (b). It is clearly that the damping coecients aect the dynamical behaviors of the solutions to these two systems. From the gures, we also see that L2-norms of the solutions to these two equations converge to some equilibrium, when h(t) =hi(t),i= 1, 2, 4, 5. For h(t) =h3=1 t+1,L2-norms of the solutions keep on oscillating as time goes to innity. And if h(t) =h6=t3 2,L2-norms of the solutions also converge to some constant, but which is not the value of some equilibrium. These numerical results are in accordance with the conclusion given in Theorem 1. How the Lojasiewicz exponent aect the convergence speeds of the solutions to these two equations? In Theorem 1, whether the convergence for the systems is exponential or polynomial depends on the Lojasiewicz exponent , but the convergence rate is in terms of the damping coecienth(t). These can be seen from gure 3 (a) and (b). Remark 4. Whenh(t)t, whether convergence results hold in unknown theoretically. From the numerical simulation as in gure 2 or 3, we know the solution converges to some equilibrium in L2-norm. Authors contributions All the authors contributed equally and signicantly in writing this paper. All authors read and approved the nal manuscript. Declaration of Competing Interest The authors declare that they have no competing interests. Acknowledgements The authors thank the anonymous referees very much for the helpful suggestions. References [1] P. Pucci, J. Serrin, Asymptotic stability for nonautonomous dissipative wave systems, Com- munications on Pure and Applied Mathematics 49 (1996) 177{216. [2] A. Cabot, H. Engler, S. Gadat, On the long time behavior of second order dierential equations with asymptotically small dissipation, Transactions of the American Mathematical Society 361 (2009) 5983{6017. [3] H. Attouch, X. Goudou, P. Reont, The heavy ball with friction method. i: The continuous dynamical system, Commun. Contemp. Math. 2 (2000) 1{34. [4] M. Daoulatli, Rates of decay for the wave systems with time dependent damping, Discrete and Continuous Dynamical Systems-Series A 31 (2011) 407{443. [5] A. Haraux, M. A. Jendoubi, Asymptotics for a second order dierential equation with a linear, slowly time-decaying damping term, Evolution Equations and Control Theory 2 (2013) 461{ 470. [6] Y. Gao, J. Liang, T. Xiao, A new method to obtain uniform decay rates for multidimensional wave equations with nonlinear acoustic boundary conditions, SIAM Journal of Control and Optimization 56 (2018) 1303{1320. [7] A. Cabot, P. Frankel, Asymptotics for some semilinear hyperbolic equations with non- autonomous damping, Journal of Dierential Equations 252 (2012) 294{322. 10[8] R. May, Long time behavior for a semilinear hyperbolic equation with asymptotics vanishing damping term and convex potential, Journal of Mathematical Analysis and Applications 430 (2015) 410{416. [9] Z. Jiao, X.-T. Xiao, Convergence and speed estimates for semilinear wave systems with nonautonomous damping, Mathematical Methods in the Applied Sciences 39 (2016) 5456{ 5474. [10] A. Haraux, M. A. Jendoubi, The Convergence problem for Dissipative Autonomous Systems, Springer, 2013. [11] R. Chill, On the lojasiewicz-simon gradient inequality, Journal of Functional Analysis 201 (2003) 572{601. [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Dierential Equations, Springer-Verlag: New York, 1983. [13] M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal. 3 (2004) 849{881. 11(a) sine-Gordon equation (b) Klein-Gordon equaiton Figure 1: Dynamical behaviors for two equations without damping. (a) h(t) =h0,b= 1 The tendency of the solution to sine-Gordon equation. (b) h(t) =h0,a= 1, p= 3 The tendency of the solution to Klein-Gordon equation. 120 50 100 150 200 time0510L2 normh1 0 50 100 150 200 time0510L2 normh2 0 50 100 150 200 time0510L2 normh3 0 50 100 150 200 time0510L2 normh4 0 50 100 150 200 time0510L2 normh5 0 50 100 150 200 time010L2 normh6(a) sine-Gordon equation 0 50 100 150 200 time05L2 normh1 0 50 100 150 200 time05L2 normh2 0 50 100 150 200 time0510L2 normh3 0 50 100 150 200 time0510L2 normh4 0 50 100 150 200 time0510L2 normh5 0 50 100 150 200 time0510L2 normh6 (b) Klein-Gordon equaiton Figure 2: Dynamical behaviors for two equations with time-dependent damping. (a) h(t) =hi,b= 1 The tendency ofL2-norms of the solution to sine-Gordon equation. (b) h(t) =hi,a= 1,p= 3 The tendency of L2-norms of the solution to Klein-Gordon equation. 130 10 20 30 40 50 60 70 80 90 100012345678910 a=0 a=-0.01 a=-0.03 a=-0.04 a=-0.1(a)h(t) =h2,p= 3 0 20 40 60 80 100 time01020L2 normh1 0 20 40 60 80 100 time01020L2 normh2 0 20 40 60 80 100 time01020L2 normh3 0 20 40 60 80 100 time01020L2 normh4 0 20 40 60 80 100 time01020L2 normh5 0 20 40 60 80 100 time01020L2 normh6 (b)h(t) =hi,a=0:1,p= 3 Figure 3: Convergence rates for dissipative Klein-Gordon equation. 14

1902.08700v1.Strongly_Enhanced_Gilbert_Damping_in_3d_Transition_Metal_Ferromagnet_Monolayers_in_Contact_with_Topological_Insulator_Bi2Se3.pdf

1 Strongly Enhanced Gilbert Damping in 3 d Transition Metal Ferromagnet Monolayers in Contact with Topological Insulator Bi 2Se3 Y. S. Hou1, and R. Q. Wu1 1 Department of Physics and Astronomy, University of California, Irvine, California 92697 -4575, USA Abstract Engineering Gilbert damping of ferromagnetic metal films is of great importance to exploit and design spintronic devices that are operated with an ultrahigh speed. Based on scattering theory of Gilbert damping, we extend the torque method originally used in studies of magnetocrystalline anisotropy to theoretically determine Gilbert dampings of ferromagnetic metals. This method is utilized to investigate Gilbert dampings of 3 d transition metal ferromagnet iron, cobalt and nickel monolayers that are co ntacted by the prototypical topological insulator Bi 2Se3. Amazingly, we find that their Gilbert dampings are strongly enhanced by about one order in magnitude, compared with dampings of their bulks and free -standing monolayers, owing to the strong spin -orbit coupling of Bi 2Se3. Our work provides an attractive route to tailoring Gilbert damping of ferromagnetic metallic films by putting them in contact with topological insulators. Email: wur@uci.edu 2 I. INTRODUCTION In ferromagnets, the time -evolution of their magnetization M can be described by the Landau -Lifshitz -Gilbert (LLG) equation [1-3] 1Meff Sdd dt dt MMM H M , where 0B g is the gyromagnetic ratio, and MSM is the saturation magnetization. The first term describes the precession motion of magnetization M about the effective magnetic field, Heff, which includes contributions from external field, magnetic anisotropy, exchange, dipole -dipole and Dzyaloshinskii -Moriya interactions [3]. The second term represents the decay of magnetization prece ssion with a dimensionless parameter , known as the Gilbert damping [4-8]. Gilbert damping is known to be important for the performance of various spintronic devices such as hard drives, magnetic random access memories, spin filters, and magnetic sensors [3, 9, 10]. For example, Gilbert damping in the free layer of reader head in a magnetic hard drive determines its response speed and signal -to-noise ratio [11, 12]. The bandwidth, insertion loss , and response time of a magnetic thin film microwave device also critically depend on the value of in the film [13]. The rapid developm ent of spintronic technologies calls for the ability of tuning Gilbert damping in a wide range. Several approaches have been proposed for the engineering of Gilbert damping in ferromagnetic (FM) thin films, by using non -magnetic or rare earth dopants, addi ng differ ent seed layers for growth, or adjusting composition ratios in the case of alloy films [9, 14-16]. In par ticular, tuning via contact with other materials such as heavy metals, topological insulators (TIs), van der Waals monolayers or magnetic insulators is promising as the selection of material combinations is essentially unlimited. Some of these materials may have fundamentally different damping mechanism and offer opportunity for studies of new phenomena such as spin -orbit torque, spin -charge conversion, and thermal -spin-behavior [17, 18]. In this work, we systematically investigate the effect of Bi 2Se3 (BS), a prototypical TI, on the Gilbert damping of 3d transitio n metal (TM) Fe, Co and Ni monolayers (MLs) as they 3 are in contacted with each other. We find that the Gilbert dampings in the TM/TI combinations are enhanced by about an order of magnitude than their counterparts in bulk Fe, Co and Ni as well as in the fr ee-standing TM MLs. This drastic enhancement can be attributed to the strong spin -orbit coupling (SOC) of the TI substrate and might also be related to its topological nature . Our work introduces an appealing way to engineer Gilbert dampings of FM metal fi lms by using the peculiar physical properties of TIs. II. COMPUTATIONAL DETAILS Our density functional theory (DFT) calculations are carried out using the Vienna Ab- initio Simulation Package (VASP) at the level of the generalized gradien t approximation [19-22]. We treat Bi -6s6p, Se -4s4p, Fe -3d4s, Co -3d4s and Ni -3d4s as valence electrons and employ the projector -augmented wave pseudopotentials to d escribe core -valence interactions [23, 24]. The energy cutoff of plane -wave expansion is 450 eV [22]. The BS substrate is simulated by five quintuple layers ( QLs), with an in -plane lattice constant of aBS = 4.164 Å and a vacuum space of 13 Å between slabs along the normal axis. For the computational convenience, we put Fe, Co and Ni MLs on both sides of the BS slab. For the structural optimization of the BS/TM slabs, a 6× 6× 1 Gamma -centered k -point grid is used, and the positions of all atoms except those of the three central BS QLs are fully relaxed with a criterion that the force on each atom is less than 0.01 eV/Å. The van der Waals (vdW) correction in the form of the nonlocal vdW functional (optB86b -vdW) [25, 26] is included in all calculations. The Gilbert dampings are determined by extending the torque method that we developed for the study of magnetocrystalline anisotropy [27, 28]. To ensure the numerical convergence, we use very dense Gamma -centered k -point grids and, furthermore, large numbers of unoccupied bands. For example, the first Bri llouin zone of BS/Fe is sampled by a 37× 37× 1 Gamma -centered k -point grid, and the number of bands for the second - variation step is set to 396, twice of the number (188) of the total valence electrons. More computational details are given in Appendix A. Mag netocrystalline anisotropy energies are determined by computing total energies with different magnetic orientations [29]. 4 III. TORQUE METHOD OF DETERMINING GILBERT DAMPING According to the scattering theory of Gilbert damping [30, 31], the energy dissipation rate of the electroni c system with a Hamiltonian, H(t), is determined by dis 2i j j i F i F j ijE E E E Euu HHuu . Here, EF is the Fermi level and u is the deviation of normalized magnetic moment away from its equilibrium, i.e., 0m m u with 00 MsM m . On the other hand, the time derivative of the magnetic energy in the LLG equation is [32] mag 3S effM dEdt MH mm . By taking dis magEE , one obtains the Gilbert damping as: 4i j j i F i F j ij SE E E EM u u HH . Note that, to obtain Eq. (4), we use mu since the eq uilibrium normalized magnetization m0 is a constant. In practical numerical calculations, FEE is typically substituted by the Lorentzian function 22 0 0.5 0.5 L . The half maximum parameter, 1 , is adjusted to reflect different scattering rates of electron -hole pairs created by the precession of magnetization M [10]. This procedure has been already used in several ab initio calculations for Gilbert dampings of metallic systems [8, 9, 32-35], where the electronic responses play the major role for energy dissipation . In this work, we focus on the primary Gilbert damping in FM metals that arises from SOC [10, 36-38]. There are two important effects in a uniform precession of magnetization M, when SOC is taken into consideration. The first is the F ermi surface breathing as M rotates, i.e., some occupied states shift to above the Fermi level and some unoccupied states shift to below the Fermi level. The second is the transition between different states across the Fermi level as the precession can be viewed as a perturbation to 5 the system. These two effects generate electron -hole pairs near the Fermi level and their relaxation through lattice scattering leads to the Gilbert damping. Now we demonstrate how to obtain the Gilbert damping due to SOC thro ugh extending our previous torque method [27]. The general Hamiltonian in Eq. (4) can be replaced by SOC j j jr H l s [4, 27] where the index j refers to atoms, and jji lr and s are orbital and spin operators, respectively. This is in the same spirit for the determination of the magnetocrystal line anisotropy [27], for which our torque meth od is recognized as a powerful tool in the framework of spin -density theory [27]. When m points at the direction of , , ,x y zm m m n , the term ls in HSOC is written as follows: 22 2211cos sin sin22 1sin sin cos 52 2 2 1sin cos sin2 2 2ii z ii z ii zs l l e l e s l l e l e s l l e l e n ls To obtain the derivatives of H in Eq. (4), we assume that the magnitude of M is constant as its direction changes [36]. The processes of getting angular derivatives of H are straightforward and the results are given by Eq. (A1) -(A5) in Appendix B. IV. RESULTS AND DISCUSSION In this section, we first show that our approach of determining Gilbert damping works well for FM metals such as 3d TM Fe, Co and Ni bulks. Following that, we demonstrate the strongly enhanced Gilbert dampings of Fe, Co and Ni MLs due to the contact with BS and then discuss the underlying physical mechanism of these enhancements. A. Gilbe rt dampings of 3d TM Fe, Co and Ni bulks Gilbert dampings of 3d TM bcc Fe, hcp Co and fcc Ni bulks calculated by means of our extended torque method are consistent with previous theoretical results [10]. As shown in Fig. 1, the intraband contributions decrease whereas the interband contributions increase as the scattering rate increases. The minimum values of have the same magnitude 6 as those in Ref. [10] for all three metals, showing the applicability of our approach for the determination of Gilbert dampings of FM metals. Figure 1 (color online) Gilbert dampings of (a) bcc Fe, (b) hcp Co and (c) fcc Ni bulks. Black curves give the total Gilbert damping. Red and blue curves give the intraband and interband contributi ons to the total Gilbert damping, respectively. B. Strongly enhanced Gilbert dampings of Fe, Co and Ni MLs in contact with BS We now investigate the magnetic properties of heterostructures of BS and Fe, Co and Ni MLs. BS/Fe is taken as an example and its atom arrangement is shown in Fig. 2a. From the spatial distribution of charge density difference BS+Fe-ML BS Fe-ML in Fig. 2b, we see that there is fairly obvious charge transfer between Fe and the topmost Se atoms. By taking the average of in the xy plane, we find that charge transfer mainly takes place near the interface (Fig.2c). Furthermore, the charge transfer induces non - negligible magnetization in the topmost QL of BS (Fig. 2b). Similar charge transfers and induced magnetization are also found in BS/Co and BS/Ni (Fig. A1 and Fig. A2 in 7 Appendix C). These suggest that interfacial interactions between BS and 3 d TMs are very strong. Note that BS/Fe and BS/Co have in -plane easy axes whereas the BS/ Ni has an out-of-plane one. Figure 2 (color online) (a) Top view of atom arrangement in BS/Fe. (b) Charge density difference near the interface in BS/Fe. Numbers give the induced magnetic moments (in units of B ) in the top most QL BS. Color bar indicates the weight of negative (blue) and positive (red) charge density differences. (c) Planer -averaged charge density difference in BS/Fe. In (a), (b), (c), dark green, light gra y and red balls represent Fe, Se and Bi atoms, respectively. Fig. 3a and 3b show the dependent Gilbert dampings of BS/Fe, BS/Co and BS/Ni. It is striking that Gilbert dampings of BS/Fe, BS/Co and BS/Ni are enhanced by about one or two order s in magnitude from the counterparts of Fe, Co and Ni bulks as well as their free-standing MLs, depending on the choice of scattering rate in the range from 0. 001 to 1.0 eV. Similar to Fe, Co and Ni bulks, the intraband contributions monotonically decrease while the interband contributions increase as the scattering rate gets larger (Fig. A3 in Appendix D). Note that our calculations indicate that there is no obvious difference between the Gilbert dampings of BS/Fe when f ive- and six -QL BS slabs are used (Fig. A4 in Appendix E). This is consistent with the experimental observation that the interaction between the top and bottom topological surface states is negligible in BS thicker than five QLs [39]. 8 Figure 3 (color online) Scattering rate dependent Gilbert dampings of (a) Fe ML, bcc Fe bulk, BS/Fe and PbSe/Fe, (b) Co ML, hcp Co bulk, BS/Co, Ni ML, fcc Ni bulk and BS/Ni. (c) Dependence of the Gilbert dampin g of BS/Fe on the scaled SOC BS of BS in the range from zero ( BS0 ) to full strength ( BS1 ). Solid lines show the fitting of Gilbert damping BS/Fe to Eq. (6). The inse t shows Gilbert damping comparisons between BS/Fe at BS0 , bcc Fe bulk and Fe ML. As is well -known, TIs are characterized by their strong SOC and topologically nontrivial surface states. An important issue is how they affect the Gilbert damping s in BS/TM systems. Using BS/Fe as an example, we artificially tune the SOC parameter BS of BS from zero to full strength and fit the Gilbert damping BS/Fe in powers of BS as 2 BS/Fe 2 BS BS/Fe BS 0 (6) . As shown in Fig. 3c, we obtain two interesting results: (I) when BS is zero, the calculated residual Gilbert damping BS/Fe BS 0 is comparable to Gilbert dampings of bcc Fe bulk and Fe free -standin g ML (see the inset in Fig. 3c) ; (II) Gilbert damping BS/Fe increases almost linearly with 2 BS , simi lar to previous results [36]. These reveal that the strong SOC of BS is crucial for the enhancement of Gilbert damping. To gain insight i nto how the strong SOC of BS affects the damping of BS/Fe, we explore the k-dependent contributions to Gilbert damping, BS/Fe . As shown in Fig. 4a , many bands near the Fermi level show strong intermixing between Fe and BS orbitals (mar ked by black arrows ). Accordingly, these k-points have large contributions to the Gilbert 9 damping (marked by red arrows in Fig. 4b). However, if the hybridized states are far away from the Fermi level, they make almost zero contribution to the Gilbert damp ing. Therefore, we conclude that only hybridizations at or close to Fermi level have dominant influence on the Gilbert damping. This is understandable, since energy differences EF-Ei and EF-Ej are important in the Lorentzian functions in Eq. (4). Figu re 4 (color online) (a) DFT+SOC calculated band structure of BS/Fe. Color bar indicates the weights of BS (red) and Fe ML (blue). Black dashed line indicates the Fermi level. (b) k - dependent contributions to Gilbert damping BS/Fe at sc attering rate 26meV . Inset shows the first Brillouin zone and high -symmetry k -points , K and . It appears that there is no direct link between the topologic al nature of BS and the strong enhancement of Gilbert damping. The main contributions to Gilbert damping are not from the vicinity around the -point, where the topological nature of BS manifests. Besides, BS should undergo a topol ogical phase transition from trivial to topological as its SOC BS increases [40]. If the topological nature of BS dictates the e nhancement of Gilbert damping, one should expect a kink in the BS curve at this phase transition point but this is obviously absent i n Fig. 3c. 10 To dig deeper into this interesting issue, we replace the topologically nontrivial BS with a topologically trivial insulator PbSe, because the latter has a nearly the same SOC as the former. As shown in Fig. 3a, the Gilbert damping of PbSe/Fe is noticeably smaller than that of BS/Fe, although both are significantly enhanced from the values of of Fe bulk and Fe free -standing ML. Taking the similar SOC and surface geometry between BS and PbSe (Fig. A5 in Appendix F) , the large difference between the Gilbert dampings of BS/Fe and PbSe/Fe suggests that the topological nature of BS still has an influence on Gilbert damping. One possibility is that the BS surface is metallic with the presence of the time -reversal protected t opological surface states and hence the interfacial hybridization is stronger. Figure 5 (color online) Comparisons between Gilbert damping of BS/Fe at 26meV and (a) total DOS, (b) Fe projected DOS and (c) BS projected DOS. Red arrows and light cyan rectangles highlight the energy windows where Gilbert damping and the total DOS and Fe PDOS have a strong correlation . In (a), (b) and (c), all DOS are in units of state per eV and Fermi level EF indicated by the vertical green lines is set to be zero. A previous study of Fe, Co and Ni bulks suggested a strong correlation between Gilbert damping and total density of states (DOS) around the Fermi level [36]. To attest if this is 11 applicable here, we show the total DOS and Gilbert damping BS/Fe of BS/Fe as a function of the Fermi level based on the rigid band a pproximation. As shown in Fig. 5a, Gilbert damping BS/Fe and the total DOS behave rather differently in most energy regions. From the Fe projected DOS (PDOS) and BS projected PDOS (Fig. 5b and 5c), we see a better correlation between G ilbert damping BS/Fe and Fe -projected DOS, especially in regions highlighted by the cyan rectangles . We perceive that although the -DOS correlation might work for simple systems, it doesn’t hold when hybridiza tion and SOC are complicated as the effective SOC strength may vary from band to band. V. SUMMARY In summary, we extend our previous torque method from determining magnetocrystalline anisotropy energies [27, 28] to calculating Gilbert da mping of FM metals and apply this new approach to Fe, Co and Ni MLs in contact with TI BS. Remarkably, the presence of the TI BS substrate causes order of magnitude enhancements in their Gilbert dampings. Our studies demonstrate such strong enhancement is mainly due to the strong SOC of TI BS substrate . The topological nature of BS may also play a role by facilitati ng the interfacial hybridiz ation and leaving more states around the Fermi level . Although alloying with heavy elements also enhances Gilbert dampings [32], the use of TIs pushes the enhancement into a much wide r range. Our work thus establishes an attractive way for tuning the Gilbert damping of FM metallic films, especially in the ultrathin reg ime. ACKNOWLEDGMENTS We thank Prof. A. H. MacDonald and Q. Niu at University Texas, Austin, for insightful discussions. We also thank Prof. M. Z. Wu at Colorado State University and Prof. J. Shi at University of California, Riverside for sharing their ex perimental data before publication. Work was supported by DOE -BES (Grant No. DE -FG02 -05ER46237). Density functional theory calculations were performed on parallel computers at NERSC supercomputer centers. 12 Appendix A: Details of Gilbert damping calcula tions To compare Gilbert dampings of Fe, Co and Ni free -standing MLs with BS/Fe, BS/Co, and BS/Ni, we use 33 supercells containing three atoms and set their lattice constants to 4.164 Å, same as that of the BS substrate. This means that the lattice constant of their primitive unit cell (containing one atom) is fixed at 2.40 Å. The relaxed lattice constants of Fe (2.42 Å), Co (2.35 Å) and Ni (2.36 Å) free -standing MLs are close to this value. Systems a (Å) b (Å) c (Å) k-point grid Fe bulk 2.931 2.931 2.931 35× 35× 35 16 36 2.25 Co bulk 2.491 2.491 4.044 37× 37× 23 18 40 2.22 Ni bulk 3.520 3.520 3.520 31× 31× 31 40 80 2.00 Fe ML 4.164 4.164 -- 38× 38× 1 24 56 2.33 Co ML 4.164 4.164 -- 37× 37× 1 27 64 2.37 Ni ML 4.164 4.164 -- 39× 39× 1 30 72 2.40 BS/Fe 4.164 4.164 -- 37× 37× 1 188 396 2.11 BS/Co 4.164 4.164 -- 37× 37× 1 194 408 2.10 BS/Ni 4.164 4.164 -- 37× 37× 1 200 432 2.16 PbSe/Fe 4.265 4.265 -- 37× 37× 1 174 376 2.16 Table A1. Here are details of Gilbert damping calculations of all systems that are studied in this work. is abbreviated for the number of valence electrons and stands for the number of total bands. is the ratio between and , namely, . Note that five QLs of BS are used in calculations for BS/Fe, B S/Co and BS/Ni. Appendix B: Derivatives of SOC Hamiltonian HSOC with respect to the small deviation u of magnetic moments Based on the SOC Hamiltonian HSOC in Eq. (5) in the main text, derivatives of the term ls against the polar angle and azimuth angle are 13 11sin cos cos22 1 1 1cos sin sin A1 ,2 2 2 1 1 1cos sin sin2 2 2ii nz ii z ii zs l l e l e s l l e l e s l l e l e ls and 22 22110 sin sin22 10 sin cos A2 .2 2 2 10 cos sin2 2 2ii n ii iis i l e i l e s i l e i l e s i l e i l e ls Note that magnetization M is assumed to have a constant magnitude when it precesses , so we have 0SOC SOC H M H m . When the normalized magnetization m points along the direction of , , ,x y zm m m n , we have: sin cosxm , sin sinym and coszm . Taking 0m m u and the chain rule together, we obtain derivatives of SOC Hamiltonian HSOC with respect to the small deviation of magnetic moments as follows: sincos cos A3 ,sinSOC SOC SOC SOC SOC x x x x x SOC SOCu m m m m H H H H H m m HH coscos sin A4 ,sinSOC SOC SOC SOC SOC y y y y y SOC SOCu m m m m H H H H H m m HH and u14 sin A5 .SOC SOC SOC SOC SOC z z z z z SOCu m m m m H H H H H m m H Combining Eq. (5) and Eq. (A1 -A6), we can easily obtain the final formulas of derivatives of SOC Hamiltonian HSOC of magnetization m. Appendix C: Charge transfers and induced magnetic moments in BS/Fe, BS/Co and BS/Ni Figure A1 (color online) Planar -averaged char ge difference BS TM ML BS TM-ML (TM = Fe, Co and Ni) of (a) BS/Fe, (b) BS/Co and (c) BS/Ni . The atoms positions are given along the z axis. 15 Figure A2 (Color online) Charge density difference BS TM ML BS TM ML (TM = Fe, Co and Ni) nea r the interface betwee n the TM monolayer and the top most QL BS of (a) BS/Fe, (b) BS/Co and (c) BS/Ni. The color bar shows the weights of the negative (blue) and positive (red) charge density differences. Numbers give the induced magnetic moments (in units of B ) in the topmost QL BS. Bi and Se atoms are shown by the purple and light green balls, respectively. Appendix D: Contributions of intraband and interband to the Gilbert dampings of BS/Fe, BS/Co and BS/Ni Figure A3 (color online) Calculated Gilbert dampings of (a) BS/Fe, (b) BS/Co and (c) BS/Ni. Black curves give the total damping. Red and blue curves give the intraband and interband contributions, respectively. 16 Appendix E: Gilbert dampings of BS/Fe with five - and six -QLs of BS slabs 8 Figure A4 (color online). Gilbert dampings of BS/Fe with five (red) and six (black) QLs of BS slabs. In the calculations of the Gilbert damping of BS/Fe with six QLs of BS, we use a 39 ×39×1 Gamma -centered k -point grid, and the number of the total bands is 448 which is twice larger than the number of the total valence electrons (216). Appendix F: Structural c omparisons between BS/Fe and PbSe/Fe 17 Figure A5 (color online) (a) Top view and (c) side view of atom arrangement in BS/Fe. (b) Top view and (d) side view of atom arrangement in PbSe/Fe. In (a) and (c), the xyz -coordinates are shown by the red arrows. In (b) and (d), the rectangles with blue dashed lines highlight the most top QL BS in BS/Fe which is similar to the Pb and Se atom laye rs in PbSe/Fe. The important Fe - Bi, Fe -Se and Fe -Pb bond length is given by the numbers in units of Å . Dark green, light green, purple -red and dark gray balls represent Fe, Se, Bi and Pb atoms, respectively. Note that computational details are given in Tab le A1. References [1] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 101 (1935). [2] T. Gilbert, Phys. Rev. 100, 1243 (1955). [3] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443 (2004). [4] D. Steiauf and M. Fä hnle, Physic al Review B 72, 064450 (2005). [5] V. L. Safonov, Journal of applied physics 91, 8653 (2002). [6] M. Fä hnle and D. Steiauf, Physical Review B 73, 184427 (2006). [7] D. Thonig and J. Henk, New Journal of Physics 16, 013032 (2014). [8] L. Chen, S. Mankovsky, S. Wimmer, M. Schoen, H. Kö rner, M. 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2002.12255v1.Ultrafast_magnetization_dynamics_in_half_metallic_Co__2_FeAl_Heusler_alloy.pdf

Ultrafast magnetization dynamics in half-metallic Co 2FeAl Heusler alloy R. S. Malik,1E. K. Delczeg-Czirjak,1D. Thonig,2R. Knut,1I. Vaskivskyi,1 R. Gupta,3S. Jana,1R. Stefanuik,1Y. O. Kvashnin,1S. Husain,3A. Kumar,3P. Svedlindh,3J. S oderstr om,1O. Eriksson,1, 2and O. Karis1 1Department of Physics and Astronomy, Uppsala University, Box 516, SE- 75120 , Uppsala, Sweden 2School of Science and Technology, Orebro University, SE- 70182 Orebro, Sweden 3Department of Materials Science and Engineering, Uppsala University, Box 534, SE- 75121 , Uppsala, Sweden Abstract We report on optically induced, ultrafast magnetization dynamics in the Heusler alloy Co 2FeAl, probed by time-resolved magneto-optical Kerr eect. Experimental results are compared to results from electronic structure theory and atomistic spin-dynamics simulations. Experimentally, we nd that the demagnetization time ( M) in lms of Co 2FeAl is almost independent of varying structural order, and that it is similar to that in elemental 3d ferromagnets. In contrast, the slower process of magnetization recovery, specied by R, is found to occur on picosecond time scales, and is demonstrated to correlate strongly with the Gilbert damping parameter ( ). Our results show that Co 2FeAl is unique, in that it is the rst material that clearly demonstrates the importance of the damping parameter in the remagnetization process. Based on these results we argue that for Co2FeAl the remagnetization process is dominated by magnon dynamics, something which might have general applicability. 1arXiv:2002.12255v1 [cond-mat.mtrl-sci] 27 Feb 2020Studies of ultrafast demagnetization was pioneered by Beaurepaire et al. [1], who demon- strated that the optical excitation of a ferromagnetic material - using a short pulsed laser could quench the magnetic moment on sub-picosecond timescales. The exact underlying mi- croscopic mechanisms responsible for the transfer of angular momentum have been strongly debated for more than 20 years [2{4]. Ultrafast laser-induced demagnetization has now be- come an intense eld of research not only from fundamental point of view but also from a technological aspect, due to an appealing possibility to further push the limits of oper- ation of information storage and data processing devices [5]. Both experiment [4{15] and theory [16{23] report that all of the 3d ferromagnets (Fe, Ni and Co) and their alloys, show characteristic demagnetization times in the sub-picosecond range, while 4f metals exhibit a complicated two-step demagnetization up to several picoseconds after the excitation pulse [4, 24]. In this work, we have made element specic investigations of the ultrafast magnetiza- tion dynamics of a half-metallic Heusler alloy. This class of alloys has been investigated intensively, especially concerning the magnetic properties, ever since the discovery in 1903, when Heusler et al. reported that alloys like Cu 2MnAl exhibit ferromagnetic properties, even though none of its constituent elements was in itself ferromagnetic [25]. The ferro- magnetic properties were found to be related to the chemical ordering [26]. One of the key features of several Heusler alloys is their unique electronic structure, where the majority spin band-structure has a metallic character while the minority spin band is semiconducting with a band gap. Such materials are also referred to as half-metallic ferromagnets (HMFs) and were initially predicted by de Groot et al. [27], based on electronic structure theory. Half-metals ideally exhibit 100% spin-polarization at the Fermi level. This exclusive prop- erty makes them candidates to be incorporated in spintronic devices, e.g. spin lters, tunnel junctions and giant magneto-resistance (GMR) devices [28{31]. One of the advantages of Heusler alloys with respect to other half-metallic system, like CrO 2and Fe 3O4, are their relatively high Curie temperature ( Tc) and low coercivity ( Hc) [32, 33]. Heusler alloys are also appealing for spintronic applications due to the low Gilbert damping, which allows for a long magnon diusion length [34{39]. It has been shown that the low value of the Gilbert damping constant is related with the half-metallicity [37, 38]. The origin of the band gap and the mechanism of half-metallicity in these materials have been studied by using rst principle electronic structure calculations [40{43]. The half-metallic property is furthermore 2known to be very sensitive to structural disorder [41{45]. From a fundamental point of view, it is intriguing to ask, how the band gap in the minority spin channel eects the ultrafast magnetization dynamics of Heusler alloys [46, 47]. It has already been reported that some of the half-metals like CrO 2and Fe 3O4exhibit very slow dynamics, involving time-scales of hundreds of picoseconds, [46, 47] while several Co-based Heusler alloys show a much faster demagnetization, similar to the time-scales of the elemental 3d-ferromagnets [48{50]. The faster dynamics of these Heuslers has been discussed in Ref.[46] to be due to the fact that the band gap in the minority spin channel is typically around 0 :30:5 eV, which is smaller than the photon energy (1 :5 eV) of the exciting laser. It is also smaller than the band gap of CrO 2and Fe 3O4. Importantly, the Heusler alloys oer the possibility to study magneti- zation dynamics, as a function of structural order, since they normally can be prepared to have a fully ordered L21phase, a partially ordered B2 phase, and a completely disordered A2 phase. The structural relationships of these phases are described in the Supplemental Material (SM) [51]. We have here studied the optically induced, ultrafast magnetization dynamics of Co 2FeAl (CFA) lms, using time-resolved magneto-optical Kerr eect (TR-MOKE) as described in Ref. [52]. By control of the growth temperature, CFA alloy forms with varying degree of structural order, in a continuous way between the A2 and B2 phases, as well as between theB2 andL21phases [53, 54]. We present data from four CFA samples, grown at 300 K, 573 K, 673 K, and 773 K respectively. We henceforth denote each sample by its growth temperature as a subscript, e.g. CFA 300K. As evidenced by X-ray diraction, the sample grown at 300 K is found to exhibit the A2 phase, while the samples grown at 573 K and 673 K predominantly exhibit the B2 phase. The sample grown at 773 K is found to exhibit a pure B2 phase [54]. The value of the Gilbert damping is found to monotonously decrease with annealing temperature and is thus lowest for the sample grown at 773 K [55]. Calculations based on density functional theory (DFT) of the magnetic moment, Heisen- berg exchange interaction and the Gilbert damping parameter are described in detail in (SM) [51]. These parameters were used in a multiscale approach to perform atomistic mag- netization dynamics simulations, described in Sec.S1 of (SM) [51]. Here we employed the two temperature model (2TM) for the temperature prole of the spin-system. In the 2TM, the spin temperature increases due to the coupling to the hot-electron bath, that is excited by the external laser pulse. In the simulations we used a peak temperature in the 2TM of 30.70.750.80.850.90.951.01.051.1M/M(0)05 1 0 1 5A2CoFe05 1 0 1 5t( p s )B2 05 1 0 1 5 2 0L21FIG. 1. (Color online) Simulations of ultrafast dynamics of Co 2FeAl in the dierent structural phasesA2 (left panel), B2 (central panel), and L21(right panel). The demagnetization is shown element resolved (blue line - Co, red line - Fe). The peak temperature is 1200 K. The dotted line indicates the equilibrium magnetization at T= 300 K. 1200 K. A full description of the 2TM and the details of all spin-dynamics simulations are described in Sec.S2 of SM [51]. The results of the simulations are shown in Fig. 1, for the A2,B2 andL21phases. It can be seen that the dierent phases react dierently to the external stimulus. In general, this model provides a dynamics that is controlled by i)the temperature of the spin-subsystem, ii)the strength of the magnetic exchange interaction and iii)the dissipation of angular momentum and energy during the relaxation of the atomic magnetic moments (Gilbert damping) [56]. Before continuing the discussion, we note that the average magnetization, M, of element Xis calculated as MX=P icX iMX i=4P icX i, wherecX iis the concentration of the particular element Xin the particular phase and iruns over the four nonequivalent sites of the unit cell. After the material demagnetizes, the spin temperature eventually drops and the average magnetization returns to its initial value after 10 20 ps (cf. Fig. 1). To estimate the time constants of the demagnetization Mand remagnetization ( R) 4processes, in an element-specic way, we t both the theoretical and experimental transient magnetizations by a double exponential function [57]. We show results of MandRin Fig. 2 for the A2 and B2 phase, as well as for alloys with intermediate degree of disorder (described in Sec.S2 of SM [51]). 4.5 4.0 3.5 3.0 2.5 2.0R (ps) 0 20 50 80 100 Amount of B2 order [%] Fe Co Remagnetization (b)2.0 1.5 1.0 0.5 0.0M (ps) Fe CoDemagnetization (a) FIG. 2. (Color online) Element resolved relaxation times of Co 2FeAl, from simulations of alloys with varying amounts of A2!B2 phase. 0 corresponds to pure A2 phase while 100 corresponds to pureB2 phase. Panel (a) shows the demagnetization time and panel (b) shows the remagnetization time. Both time constants are obtained from tting the time trajectory of MX(t) by a double- exponential function (see text). The theoretical demagnetization time is seen from Fig. 2 to typically be around 1 ps, whereas the remagnetization time is 2 5 ps. Going from the A2 to the B2 alloy, both times increase, albeit the simulations show a stronger increase of the remagnetization time 5as function of alloy composition. We also note that the relevant time scale is somewhat larger for Fe than for Co, and the ratio between them,Fe=Co, grows when going from A2 to B2 phase. Figures 3 (a-d) shows the measured magnetization dynamics of CFA lms that were grown at dierent temperatures (see SM, Sec.S3 for thin lms synthesis, and Sec.S4 for details on the experimental measurements [51]). The inset shows the observed magnetization dynamics up to1 ps. For all samples, the data for Fe (red) and Co (blue) show similar demagnetization dynamics in the rst few hundred femtoseconds, whereupon dierences in the magnetization dynamics become visible, especially on the picosecond timescale. 1.000.950.900.850.800.75 Normalized Asymmetry (arb.units)1086420 Time Delay (ps) A2(a) 1.000.950.900.850.800.75A/A00.80.40.0FeCo1086420 Time Delay (ps) 90% B2(b) 1.000.950.900.850.800.75A/A00.80.40.0FeCo 1086420 Time Delay (ps) 90% B2(c) 1.00.90.80.7A/A00.80.40.0 Fe Co1086420 Time Delay (ps) 100% B2(d) 1.00.90.80.7A/A00.80.40.0FeCo FIG. 3. Measured element-specic Fe (red) and Co (blue) magnetization dynamics of Co 2FeAl. Samples are denoted by the growth temperature in each case. The red and blue lines correspond to tted data (see text). (a) 300 K (100% A2phase), (b) 573 K (90% B2 phase), (c) 673 K (90% B2 phase), and (d) 773 K (100% B2 phase). The insets show the demagnetization dynamics up to 1 ps. All of the measurements were performed with similar pump- uence (for details, see Sec.S4 of SM). Figure 4 (a-b) shows the measured values of the demagnetization and remagnetization time constants, for the four dierent growth temperatures, representing dierent degree of disorder in Co 2FeAl, along the alloy path A2!B2. It may be seen that the Mfor Fe and Co is the same within the error bars for all four samples, regardless of the degree of structural ordering (Fig. 4a). It may also be noted that the measured Mfor CFA is similar to that of 3d transition metals [48, 49] and very much shorter than that of CrO 2or Fe 3O4. Demagnetization times that are independent on degree of structural ordering is interest- 6ing, since it can be expected that the presence of structural disorder in Heusler alloys ought to result in a lower degree of spin polarization of the electronic states (i.e. an increased density of states (DOS) at the Fermi level in the minority band). This is expected to en- hance spin- ip scattering, with an accompanying speed-up of the demagnetization dynamics [46, 47]. The electronic structure calculation of CFA also shows that the DOS at the Fermi level varies with dierent structural phases (analyzed in the Sec.S1 of SM [51]). The A2 phase has a large number of states at the Fermi level, while the L21phase, and to some extent the B2 phase, has a low amount [54]. Despite these dierences in the electronic struc- ture, the measured demagnetization dynamics shown in Fig. 4(a) is essentially independent on degree of structural ordering. On longer time-scales, there is a signicant eect of structural ordering on the observed magnetization dynamics, which becomes particularly relevant for the remagnetization pro- cess. As seen in Fig. 4b, there is a monotonous increase of remagnetization time, R, with increasing growth temperature and hence the degree of ordering along the A2!B2 path. The sample grown at 300 K with A2 phase, exhibits the fastest remagnetization dynam- ics (R). With increasing growth temperature and corresponding increase in the structural ordering along the A2!B2 path, a distinct trend of increasingly slower remagnetization dynamics is observed. The most conspicuous behaviour of the measured magnetization dynamics, and its depen- dence on the degree of ordering, concerns the remagnetization time (Fig. 4b). The time-scale of the remagnetization process is suciently long to allow for an interpretation based on atomistic spin-dynamics. Two materials specic parameters should be the most relevant to control this dynamics; the exchange interaction, as revealed by the local Weiss eld, and the damping parameter. In the Sec.S2 of SM [51], we report on the calculated Weiss elds and damping parameters. It is clear from these results that the trend in the experimental data shown in Fig. 4b, can not be understood from the Weiss eld alone, whereas an expla- nation based on the damping is more likely. In order to illustrate this, we show in Fig. 5 the inverse of the measured remagnetizatiom time compared to the theoretically calculated damping and experimental measured damping through ferromagnetic resonance (FMR) (de- scribed in Sec.S6 of SM)[51]. The gure shows that the damping is large in the completely disordered A2 phase and for a large range of structural orderings, which comes out from both theory and experiment. The gure also demonstrates that the inverse of the measured 7300250200150100500 τM (fs) Fe Co(a) 3.02.52.01.5 τR (ps)CFA300 K CFA573 K CFA673 K CFA773 K (b)FeCo FIG. 4. Measured magnetization times for the investigated Co 2FeAl alloys. In (a) the demagneti- zation time, M, is shown and in (b) the remagnetization time, R, is plotted. remagnetization time scales very well with both the calculated damping and experimentally measured damping . According to the gure, a large damping parameter corresponds to faster remagnetization dynamics in the measurements. Co2FeAl is, to the best of our knowledge, the rst system where experimental observations and theory point to the importance of damping in the process of ultrafast magnetization dynamics. We note that this primarily is relevant for the remagnetization process; the initial part of the magnetization dynamics (rst few hundred fs) is distinctly dierent. In the demagnetization we observe a similar behaviour for Fe and Co in all samples, and an insensitivity of the demagnetization times in relation to structural ordering. Also, the 80.550.500.450.400.350.300.25 (τr )-1 (1/ps) 43210Damping ( x 10-3) 100806040200Amount of B2 ordering (%)Remagnetization timeTheoretical dampingExperimental dampingFIG. 5. The relationship of inverse of the measured remagnetization time (right y-axis) and the- oretically calculated and experimentally measured Gilbert damping (left y-axis) in Co 2FeAl for varying amount of B2 order along the A2!B2 path, i.e. 0 corresponds to pure A2 phase while 100 corresponds to pure B2 phase. measured and theoretical demagnetization times evaluated from atomistic spin-dynamics simulations, do not agree. Other mechanisms, of electronic origin, most likely play role in this temporal regime. The remagnetization process of Co 2FeAl alloys with varying degree of structural order, highlights clearly the importance of the Gilbert damping and that magnon dynamics domi- 9nates the magnetization at ps time-scales. The relevance of the Gilbert damping parameter for ps dynamics is natural, since this controls angular momentum (and energy) transfer to the surrounding. What is surprising with Co 2FeAl is the fact that other interactions (e.g. the Weiss eld) show such a weak dependence on the amount of structural diorder. This is fortuitous, since it allows to identify the importance of the Gilbert damping. A picture emerges from the results presented here, that the magnetization dynamics in general have two regimes; one which is primarily governed by electronic processes, and is mainly active in the rst few hundered fs ( M), and a second regime where it is primarily magnons that govern the remagnetiztion dynamics ( R). 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2312.04202v2.Probing_levitodynamics_with_multi_stochastic_forces_and_the_simple_applications_on_the_dark_matter_detection_in_optical_levitation_experiment.pdf

Probing levitodynamics with multi-stochastic forces and the simple applications on the dark matter detection in optical levitation experiment Xi Cheng, Ji-Heng Guo, Wenyu Wang1,∗and Bin Zhu2,† 1Beijing University of Technology, Beijing 100124, China 2School of Physics, Yantai University, Yantai 264005, China If the terrestrial environment is permeated by dark matter, the levitation experiences damping forces and fluctuations attributed to dark matter. This paper investigates levitodynamics with multiple stochastic forces, including thermal drag, photon recoil, feedback, etc., assuming that all of these forces adhere to the fluctuation-dissipation theorem. The ratio of total damping to the stochastic damping coefficient distinguishes the levitodynamics from cases involving only one single stochastic force. The heating and cooling processes are formulated to determine the limits of temperature change. All sources of stochastic forces are comprehensively examined, revealing that dark matter collisions cannot be treated analogously to fluid dynamics. Additionally, a meticulous analysis is presented, elucidating the intricate relationship between the fundamental transfer cross- section and the macroscopic transfer cross-section. While the dark damping coefficient is suppressed by the mass of the levitated particle, scattering can be coherently enhanced based on the scale of the component microscopic particle, the atomic form factor, and the static structure factor. Hence, dark damping holds the potential to provide valuable insights into the detection of the macroscopic strength of fundamental particles. We propose experimental procedures for levitation and employ linear estimation to extract the dark damping coefficient. Utilizing current levitation results, we demonstrate that the fundamental transfer cross section of dark matter can be of the order σD T∼<O(10−26)cm2. PACS numbers: 37.10.Mn, 95.35.+d, 84.71.Ba, 05.40.Ca, 05.10.Gg I. INTRODUCTION Observations in cosmology and astrophysics have con- firmed the presence of dark matter (DM) [1]. One of the most promising experimental approaches to detect DM on Earth is to search for small energy depositions resulting from elastic scattering of DM in sensitive de- tectors. Searches for weakly interacting massive parti- cles (WIMPs) represent some of the most advanced tech- niques in this field. However, stringent constraints exist on the cross section for DM particles with masses ex- ceeding 1 GeV. DM particles may have eluded detection if the resulting energy deposits fall below the threshold of current detectors. Therefore, it is imperative to ex- plore alternative strategies and develop novel detection methods to investigate the remaining parameter space of DM i.e., light DM [2–19]. In recent years, a fundamentally different approach has been proposed to increase the sensitivity to the energy transfer from DM particles to specific microscopic inter- nal degrees of freedom within a detector. This approach involves optically monitoring the center-of-mass (COM) motion of a levitated macroscopic object. The levitation of micrometer-sized objects in vacuum was first demon- strated in the pioneering work of Arthur Ashkin in the 1970s [20]. Levitated nanoparticles have since been uti- lized in various fields, including trapping and cooling of atoms, control mechanisms based on optical, electrical, ∗guojiheng@buaa.edu.cn, wywang@bjut.edu.cn, †zhubin@mail.nankai.edu.cnand magnetic forces, and investigations of light-matter interactions [21–31] Laboratory-scale levitation experiments aiming to reach the quantum regime require a substantial degree of isolation and control, making them highly sensitive to forces or phenomena that couple to mass. Recently, a study by Ref. [32] demonstrated that levitated nanoscale mechanical devices operated around the standard quan- tum limit for impulse sensing can detect the deposition of energy transfer from DM particles. Levitation devices of- fer a unique opportunity for directional searches for DM masses in the keV-GeV regime, which currently have lim- ited direct detection constraints. Moreover, they exhibit remarkable complementarity to other proposals due to coherent interaction at low momentum detection thresh- olds. [33] The results of Ref. [33] demonstrate the poten- tial of optomechanical sensors in searching for DM. Large arrays of sensors could facilitate the reconstruction of track-like signals from DM particles while possessing suf- ficient sensitivity to detect and count individual collisions of latent gas in ultra-high vacuum environments. Cavity optomechanical systems aim to operate in the shot-noise dominant regime to observe macroscopic quantum phe- nomena. In this study, we focus on the detection of DM us- ing levitated particles and investigate the thermal dy- namics of Langevin systems subjected to multi-stochastic forces. The presence of distinct stochastic forces can induce fluctuation-dissipation from an unknown source, making it challenging to detect the DM-induced collision signal amidst the noise. By examining the properties of these stochastic forces and studying the recoil of levi-arXiv:2312.04202v2 [cond-mat.stat-mech] 16 Dec 20232 tated particles, we aim to unveil their characteristics and provide insights into the presence of additional unique sources. The rest of this paper is organized as follows: Sec. II presents the general formulation of the Langevin system with multi-stochastic forces. Sec. III examines the properties of various stochastic forces. Sec. IV dis- cusses the application of these forces in detecting dark matter. Finally, Sec. V concludes the paper. II. LEVITODYNAMICS WITH MULTI-STOCHASTIC FORCES A. Why we go beyond the framework of fluctuation theorem Nanoparticle levitation or trapping can be achieved through laser cooling techniques, as illustrated in panel A of Fig. 1 [21]. In an atomic beam, collimation selects atoms moving in a specific direction that can be deceler- ated using a single laser beam molasses. Atoms in a gas move in random directions, and reducing their tempera- ture necessitates laser cooling in all three spatial dimen- sions by using three orthogonal standing waves. How- ever, for a moving atom, the Doppler effect introduces an imbalance in the forces. The Doppler shift brings the light closer to the atom’s resonance, increasing the rate of absorption from the laser beam and resulting in a force that decelerates the atom. This Doppler cool- ing, also known as the feedback mechanism in levitation, can be controlled using a switch, which is depicted in panel A of Fig. 1. By enabling or disabling the feed- back loop, the heating (relaxation) or cooling processes of the levitated particle can be measured. Apart from the molasses forces, the levitated particle can interact with the environment through various other forces, as illustrated in panel B of Fig. 1 [34]. Noise in the trap- ping potentials generates fluctuations and dissipation in particle motion. In the quantum regime, these mecha- nisms induce decoherence in the motional quantum state by leaking information about the particle’s position to the environment. The primary sources of noise in lev- itodynamics include (i) collisions with gas molecules in the vicinity; (ii) emission, absorption, and scattering of thermal electromagnetic radiation; (iii) noise in the trap center due to vibrations and/or stray fields; (iv) noise in the trap frequency caused by fluctuations in the elec- tromagnetic fields generating the trapping potentials; (v) cross-coupling to other thermal degrees of freedom in the particle’s motion along orthogonal directions, rotation, or internal excitation; and (vi) the unknown interaction which can be dark interaction talked about in this pa- per. Further analysis of these noise sources can provide a deep understanding of levitated particles and enhance sensor accuracy. In the next section, we will explore the thermal dynamics of this noise. Although the system may have numerous degrees of freedom, a good approximation is that these freedomsare well decoupled. Each degree of freedom exhibits in- dependent harmonic oscillator motion with frequency ω0 and damping coefficient γ. The system can be described by a one-dimensional Brownian particle with mass mfol- lowing the equation of motion d2x dt2+γdx dt+ω2 0x=F(t) m. (1) It is known as the Langevin system. The force F(t) can be a time-dependent external force used to explore re- sponse functions or a time-dependent random force re- sulting from molecular collisions with the environment. For example, in the case of Brownian motion in a fluid, γrepresents the damping coefficient of spherical parti- cles. The stochastic force F(t) has an expectation value ⟨F⟩= 0. Its correlation function is given by ⟨F(t)F(t′)⟩= 2mγk BTδ(t−t′), (2) where kBis the Boltzmann constant and it is propor- tional to the temperature of the environment TE. The system establishes a fundamental relationship between fluctuation and dissipation. The fluctuating force is re- lated to the coefficient of viscosity of the fluid, represent- ing dissipative forces in the system, and it depends on the temporal characteristics of molecular fluctuations. The energy of the levitated particle changes due to its inter- action with the environment, and the time evolution of the average energy ⟨E⟩is described by the Fokker-Planck equation. The energy flow can be understood as fluc- tuations driving the levitated particle while dissipation dampens it. In the equilibrium state, the energy from both sides compensates each other. An effective temper- ature, equal to the temperature of the environment, can be assigned to the system. However, as mentioned earlier, the levitated particle interacts with multiple stochastic forces simultaneously, resulting in the presence of multiple force sources. In particular, these different stochastic forces don’t need to be in equilibrium, which leads to the failure of the the- oretical formulation mentioned above. Nevertheless, not every damping coefficient needs to correspond to a single fluctuation source, thereby altering the thermal dynam- ics of the levitated particle. While it is possible to achieve a steady state for the levitated particle, this state does not indicate the temperature of the environment due to the absence of a uniform temperature. For instance, in a cold atom system, the temperature in the correlation function of the feedback cooling force (as described by Eq. (2)) must be significantly lower than the environ- ment temperature due to collisions with surrounding gas molecules. Otherwise, the cooling process will not be effective. Consider the possible dark matter interaction studied in this paper, where the dark matter decoupled from the heat bath during the early epoch of the universe. There is no reason for the correlation function of the dark force to be identical to that of the gas environment. The details of the Langevin system with multiple stochastic3 A B Gas Cxy zFeedback Dark interaction rad FB DM Gas rad FB DM Fluctuation Dissipation Field fluctuationsCoupling to other degrees of freedomGas moleculesBlack body radiation FIG. 1. A: Sketch map of the levitation of particle conducted by laser cooling techniques, The feedback loop can be turned on or off in the experiments. B: The noise sources in levitodynamicsa: (i) collisions with molecules of the surrounding gas; (ii) emission, absorption, and scattering of thermal electromagnetic radiation; (iii) noise in the trap center from vibrations and/or stray fields; (iv) noise in the trap frequency from fluctuations of the electromagnetic fields that generate the trapping potentials; (v) cross-coupling to other thermal degrees of freedom in the particle either motion along orthogonal directions, rotation, or internal excitation; and (vi) the unknown interaction which can be dark interaction talked about in this paper. C: The fluctuations and dissipation of the Langevin system with multi-stochastic sources. The arrows show the energy flow. aThe figure is adapted from Fig. 4 of Ref. [34] forces should be investigated. In such scenarios, a steady state can be achieved, and the fluctuation and dissipa- tion can be observed, as shown in panel C of Fig. 1. The levitated particle is driven by fluctuations from var- ious sources and dissipates through similar multi-sources. While the total energy flow in the fluctuations is equal to that of the dissipation in steady states, this relation- ship may not hold for individual pairs of fluctuations and dissipation. B. New framework In this work, we extend the framework of fluctuation theorems, as introduced in Ref. [35], to investigate the Langevin system with multiple stochastic forces. This framework enables the study of relaxation from a non- equilibrium state towards equilibrium. The trapped par- ticle experiences a trap force generated by the laser aswell as stochastic forces arising from random impacts of gas molecules, radiation, and dark interactions. As demonstrated in Ref. [35], the platform established in this study enables experimental investigations of non- equilibrium fluctuation theorems for arbitrary steady states. Furthermore, it can be extended to explore quan- tum fluctuation theorems and systems that do not adhere to detailed balance. To simplify the analysis, we focus on four specific types of forces: 1. The force arises from collisions between the levi- tated particle and the thermally moving molecules in the atmosphere. This force is characterized by a damping coefficient γTHand a stochastic driv- ing force FTH, as shown in Eq. (2). The damping coefficient can be obtained from the collisions of microscopic components in fluid dynamics. 2. The heating rate is attributed to the stochastic driving force FREarising from photon recoil kicks.4 Additionally, a damping coefficient γRE(denoted asγϵin Ref. [36]) arises from the back-reaction of the scattered field on the motion of the levitated particle and can be evaluated accordingly. [37] Pho- tons, the quanta of the radiation potential, trap the particle and form the optical position sensor. In- creasing the optical power raises the kick rate from individual photons, resulting in a force caused by shot noise of radiation pressure. As demonstrated in Ref. [36], the damping coefficient can be directly measured from the photon recoil during reheating. 3. We consider the feedback optical force FFBand the corresponding damping coefficient γFB. The optical force, along with the Doppler effect, decelerates the levitated particle, making it a crucial mechanism in levitation and cold atom physics. Importantly, the optical force FFBis a deterministic force that does not fluctuate, but rather damps the motion of the levitated particle. This is analogous to other damping effects. The main advancement of this paper lies in distinguishing between stochastic and deterministic forces in thermal dynamics. 4. An additional stochastic force can be generated through collisions with the surrounding dark mat- ter. Within the Langevin system, the interaction with dark matter is described by the damping co- efficient γDMand the driving force FDM. It should be noted that laser trapping techniques gener- ate a force where the driven force acts as the restoring force for the particle’s oscillation. Furthermore, it is im- portant to note that the damping coefficient γREdiffers from γFBsince they arise from different optical trapping setups. Dissipation and fluctuations can also occur due to self-emission, absorption, and scattering with ambient radiation from the environment. These dissipative and fluctuating effects are not considered in this paper. Nat- urally, the formulation in this section can be straightfor- wardly extended to include systems with other additional stochastic forces. It is important to keep in mind that the correlation function of the added stochastic force does not depend on a single uniform temperature. We consider a scenario where a levitated particle with mass mis created using a strongly focused laser beam of frequency Ω under ambient temperature and gas pressure conditions. In this setup, the particle is kicked by the incident photons from the environment, each carrying an energy of ℏΩ, resulting in the formation of a harmonic oscillator with a spring constant kand a frequency ω0= p k/m. Typically, the frequency of the incident laser beam, Ω, is significantly higher than the frequency of the levitation potential, ω0. The three spatial dimensions can be treated as decoupled from one another. Consequently, the motion of the Langevin system can be described as. d2q dt2+γTdq dt+ω2 0q=FTH(t) +FRE+FDM(t) +FFB(t) m. (3)The total damping coefficient γT, represented by the sec- ond term on the left side of the above equation, accounts for the sum of damping coefficients arising from various sources γT=γTH+γRE+γFB+γDM. (4) The right-hand side represents the driven forces in the system. Generally, each damping coefficient corresponds to a term on the right-hand side representing the driven forces. However, as mentioned earlier, the feedback force in the laser cooling technique is deterministic. This force FFB=−ω0ηq2pis a time-varying, non-conservative damping optical force introduced through parametric feedback with a strength η, where p=m˙qrepresents the momentum. In Langevin systems, the average value of any stochas- tic force Fiis zero. The correlation functions should fol- low a similar form as Eq. (2), which is proportional to the damping coefficients and temperature if they satisfy the fluctuation dissipation theorem. Here, Tidoes not rep- resent a uniform temperature but should be regarded as an effective parameter that characterizes the correlation function of each individual stochastic force. Naturally, TTHcorresponds to the temperature of the ambient en- vironment TE. The sum of all stochastic forces is denoted as. FS(t) =FTH(t) +FRE(t) +FDM(t). (5) It is evident that the equilibrium of the steady state of the levitated particle does not imply that its temperature is equal to that of the environment. A new stochastic temperature, denoted as TS, can be defined as TS=γTHTTH+γRETRE+γDMTDM γTH+γRE+γDM, (6) which accounts for the temperature of the system in the correlation function ⟨FS(t)FS(t′)⟩= 2mγSkBTSδ(t−t′). (7) In the above equation, γSrepresents the sum of all damp- ing coefficients associated with the respective stochastic forces γS=γTH+γRE+γDM. (8) It is important to note that the system temperature TS does not describe the thermal state of the Langevin sys- tem since the levitated particle represents a single de- gree of freedom. The parameter TS, along with all the Tiin Eq. (6), characterizes the nature of the stochas- tic forces. In the absence of a feedback loop, TSrepre- sents the temperature of the environment in an ordinary Langevin system. In experiments, the dynamical relax- ation can be measured by manipulating the feedback loop switch. [36] The heating or cooling trajectories can pro- vide detailed information about the stochastic forces. As5 mentioned in the introduction, the Langevin system con- tains multi-stochastic forces from different sources. The phenomenology of heating or cooling processes may devi- ate from predictions if only two parameters, γTandTS, are used to account for all the stochastic forces. There- fore, the following section discusses parametric feedback cooling with multiple stochastic forces. As demonstrated in Ref. [35], the energy of the Langevin system can be expressed in a form that resem- bles under-damped Brownian motion in the energy space. Expressing Eq. (3) as a stochastic differential equation (SDE) is more convenient. dq=p mdt , dp= −mω2 0q−ω0ηq2p−γTHp −γREp−γFBp−γDMp) dt (9) +p 2mγTHkBTTHdWTH +p 2mγREkBTREdWRE +p 2mγDMkBTDMdWDM, where Wi(t) are Wiener processes with ⟨Wi(t)⟩= 0,⟨Wi(t)Wj(t′)⟩=δij(t′−t). (10) Here only the thermal and additional dark stochastic forces are taken into account. For a short (infinitesimal) time interval d twe have ⟨dWi⟩= 0, (dWi)2 = dt. (11) Next, the energy change d Eis determined by dE=∂E ∂q dq+∂E ∂p dp+1 2∂2E ∂p2 (dp)2.(12) It is important to note that the Wiener process implies that (d p)2is of the same order as d t. Therefore, the second-order partial differential of the energy Eneeds to be considered, while (d q)2and d qdpcan be neglected. By taking the derivatives of the energy with respect to q andp, we obtain dE=mω2 0qdq+p mdp+1 2m(dp)2. (13) Inserting d q,dpand neglecting all terms of order (d t)3/2 or higher yields dE= −γTp2 m−ηE2 2mω0 dt+γSkBTS 2dt +p 2EγTHkBTTHdWTH (14) +p 2EγREkBTREdWRE +p 2EγDMkBTDMdWDM. Note that an additional factor of 1 /2 arises in the second term on the right-hand side when performing stochas- tic integration over the time evolution of the harmonicoscillation. This formula elucidates the fundamental dis- tinction between energy increments driven by single and multiple stochastic forces. For a single stochastic force, the formulation comprises a single damping coefficient and its corresponding temperature. However, the pres- ence of multiple stochastic forces considerably compli- cates the formulation, impacting the energy distribu- tion, dissipation, relaxation, and other factors. Each term γikBTidtarises in association with a correspond- ing stochastic force. However, additional contributions to the damping terms, γip2dt/m, arise from the feedback mechanism. This additional damping force can, in fact, be any dissipative force that does not involve fluctua- tions to drive the levitated particle. Consequently, the total damping coefficient γT, which incorporates the ad- ditional contribution of γFB, is not equal to the stochastic damping coefficient γS. As demonstrated below, this dif- ference in damping coefficients gives rise to distinct ther- mal dynamics compared to those resulting from a single stochastic force, even when TSis erroneously assumed to be the environmental temperature TE. Following a similar procedure outlined in the appendix of Ref. [35], we adopt ϵ=√ Eto study the energy incre- ment and prevent the emergence of multiplicative noise in Eq. (14) dϵ= −γTϵ 2−ηϵ3 4mΩ0+γSkBTS 4ϵ dt (15) +r γTHkBTTH 2dWTH+r γREkBTRE 2dWRE +r γDMkBTDM 2dWDM. The variable ϵevolves at temperature TSunder the in- fluence of an external force f(ϵ) in the presence of high friction characterized by ν= 4/γS dϵ=1 νf(ϵ)dt+r γTHkBTTH 2dWTH (16) +r γREkBTRE 2dWRE+r γDMkBTDM 2dWDM. We can denote a new parameter ζ=γT γS(17) to account for the difference between stochastic damping and the total damping. Then the external force f(ϵ) =−2ζϵ−ηϵ3 mω0γS+kBTS ϵ =−dU(ϵ) dϵ. (18) Then the potential U(ϵ) should be U(ϵ) =ζϵ2+η 4mω0γSϵ4−kBTElnϵ . (19)6 -12-10-8-6-4-20 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Eη=1.000η=0.100η=0.010η=0.001ζ = 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Eη=1.000η=0.100 η=0.010 η=0.001ζ = 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Eη=1.000η=0.100 η=0.010η=0.001ζ = 5 FIG. 2. The logarithm of the energy distribution ρ(E, α, ζ ) is compared for different ζvalues. The distributions are normalized with ln ρ(0, α, ζ ) = 0. The parameters are set as kSTS= 1,m= 1,k= 1, and γS= 0.01. Each curve corresponds to a different feedback strength ηshown in the plot. The plots for ζ= 2 and ζ= 5 depict the distributions with an additional damping factor. Denote α=η mω0γS, β S=1 kBTS. (20) Then the distribution of ϵcan be easily derived by Boltzmann-Gibbs distribution ρ(ϵ, α, ζ )∝exp −U(ϵ) kBTS ∝ϵexpn −βS ζϵ2+α 4ϵ4o . (21) The distribution function of energy can be got by recov- ering variables from ϵtoE=ϵ2 ρ(E, α, ζ ) =1 Zαζexpn −βSζE−βSα 4E2o , (22) where the normalisation factor Zαζ=R dEρ(E, α) is given by Zαζ=rπ αβSeζ2βS/αErfc r ζ2βS α! . (23) The distribution clearly reverts to the form of the single stochastic force when ζ= 1. However, the tempera- tureTSremains determined by multiplicative forces. To demonstrate the modification of the energy distribution, we numerically calculated the logarithm of the distribu- tionρ(E, α, ζ ) for different ζvalues. The results are de- picted in Fig. 2. From the figure, it is evident that in- creasing the ζparameter sharpens the distribution as the energy increases. This can be easily understood as a con- sequence of the additional damping factor ( γFB) leading to increased energy dissipation. The modification of the distribution can be observed through the average energy or the effective tempera- ture of the levitated particle. The average energy is ob- tained by integrating over the energy distribution given by Eq. (22). ⟨E⟩=Z−1 αZ dEEexp −βSζE−βEα 4E2 .(24)Evaluation of the integral yields ⟨E⟩=2 βS r βS αe−ζ2βS/α √πErfcq ζ2βS α−ζβS α .(25) If defined an effective temperature obtained by applying the feedback mechanism ⟨E⟩=kBTeff, then Teff=TS 2√αkBTSe−ζ2/αkBTS √πErfc ζ√αkBTS−2ζ αkBTS . (26) 00.10.20.30.40.50.60.70.80.91 0 1 2 3 4 5 6 7 8 9 10 αTeff / TS ζ=5ζ=2ζ=1 FIG. 3. The ratio of the effective temperature to the stochas- tic temperature Teff/TSversus α.ζ= 2 and ζ= 5 curves show the depression of effective temperature with additional damp- ing factor. Fig. (3) shows the numerical results of the ratio Teff/TS as a function of the parameter α. It can be observed7 that the effective temperature decreases with increasing α. Moreover, the effective temperature is further reduced by the additional damping factor introduced by the feed- back mechanism. This yields a highly interesting result. lim α→0Teff TS=1 ζ. (27) When α→0, it signifies the absence of the driven force from the feedback. Hence, the term 1 /ζquantifies the dampening effect of the additional damping factor. This mechanism effectively cools the levitated particle by in- teracting with the environment. Additionally, a non-zero feedback strength enhances the cooling effect. The averages over the last two terms of Eq. (14) vanish ⟨dWTH⟩= 0,⟨dWDM⟩= 0. Thus the time evolution of the average energy is d⟨E⟩ dt=−γT⟨E⟩ −η 2mω0⟨E2⟩+γSkBTS 2(28) =−γT⟨E⟩ −γSα 2⟨E2⟩+γSkBTS 2.This equation describes the cooling process in which the average energy decreases with the evolution due to the presence of feedback. In the steady state, the time deriva- tive on the left side of Eq. (28) becomes zero. α⟨E2⟩=kBTS−2ζ⟨E⟩. (29) The average dissipation rate ¯Pof the feedback mecha- nism can be formulated as ¯P=γFB⟨E⟩+γSα⟨E2⟩ 2. (30) Since the second moment of the energy ⟨E2⟩in Eq. (28) cannot be expressed in terms of the average energy ⟨E⟩, the equation cannot be easily solved for ⟨E⟩. However, in the equilibrium state for the exponential distribution, we have ⟨E2⟩= 2⟨E⟩2. Therefore, the cooling process can be approximately derived. ⟨E(t)⟩=ζ 2α−1 +p 1 + 2 αkBTS/ζ2+ 1 +p 1 + 2 αkBTS/ζ2 Cexp −γTp 1 + 2 αkBTS/ζ2t 1−Cexp −γTp 1 + 2 αkBTS/ζ2t . (31) Here, Cis an arbitrary constant determined by the ini- tial condition. It is reasonable to assume that the cool- ing process initiates when the levitated particle reaches a steady state with the ambient stochastic forces in the absence of feedback. In the levitation experiment, the feedback force can be toggled on or off, allowing for the measurement of both the cooling and heating processes of the levitated particle. The heating process is conducted by turning off the feedback loop, resulting in the evolu- tion of the average energy according to d⟨E⟩ dt=−γS⟨E⟩+γSkBTS 2. (32) It is important to note that when the feedback is turned off, the damping and fluctuation arise from the collec- tive ambient stochastic forces discussed earlier. There- fore, in Eq. (28), γTmust be replaced by γS. However, the presence of multiple sources implies that the system temperature does not necessarily equal the environmen- tal temperature. The relaxation evolution can be easily solved ⟨E(t)⟩=kBTS 2+ ⟨E(0)⟩ −kBTS 2 e−γSt =kBTS 2+kBTL 2−kBTS 2 e−γSt.(33) The temperature TLin the above equation represents the newly introduced effective temperature that character-izes the initial heating state. By examining Eq. (33), it becomes apparent that as time progresses, the levi- tated particle will tend to reach the ambient temperature TH=TS. This result can also be obtained from Eq. (32), where the time derivative on the left-hand side vanishes in the equilibrium state. At this point, the energy flow from fluctuations compensates for the dissipation. If the feedback is turned on, the system will be cooled and will no longer be in equilibrium with the ambient stochastic forces. By requiring ⟨E(0)⟩=kBTH/2, we obtain C=αkBTH+ζ−p ζ2+ 2αkBTH αkBTH+ζ+p ζ2+ 2αkBTH. (34) It is straightforward to determine that 0 < C < 1 based on the condition ζ >1. By examining Eq. (31), we can determine the final steady state of the cooling process, where the energy flow from the oscillator to the stochastic source is counterbalanced by the energy extracted from the feedback mechanism ⟨E(t→ ∞ )⟩=ζ 2α s 1 +2αkBTH ζ2−1! . (35) Therefore, the approximate effective temperature at the8 cooling limit should be. Tlimit L=ζ αkB s 1 +2αkBTH ζ2−1! . (36) It can be observed that the ratio β≡2αkBTH/ζ2de- termines the simple expression for the cooling limits. If β≪1, the cooling limit is expected to be Tlimit L≃TH ζ. (37) This indicates that the cooling process is dominated by the feedback damping γFB. The cooling limit is in agree- ment with the effective temperature given by Eq. (27). However, in the case of β≫1, Tlimit L≃r 2TH αkB, (38) This is independent of γFB. This implies that the cool- ing process is dominated by the feedback optical force, and it also necessitates 2 /αk B< TH. Otherwise, cooling will not be achieved. These limits differ from the results in Ref. [35], which are solely determined by the balance between single fluctuation and feedback optical force. 00.20.40.60.81 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6ζ=1.1 heatingζ=2 ζ=5 ζ=10 Log10(t+0.1 τ)T(t) / TS FIG. 4. The numerical results for the heating and cooling processes are presented. The parameters αkBandγSare nor- malized to unity in the numerical calculation. The dashed line represents the heating process, which evolves exponen- tially as exp( −γSt) with time. The solid lines represent the cooling processes for different values of the parameter ζ. The time evolution is scaled by a factor of 0 .1τto enable a loga- rithmic representation. Fig. 4 illustrates the numerical results for the heating and cooling processes. Parameters αkBandγSare nor- malized to unity in the numerical calculation, and the ratio of the effective temperature to the ambient tem- perature TSis presented. The heating process evolves exponentially as exp( −γSt) with time, while the coolingrate is determined by the parameter ζand evolves as exp(−γTp 1 + 2 αkBTS/ζ2t). A larger value of ζimplies faster cooling. Here, it is important to address the dif- ferences between our work and the theory proposed in Ref. [35]. In our work, cooling can be achieved through both feedback damping and feedback optical force. When γFB≫γS, the feedback damping from the Doppler effect dominates the dissipation process. However, in the limit γFB→0 or equivalently ζ→1, the cooling process is the same as the one proposed in Ref. [35], where the optical force cools the system. In the following section, we will discuss the distinction between the optical feedback force and the Doppler damping force. III. THE DAMPING FORCES FROM DIFFERENT SOURCES Based on the thermal dynamics presented in the pre- vious section, it is evident that the damping factor and temperature are the two fundamental parameters for de- scribing properties, such as correlation functions of the stochastic forces. The damping coefficients can be viewed as macroscopic phenomena that arise from collective in- teractions among the microscopic particles comprising both the levitated particle and the environment. These interactions can be attributed to mechanical or electro- magnetic dynamics in the microscopic realm. To gain a comprehensive understanding of the physics of levita- tion and enhance the sensor’s sensitivity, it is crucial to thoroughly investigate the collective interactions. This will allow us to modify experimental setups and unveil the underlying mechanism. Therefore, in this section, we explore the microscopic origins of the three damping co- efficients discussed in the previous section. Since γTH, γRE, and γFBhave been extensively studied in the liter- ature, and this paper aims to explore new methods for dark matter detection, we will provide a brief introduc- tion to γTH,γRE, and γFB. The formulation of γDMand the distinctions between this coefficient and the other three will be discussed in detail. A. The isothermal drag force The first damping rate we need to consider is the isothermal drag force, γTH, which arises from the in- teraction between a spherical particle and rarefied gas molecules. A single aerosol particle suspended in a non- equilibrium gas can experience various forces due to non-uniformity, including isothermal drag, thermal force, photophoretic force, diffusion force, and other forces re- sulting from combined flows of heat, mass, and momen- tum. Detailed investigations of these forces are beyond the scope of this paper, but they can be found in stan- dard fluid dynamics textbooks. [38] Clearly, the isother- mal drag force is the primary contributor to the damping force experienced by the levitated particle in the fluid. As9 A 𝒗𝑥𝑦𝑧𝑅!𝑈"𝑂B FIG. 5. (A) Figure illustrating the Knudsen number in fluid dynamics: a stationary flow past a particle in its own saturated vapor. The damping is determined by the mean free path of the molecules and the length scale of the particle. (B) Calculation scheme for isothermal drag: a spherical ball with a radius R0moving through the fluid with a vapor velocity of U∞. depicted in panel A of Fig. 5, a particle in its own sat- urated vapor experiences a stationary flow. The phys- ical phenomena are attributed to phase changes occur- ring on the particle surface. Theoretical fluid dynamics, as a subdivision of continuum mechanics, does not aim to describe the molecular structure of a medium or the motion of individual molecules. Continuum models rep- resent matter that is sufficiently dense to allow averaging over a large number of molecules, enabling the definition of meaningful macroscopic quantities. However, this ap- proach has inherent limitations, which can be expressed in terms of the Knudsen number, denoted as Kn Kn=λ L. (39) Here, λrepresents the mean free path of the microscopic component particles in the surrounding matter, while L refers to the characteristic length scale of the levitated particle. The drag force can be analyzed by examin- ing the low-speed motion of a volatile spherical parti- cle in its saturated vapor, regardless of the Knudsen number value. It was assumed that the particle main- tains its spherical shape, compensating for any defor- mations caused by evaporation-condensation processes through the surface-tension force that tends to maintain a spherical shape. This assumption holds true for slow evaporation-condensation processes, allowing the parti- cle radius to be treated as a constant. The schematic diagram can be observed in plot B of Fig. 5, illustrat- ing a spherical particle with radius R0surrounded by a rarefied gas consisting of molecules with mass mM. The number density of molecules is denoted as n∞, while the vapor velocity is denoted as U∞. Here, rrepresents the radius vector from the particle’s center, and vrepresents the velocity vector of the molecules. Typically, the vapor velocity U∞is significantly smaller than the velocity ofthe molecules, resulting in a ratio u∞. u∞=U∞mM 2kBTTH ≪1. (40) is denoted and the molecular distribution can be lin- earized as f(r,v) =f∞[1 + 2 c·u∞+h(r,v)], (41) where f∞is the Maxwell-Boltzmann distribution f∞=n∞mM 2πkBTTH3 2 exp −c2
, (42) withc=vmM 2kBTTH1 2 . h(r,v) represents the disturbance in the distribution function near the particle. It is important to empha- size that the linearization of the velocity distribution is a crucial step in the analysis. However, when the levi- tated particle collides with the surrounding dark matter, the condition given by Eq. (40) fails, and the lineariza- tion operation is not applicable. The temperature TTH in Eq. (40) has the same meaning as described in the pre- vious section, referring to the temperature of the heating bath. A comprehensive investigation of the isothermal drag force can be found in Ref. [39]. Here, we provide a sum- mary of the coefficient results for different cases as fol- lows. When Kn≪1, the molecules in the surrounding matter can be treated as material fragments of the mov- ing medium. This region corresponds to the fundamental concept of continuum description in fluid flows, known as the viscous slip-flow regime. The analytical expression10 for the damping coefficients can be written as γTH=FV mU∞(43) =6π mηgR0(1 +aKn +bKn2+O(Kn3)), where Kn=λ/R 0.ηgrepresents the gas viscosity at temperature TTH, while aandbare numerical coefficients that depend on the momentum accommodation during collisions between the molecules and the spherical parti- cle. The damping coefficient can be related to the Stokes formula used in statistical physics. When Kn≫1, the surrounding molecules can be treated as freely moving, leading to what is known as the free-molecular regime. In this regime, the damping coefficients are γTH=FF mU∞(44) =8π1 3 3mR2 0PTHmM 2kBTTH1 2 × 2 +ατ−αn32−π(9−αE) 32−π(1−αn) (9−αE) , where PTHdenotes the pressure of the surrounding gas. The momentum and energy accommodation coefficients αr, αn, αEcan be found in tables listed in Ref. [39] and the accompanying references. It can be observed that ad- justing the pressure leads to different thermal damping coefficients. Since nanoparticle levitation experiments are typically conducted in ultra-high vacuum, this prop- erty can be utilized to mitigate or eliminate thermal damping and fluctuations in the levitation, as demon- strated in the next section. B. The photon recoil in the optical levitation The trapping potential in the levitation is achieved through the use of a strongly focused laser, which im- parts a photon kick to the levitated particle. Subse- quently, fluctuations occur due to classical noise in the laser intensity, causing modulation of the trapping po- tential. Additionally, a damping force arises during colli- sions to prevent the energy of the levitated particle from diverging. The damping coefficient can be deduced from the fluctuations of the trapping potential. The Hamilto- nian for the trapped particle is given by [40] H=p2 2m+1 2mω2 0(1 +ϵ(t))q2, (45) where ϵ(t) represents the newly introduced time- dependent factor that accounts for the fluctuation of the laser intensity. The rate is reduced by the ratio of the parametric resonance linewidth, ϵ0ω0, to the band- width of the fluctuations, ∆ ω0. Here, ϵ0denotes the root-mean-square fractional fluctuation in the spring con- stant k. Consequently, the rate scales as ω2 0S, withS≃ϵ2 0/∆ω0representing the noise spectral density in fractions squared per rad/sec. The damping rate can be computed classically, which is consistent with expectations for a harmonic-oscillator potential. First-order time-dependent perturbation the- ory can be employed to calculate the average transition rates between quantum states of the trap. The time- evolving perturbation of the quantum-mechanical Hamil- tonian given by Eq. (45) is H′(t) =1 2ϵ(t)m2ω2 0q2. (46) The damping of the levitated particle can be understood as the transition from a higher energy level |n⟩to a lower level|m⟩induced by the perturbation. The rate can be calculated by taking the average over a time interval T Rn→m=1 T−i ℏZT 0dtH′ mn(t)eiωmnt2 (47) =mω2 0 2ℏ2Z∞ −∞dτeiωmnτ⟨ϵ(t)ϵ(t+τ)⟩⟨m|q2|n⟩2. Here, we assume that the averaging time Tis short com- pared to the time scale over which the level populations vary, but large compared to the correlation time of the fluctuations. This allows the range of τto formally ex- tend to ±∞. ⟨ϵ(t)ϵ(t+τ)⟩=1 TZT 0dtϵ(t)ϵ(t+τ). (48) Using the transition matrix elements of q2andωn±2,n in Eq. (47), we can get the transition rate Rn±2→n=π2ω2 0 16Sk(2ω0)(n+ 1±1)(n±1),(49) in which Skis the one-sided power spectrum of the frac- tional fluctuation in the spring constant Sk(ω0) =2 πZ∞ 0dτcos(ω0τ)⟨ϵ(t)ϵ(t+τ)⟩. (50) The one-sided power spectrum is defined so that Z∞ 0dωSk(ω) =⟨ϵ(t)2⟩=ϵ2 0. (51) Assuming that the levitated particle occupies the state |n⟩with a probability P(n, t) at time t, the average damp- ing rate can be calculated as d⟨E(t)⟩ dt=X nP(n, t)2ℏω0(Rn→n−2−Rn→n+2) =π 2ω2 0Sk(ω0)⟨E(t)⟩. (52) The average energy of the oscillator, denoted as ⟨E(t)⟩, can be expressed as ⟨E(t)⟩=P n(n+1 2)P(n, t)2ℏω0. It11 is evident that the damping coefficient can be extracted from the above equation in the form, d⟨E(t)⟩ dt=−γRE⟨E(t)⟩, (53) where γRE=π 2ω2 0Sk(ω0). (54) Note that the derivation of the damping coefficient in this work differs from that of Ref. [40], where the heat- ing rate is calculated. The rationale behind this work is based on the principle that every damping and fluctua- tion adhere to the fluctuation and dissipation theorem. By applying Eq. (2), the fluctuation correlation function, one can determine the fluctuation (or the heating rate as in Ref. [40]). Consequently, the damping coefficient should be calculated first, which in turn determines the fluctuation. Here, we will refer to the damping coefficient as γRE=ω2 0SRE. (55) in which SREis a newly defined parameter that absorbs all the constants, including ϵ2 0present in the expression. In the levitation experiment, we assume that SREcan be finely adjusted to remain nearly constant within the vicinity of ω0. This allows for the straightforward extrac- tion of the photon recoil contribution to the signature. The recoil temperature TREcan be derived as follows. The force fluctuations acting on the levitated particle and their spectral density, as given in Eq. (2), can be determined using the Wiener-Khinchin theorem SFiFj(ω) =Z∞ −∞D ˆFi(ω)ˆF∗ j(ω′)E dω′(56) =1 2πZ∞ −∞⟨Fi(t)Fj(t+t′)⟩eiωt′dt′ =δijmγREkBTRE π, where ˆFi(ω) represents the Fourier transform of Fi(t). The force exerted on the particle by a focused laser with frequency Ω is given by Fi(Ω) = Pscatti(Ω)/c, where Pi scatt represents the power scattered in the direction i. Therefore, D ˆFi(Ω)ˆF∗ j(Ω′)E =1 c2D ˆPi(Ω)ˆP∗ j(Ω′)E . (57) The dominant source of the fluctuation is the short noise, thus the power spectral density is [41] SFiFj(Ω) =ℏΩ 2πc2Pij scatt(Ω). (58) Compared it with the density Eq. (56), we can get the effective temperature of the recoil of the quanta ℏΩ in idirection TRE=ℏΩ 2mc2kBγREPscatt(Ω) (59) =ℏΩ 2mc2kBω2 0SREPscatt(Ω). There exists a natural linear relationship between TRE andPscatt. As discussed earlier, TREdoes not necessarily equal the temperature of the environment. C. Differences between feedback optical force and Doppler damping force This subsection discusses the differences between the work of Ref. [35] and Ref. [36]. This serves as one of the motivations for our work. Ref. [35] studied the levitodynamics and relaxation of a levitated particle, where cooling is achieved through the equilibrium be- tween stochastic fluctuations and the optical damping force FFB=−ω0ηq2p. However, as shown in Ref. [36], when the feedback loop is activated, an additional damp- ing coefficient γFBappears in the Langevin system. It is conceivable that cooling can be achieved even without the presence of the optical damping force FFB. The de- tails of the cooling are already presented in Fig. 4 and the corresponding discussion in the preceding section. Here, we provide a unified description of the two cooling ef- fects. In fact, the Doppler effect is caused by the motion of the levitated particle, which also alters its position. If a feedback loop is set up in an experiment, neglecting the higher-order O(p2) terms, the damping force can be generally expressed as Fdamping =−G(q2)p . (60) G(q2) is a function of the square of the coordinate, q2. This is due to the motion-induced Doppler effect. It should be noted that the linear term in qdoes not con- tribute to damping. Therefore, the leading two terms in the Taylor expansion are Fdamping ≃ −G(0)p−G′(0)q2p . (61) The first term represents the feedback damping coeffi- cient, while the second term corresponds to the optical damping force. Both the feedback damping and optical damping forces exist and are deterministic. As discussed in the previous section on levitodynamics, it is impor- tant to note that the total damping γTis not necessarily equal to the stochastic damping γS. However, the feed- back damping can dominate the cooling process of the levitation. In this section, we provide a brief introduction to Doppler cooling, which can also occur in the interac- tion between a levitated particle and the radiation back- ground. The feedback mechanism, also known as optical molasses, is a theoretical framework for the laser cooling12 𝜔𝜔𝜔−𝑘𝑣𝜔+𝑘𝑣⟩|1⟩|2⟩|1⟩|2 𝐹=−𝛼𝑣On resonance(a)(b) FIG. 6. The schematic diagram of Doppler cooling involves a two-level atom interacting with a pair of counter-propagating beams, forming a laser with a frequency below the atomic resonance frequency. (a) The atom is stationary. (b) The Doppler effect causes an increase in the frequency of the laser beam propagating in the direction opposite to the atom’s velocity. technique that utilizes the configuration of three orthog- onal pairs of counter-propagating laser beams along the Cartesian axes. As shown in plot (a) of Fig. 6, a station- ary atom in a pair of counter-propagating laser beams experiences no resultant force due to identical scatter- ing from each laser beam. However, for a moving atom, as depicted in plot (b), the Doppler effect causes an in- crease in the frequency of the laser beam propagating op- posite to the atom’s velocity. This Doppler shift brings the light closer to resonance with the atom, increasing the rate of absorption from this beam and resulting in a force that decelerates the atom. The damping coefficient of the feedback can be derived from the molasses force Fmol Fmol=Fscatt(ω−ω0−kv)−Fscatt(ω−ω0+kv) ≃Fscatt(ω−ω0)−kv∂F ∂ω− Fscatt(ω−ω0)−kv∂F ∂ω ≃ −2∂Fscatt ∂ωkv , (62) where Fscattrepresents the scattering force exerted by the incident laser, and kdenotes the wavevector. Therefore, the damping coefficient can be expressed simply as γFB=2k m∂Fscatt ∂ω. (63) The differentiation of the molasses force provides the precise value of the coefficient, which can be found in Ref. [42]. In this paper, we focus on extracting the dark damping coefficient in the steady state without a feed- back loop. Therefore, we do not present the detailed differentiation of the molasses force here.D. The collision between levitated particle and dark matter Evidence of the existence of dark matter arises from the gravitational effects observed in the behavior of galax- ies and clusters, as demonstrated by various astrophysi- cal observations. The accumulation of evidence increas- ingly clarifies that a significant portion of the universe’s matter exists in a non-luminous form, which could be weakly interacting with elements of the standard model (SM) and thus challenging to detect in terrestrial labo- ratories. Direct detection aims to identify signatures of dark matter scattering off a terrestrial target in labora- tory settings. Leveraging the levitation of nanoparticles provides a means to employ macroscopic force sensors for probing long-range interactions between dark and visible matter, including gravitational interactions. However, realizing such ambitious experiments would require sub- stantial advancements beyond the current state of the art. Nevertheless, similar concepts for searching for dark matter that might interact through stronger long-range interactions are already feasible. Naively, one might expect the damping rate caused by dark matter collisions to be the same as the Kn≫1 case of the isothermal drag force since the mean free path of dark matter is much greater than the size of the levi- tated particle. However, a detailed analysis reveals that damping from dark matter differs from the isothermal drag force. Although the dark matter distribution can be approximated as having a similar Maxwell-Boltzmann distribution, the linearization condition in Eq. (41) fails as the vapor velocity of the dark matter becomes com- parable to the dark matter particle velocity. More de- tails regarding the velocity distribution can be found in Ref. [43, 44].13 Levitated particle𝒖 FIG. 7. The sketch map of the dark matter scattering on a levitated particle from in one direction. The red ball is the final scattering state of the DM. Counting the transfer momentum to the levitated particle gives the damping force of the incident DM flux. Next, we provide a brief summary of the kinetic se- tups of galactic dark matter particles, which are grav- itationally bound to the halo of our galaxy. The local distribution of dark matter is formulated using the sub- halo model of the galaxy, with a density of approximately ρ= 0.3,GeV/cm3. Assuming only one type of dark mat- ter particles, the numerical density of these particles near Earth decreases as the dark matter mass mχincreases. The scattering rate of dark matter depends on time due to variations in the dark matter flux on Earth caused by the Earth’s motion around the Sun. Consequently, the dark matter signals are expected to exhibit annual mod- ulation. The velocity of dark matter at Earth’s location is anticipated to be a few hundred km/s, limited by the galactic escape velocity. In the galactic rest frame, the ve- locity distribution follows the Maxwell-Boltzmann form, with the most probable velocity v0typically chosen as 220 km/s. While the circular velocity of the Sun around the galaxy’s center is approximately 240 km/s, and the circular velocity of the Earth around the Sun is about 30 km/s. Although the relative velocity of the levitated particle with respect to Earth can be much smaller, the drift velocity of the dark matter flux is several hundred kilometers per second. Therefore, the motion of the solar system implies that the linearization of the expansion of the drift wind U∞in fluid dynamics, as in Eq. (41), is no longer applicable. Defining the pressure of the dark mat- ter flux also becomes challenging. Consequently, we need to find an alternative approach to calculate the damping coefficient caused by dark matter. Indeed, the picture of a saturated particle in a liquid, along with the concept of isothermal drag, belongs to the realm of macroscopic physical systems within fluid dy- namics. On the other hand, when considering dark mat- ter scattering on the levitated particle, we need to delve into concepts from fundamental particle physics, such asscattering cross sections, particle masses, and coupling strengths. These concepts are formulated within the framework of quantum scattering theory. While there are similarities between these two physical pictures, a direct calculation from quantum scattering theory to the macro- scopic levitated particle is required. Therefore, we move away from fluid approaches in the subsequent study. The damping coefficient can be straightforwardly de- rived from the collisions between the dark matter par- ticle and the levitated particle. Let’s consider a one- dimensional collision as an example to illustrate the de- tails. As depicted in Fig. 7, both the levitated particle and the ambient dark matter are moving along the x axis. The velocity of the dark matter particle in the flux follows a Maxwell-Boltzmann distribution. f(vx, TDM) =mχ 2πkBTDM1 2 exp −mχv2 x 2kBTDM .(64) Here the effective temperature corresponds to the average velocity v0of the DM TDM=πmχ 8kBv2 0. (65) The levitated particle will experience a damping force resulting from the transfer of momentum in the opposite direction from the dark matter particles. The strength of this force should be proportional to the velocity of the levitated particle, denoted as u, assuming it is much smaller than the velocities of the dark matter particles. However, it is important to note that in this context, u cannot be equated to U∞as in Eq. (40) of fluid dynamics. In Fig. 7, it is evident that the momentum transferred to the levitated particle can be computed by considering the scattering of dark matter particles within the final solid angle dΩ. The number density of the incident dark14 matter particles is given by ρχ/mχ. Utilizing the differ- ential elastic cross section d σ/dΩ, the momentum trans- ferred to the levitated particle during a time interval ∆ t can be expressed as ∆P=Z Zρχ mχmχ 2πkBTDM1 2 exp −mχv2 x 2kBTDM ×dσ dΩ|vx−u|∆t×mχvx(1−cosθ) dvxdΩ =−ρχ∆tmχ 2kBTDM−11 4 1 + Erfrmχ 2kBTDMu −Erfrmχ 2kBTDMuZdσ dΩ(1−cosθ) dΩ ≃ −ρχ∆tmχ 2kBTDM−1q mχ 2kBTDMu √πσT =−ρχu∆tπmχ 2kBTDM−1 2 σT. (66) In the last line of Eq. (66), we employ Taylor expansion to obtain the dominant term of the transferred momentum. Here, σTrepresents the transfer cross section defined for collisions between dark matter particles and the levitated particle σT=Zdσ dΩ(1−cosθ) dΩ. (67) We can see that the damping coefficient in one dimension is γ1D DM=−∆P ∆t1 mu=ρχ mπmχ 2kBTDM−1 2 σT. (68) The 3D result of γDMcan be obtained by using the afore- mentioned result. The momentum transfer from the dark matter particles to the levitated particle within an in- finitesimal solid angle dΩ′/2πon a hemispherical surface can be approximated using the one-dimensional result given by Eq. (68). By integrating over dΩ′/2πfor all momentum components along the direction of motion of the levitated particle, we can derive the 3D result as fol- lows γDM=Z γ1D DMcos2θ′dΩ′ 2π(69) =γ1D DM 3=ρχ 3mπmχ 2kBTDM−1 2 σT. Substituted the effective temperature TDMEq. (65), we can get a very simple expression of the damping coeffi- cient γDM=ρχv0 6mσT. (70) The damping coefficient mentioned above is suppressed by the mass of the levitated particle. However, as men- tioned earlier, the transfer scattering cross section σTrepresents the scattering between dark matter and a macroscopic particle. The connection between this inter- action and the interaction between DM and fundamental particles can be elucidated as follows. When comparing with the isothermal drag force dis- cussed in the previous subsection, we observe that the transfer cross section σTin Eq. (70) corresponds to the areaR2 0in Eq. (44). This area provides a macroscopic de- scription of the scattering process. The scattering cross section and the mass of the DM are typically the two most significant parameters in dark matter detection. How- ever, in the case of the levitation experiment, there are some misunderstandings regarding the interactions be- tween the DM and the levitated particle due to the com- posite nature of the levitation system. The levitated par- ticle is composed of millions of microscopic molecules or atoms, depending on the scale of the levitation. The de- tection of levitation aims to observe the collective motion of these composite particles. Therefore, the differential cross section in Eq. (66) and the transfer cross section σT in Eq. (67) do not represent fundamental dark matter interactions as commonly understood in the literature, which refer to interactions between elementary particles. To establish the relationship between fundamental in- teractions and levitation interactions, we need to derive this connection. The key aspect of this derivation lies in the coherence that exists from the fundamental inter- action to the levitation level. The collective movement of the levitated particles can enhance the scattering rate through coherent effects. This enhanced scattering rate gives rise to a characteristic scattering pattern known as the static structure factor. The static structure factor results from the collective interference of waves scattered by particles in the system. This interference is sensitive to the relative separation between the particles, and the static structure factor can be expressed as the spatial Fourier transform of the particle structure, represented by the density-density correlation function. Let’s consider the dark interaction with the nucleon as an example. The scattering rate can be derived from the amplitude Mχn, which originates from the funda- mental dark interaction. The scattering of the levitated particle involves atoms, which are composites composed of nucleons and electrons. Thus, the total scattering is determined at two levels: the distribution of nucleons within the atom and the configuration of atoms within the levitated particle. This physical picture resembles X- Ray diffraction in a crystal, as depicted in Fig. 8 (Ref. [45]). Firstly, the spatial distribution of matter can be formulated as the nucleus form factor f(P) in momen- tum space, which is obtained by performing the Fourier transformation of the spatial distribution. f(P) =Z ρ(r)e−iP·rd3r . (71) It should be noted that an isotropic matter distribution is assumed in this case, simplifying the situation. There- fore, f(P) is solely dependent on P. Typically, f(P) is15 𝑃!"≫𝑠𝑦𝑠𝑡𝑒𝑚𝑃!"≈∆𝑟𝑃!"≪∆𝑟 (a)(b)(c) ∆𝑟 ∆𝑟 ∆𝑟 FIG. 8. S(P) exhibits three distinct regimes depending on the particle spacing in relation to the scattering length. In the case where the scattering length is significantly larger than the particle collection, interference manifests as a sum of nearly equivalent phases, resulting in a proportional relationship between S(P) and the number of particles. When the scattering length is of comparable magnitude to the particle spacing, significant angular variations arise in the scattered intensity. In the scenario where the scattering length is significantly smaller than the particle spacing, the phases become randomized, resulting in interference that causes S(P) to approach unity, with the P-dependence solely determined by the particle form factor. normalized to unity to facilitate the calculation of the to- tal scattering amplitude with the atom. This calculation involves counting the atomic number A. MχA=MχnAf(P). (72) Next, we proceed to calculate the scattering on the lev- itated particles. In this context, the term “levitated par- ticle” represents a generic term encompassing the objects comprising condensed matter. Regardless, the atoms within the collection occupy distinct relative positions, giving rise to interference during the levitation process, particularly in the context of DM scattering. The dis- placement of the atom can be expressed as an ampli- tude, incorporating an additional unitary transformation ˆU= exp ( −ir·P), yielding the total amplitude as a re- sult MχL=all the atomsX iMχnAfi(P)e−iP·ri. (73) Here, rirepresents the position of the center of the ith nucleus. From the formulations above, we can establish a relation between the fundamental interaction and the macroscopic cross section as utilized in Eq. (69) from the damping σT∝ |M χn|2A2(74) × X ifi(P)e−iP·ri! X jfj(P)e−iP·rj ∗ . Integrating |Mχn|2yields the standard scattering cross section σD Tcommonly studied in the literature on darkmatter. Assuming all atoms are identical and possess identical form factors. σT=σD T|f(P)|2A2N 1 NX iX je−iP·(ri−rj) =σD Tf(P)|2A2NS(P), (75) where Nis the total number of the component particle of the levitated particle. The static structure factor is defined by S(P)≡1 N*X iX je−iP·(ri−rj)+ , (76) where the angled brackets indicate an average taken over appropriate ensembles of the structure. The quantity S(P) in Eq. (76) serves as an overall measure of the phase differences in the scattered field, which arise due to the relative separation of the parti- cles. In this context, the scattering wave vector plays a crucial role. The reciprocal of the scattering wave vec- tor represents a significant scattering length scale, given byl= 2π/P, which determines the severity of interfer- ence effects in comparison to the mean particle spacing. This is illustrated in Fig. 8, which displays three distinct regimes. When the scattering length scale is significantly larger than the distances between atoms within the sys- tem (as shown in the left panel of Fig. 8), the phase differences, P·(ri−rj), between neighboring scattered waves are nearly identical, leading to constructive inter- ference. S(P) =1 N*X i,je−iP·(ri−rj)+ ≈N2 N=N. (77)16 Conversely, when the scattering length scale is small com- pared to the particle spacing, (right panel of Fig. 8) the phase differences, P·(ri−rj), between waves scattered by neighboring atoms are randomized and produce S(P) =1 N*X i,je−iP·(ri−rj)+ ≈N N= 1. (78) In any case, the levitation experiments exhibit significant enhancements of the microscopic transfer cross section σD Tby factors of A2N2orA2N. This sensitivity to the microscopic forces is the reason behind the high sensitiv- ity of levitation. Additionally, it should be noted that the measured momentum pin the levitation corresponds to the collective motion of the levitated particles. P=p NA∼p m⟨E⟩ NA. (79) In this scenario, the measured energy should be signif- icantly larger than the energy scale ℏΩ. Consequently, the atomic form factor f(P) can be safely approximated asf(0), and the static structure factor can also be ap- proximated as N. Further studies on DM detection are discussed in the following section. Substituted Eq. (75) to Eq. (70), the dark damping coefficient is γDM=ρχv0 6mσD TA2N|f(P)|2S(P). (80) It is evident that the damping coefficient, despite being suppressed by the mass of the levitated particle, is en- hanced by the number of fundamental particles. When both f(P)→1 and S(P)→Noccur, the coefficient is enhanced to the point of resembling a macroscopic inter- action. This distinction arises from the coherence of the scattering and sets apart dark damping from all other types of damping discussed in the preceding subsections. Comparing all the discussed damping coefficients in this section, we can observe that the distinct properties of these coefficients provide valuable clues for extract- ing each damping effect and its corresponding fluctua- tion from specific experimental setups and appropriate conduction in levitation experiments. In the next sec- tion, we revisit the initial motivation behind our work, which is to extract the dark interaction from the afore- mentioned damping effects. We aim to provide an esti- mation of the DM parameters, including the cross-section and interaction strength, among others. IV. EXPLORATION ON THE DARK DAMPING IN THE LEVITATION EXPERIMENTS A. The linear response and the extraction of the damping coefficient Before delving into the experimental exploration, it is crucial to ascertain the reactions that occur in the lab- oratory. Typically, the linear response of the levitatedparticle to driving forces is assumed. Subsequently, the response function can be derived as a perturbation of the system. The analyticity, causality, and the Kramers- Kronig relation in the response can be found in standard textbooks on general kinetic theory. [46] In this section, we provide a brief introduction to the reaction and dissi- pation based on Eq. (1), incorporating a general damping coefficient ( γ) and temperature ( T). Assuming that the Langevin system was in equilibrium in the distant past, the position at time tis determined by the following equa- tion q(t) =Zt −∞χt(t−t′)F(t′)dt′, (81) where χ(t) =1 2πZ∞ −∞e−iωt˜χ(ω)dω . (82) Then q(t) =Z∞ −∞dt′Z∞ −∞1 2πe−iω(t−t′)F(t′) ˜χ(ω) dω .(83) Substituted it into the equation of motion Eq. (1) F(t) =Z∞ −∞dt′Z∞ −∞1 2π −ω2−iωγ+ω2 0
×e−iω(t−t′)F(t′) ˜χ(ω) dω . (84) We can see that the response function is ˜χ(ω) =1 (ω2 0−ω2)−iωγ(85) χ(t) =1 2πZ∞ −∞e−iωt (ω2 0−ω2)−iωγdω (86) =1 ω∗e−γt 2sin (ω∗s), where ω∗=q ω2 0−γ2 4. The real part of the response function ˜ χ(ω) is the reactive part of the system Re˜χ(ω) =ω2 0−ω2 (ω2 0−ω2)2+γ2ω2. (87) The higher of this function, the more the system will respond to a given frequency. While the imaginary part of the response function is the dissipative part of the system Im˜χ(ω) =γω (ω2 0−ω2)2+γ2ω2, (88) which is proportional to the damping coefficient. For the case of natural fluctuations in the position and velocity of the particle in equilibrium, the stochastic force averages to zero and is assumed to have delta-function time correlation, as depicted in Eq. (2). The average of17 the position and velocity squares obeys the equipartition theorem ⟨q2(t)⟩=kBT mω2 0,⟨p2(t)⟩=mkBT . (89) The power spectrum of q2is the Lorentzian peak ˜Sq2(ω) =2γkBT m1 (ω2 0−ω2)2+γ2ω2. (90) This spectrum can be measured by counting the occupa- tion number in a levitation experiment. The dissipative part of the response function (Eq. (88)) is related to Im˜χ(ω) =m 2kBT˜Sq2(ω). (91) This result suggests that the dissipation caused by driv- ing a system out of equilibrium with an external force is proportional to the power spectrum of the natural fluc- tuations that arise in equilibrium. ∆ω2 ωp ωSqq(ω) FIG. 9. The power spectrum ˜Sq2(ω) is characterized by the peak frequency ωpand the half width ∆ ω2. The power spectrum ˜Sq2(ω) depicted in Fig. 9 can be measured in levitation experiments by counting the oc- cupation number of quanta ℏω0inω-space. It provides comprehensive information that allows for the precise ex- traction of the damping coefficient γand temperature of the levitated particle. Various approaches exist for this purpose, and we adopt the simplest one. The primary measurable parameter is the peak frequency ωp, from which one can readily derive the relation. ω2 p=ω2 0−γ2 2. (92) However, the damping coefficient cannot be obtained directly from this relation since ω0is not a precisely pre- dicted or measured variable in real experimental studies. Therefore, in this paper, we propose measuring the half- width ∆ ω2of the peak, which represents the differencebetween the squared frequencies at half the peak height. This allows us to establish the relation. ω2 0=ω2 ps 1 +∆ω2 2ω2p2 , (93) to determine ω0more precisely. This relationship reveals that, in the presence of a very sharp peak, we can obtain an approximate damping coefficient. γ=∆ω2 2ωp. (94) With the precisely determined coefficient, the tempera- ture can be derived at the peak T=˜Sq2(ωp) ω4 0−ω4 p
γm kB. (95) By utilizing this measured damping coefficient γand temperature T, we can conduct a comprehensive study of the correlation function and the fluctuation dissipation theorem. Additionally, we can search for evidence of dark interaction through the investigation of dark damping. B. The verification of the multi-stochastic force theory and the searching for the dark damping In this subsection, we present the experimental explo- ration of the theory proposed in this work and the search for dark interactions hidden in the noise of the levitation. Firstly, we list all the new ideas proposed in the above context: 1. Multiple sources of stochastic force exist, each inde- pendently adhering the fluctuation-dissipation the- orem. These sources collectively interact with the levitated particle. 2. The cooling feedback mechanism provides a damp- ing coefficient and an optical force that also damps the levitated particle. Consequently, the forces within the feedback mechanism are deterministic. 3. The levitated particle is assigned an effective tem- perature. When the feedback is turned off, the ef- fective temperature differs from the temperature of the environment and is denoted as TS, which may include contributions from dark interactions. 4. The motion of the levitated particle arises from the collective motion of microscopic particles, thereby enabling the dark damping to unveil interactions between elementary and macroscopic particles. The first two items provide physical insights into levita- tion physics, while the other two items represent phys- ical relations and quantities that can be experimentally verified or measured. Fortunately, most of the relations18 discussed in the previous section are linear, allowing the use of linear estimation methods in experimental studies. To demonstrate the effectiveness of linear estimation, the inequality between the stochastic temperature TS and the environment temperature TEis initially verified. In the actual levitation experiment, we can disable the feedback at various environment temperatures TEiwhile keeping all other levitation setups, such as laser intensity, unchanged. Subsequently, we can measure Nsamples of the stochastic temperature TSiby analyzing the power spectrum and simulating the relation given by Eq. (95). By utilizing Eq. (6), we can define a linear relationship TS=θ1+θ2TE. (96) If only TEivaries in the experiment, we can treat θ1and θ2as constants. Non-zero θ1and the inequality θ2̸= 1 serve as verification for our proposal. We define χ2 Tas χ2 T=NX i(TSi−θ1−θ2TEi)2. (97) The minimum χ2 Tcorresponds to the best fit with the experimental data. Therefore, it is necessary to solve partial differential equations ∂χ2 T ∂θ1=NX i(−2) (TSi−θ1−θ2TEi) = 0 , (98) ∂χ2 T ∂θ2=NX i(−2TEi) (TSi−θ1−θ2TEi) = 0 .(99) The Least Square Method gives the best matching result θ1=PN iT2 EiPN iTSi −PN iTEiTSiPN iTEi NPN iT2 Ei −PN iTEi2, θ2=NPN iTEiTSi −PN iTEiPN iTSi NPN iT2 Ei −PN iTEi2. It is evident that θ1= 0 and θ2= 1 when every environ- ment temperature is equal to the stochastic temperature TEi=TSi. Therefore, the agreement of θ1andθ2pro- vides a direct test of the proposal in this work. Our objective is to search for the dark interaction in the damping coefficient. However, detecting the dark interaction requires precise tuning of the experimental setup and conduct. We propose two procedures for con- ducting the experiment and exploring dark damping, considering all the analyzed damping coefficients in the previous section. 1. Firstly, we propose measuring the steady state without feedback in a high vacuum environment. Although cooling and heating measurements are possible, measuring the steady relaxation state 012345678910 012345678910P𝟏 PTH𝛾𝛾!"S𝟏S𝟐S𝟑S = 0P𝟐P𝟑FIG. 10. The sketch map using linear estimation to extract the dark damping γDM. should be easier and more precise. In a high vac- uum environment, where the Knudsen number Kn is much greater than 1, the damping coefficient γTHin Eq. (44) is proportional to the pressure of the ambient gas. The zero pressure limit can be approached using linear estimation. Additionally, by varying the noise spectral density of the opti- cal trapping potential, denoted as SREin Eq. (55), it is possible to control the range of γRE. Then, the damping coefficient and temperature can be measured, and linear estimation can be used to ap- proach the zero optical fluctuation limit. 2. In the absence of optical fluctuation and pressure, non-zero damping coefficients provide upper limits on the transfer cross section between dark matter and the levitated particle. Subsequently, by uti- lizing the static structure factor S(P) and atomic factor f(P), we can obtain constraints on the mi- croscopic transfer cross-section between dark mat- ter and fundamental particles. Fig. 10 illustrates a schematic diagram showcasing the use of linear estimation to extract the dark damping γDM from the precisely measured γS. The actual experiment can be conducted under different vacuum pressures, al- lowing for the measurement of γSat various SREvalues. Linear estimation can then be employed to obtain the value of γSin the limit SRE→0. By performing linear estimation of lim PTH→0γSRE→0 S , the value of γSat zero pressure can be determined. Alternatively, if SREcan be adjusted to the same value, the linear estimation can be initially performed on the vacuum pressure, as depicted in Fig. 10. These two different procedures can serve as a cross-check to validate the final results of the damping coefficient. The final step involves extracting constraints on the dark matter side, such as the transfer cross section σD T,19 based on the previously derived dark damping coefficient γDMfrom Eq. (80). The levitation experiment offers a high-precision sensor that enables the measurement of the coupling between a fundamental particle and a macroscopic particle. Since the focus of this paper is on studying levitodynamics and determining the dark damping based on sensor measurements, the detailed constraints on the cross-section and dark matter mass are beyond the scope of this work. For simplicity, we assume thatσD Tis constant, the atomic form factor f(P) = 1, and the static structure factor S(P) =N. Additionally, we assume identical couplings between the dark matter and nucleons. By substituting the DM density, solar system velocity, and mass of the levitated particle into Eq. (80), the transfer cross section can be obtained. σD T∼<6γDM 220×105s−1m2 proton 0.3GeV ×mlevitated particlecm2. (100) Though our proposal has not been applied in the exper- iments, the error of current measured results in levitation experiments could give us an estimation on the order of the non-zero value of dark damping γDM. Thus we ex- amine the constraints on the transfer cross-section from current levitation experiments by utilizing the results of two experiments. Fused silica (SiO 2) particles are consis- tently employed in these experiments. In the experiment described in Ref. [36], the levitated SiO 2particles have radii on the order of 50 nm, enabling direct measurement of the photon recoil. The rate Γ RE=γREn∞represents the product of the damping coefficient and the total num- ber of quanta in the final steady state. The error in Γ RE is approximately O(0.1) kHz, while n∞is on the order ofO(105). Therefore, an estimation of the limit on γDM can be made, placing it on the order of O(10−3) Hz. By inputting the corresponding parameters for fused silica, we can derive the limit σD T∼<O(10−20)cm2. In the ex- periments described in Ref. [33], the particle radii are on the order of 5 µm, and the error of the Voigt profile is also on the order of O(10−3) Hz. Consequently, the limit on the transfer cross section could be σD T∼<O(10−26)cm2. However, conducting an actual experiment to search for dark damping would yield a much more precise limit. The linear estimation of the damping coefficient would extend to a further non-zero damping limit, potentially serving as evidence of dark matter. V. CONCLUSION If the terrestrial environment is filled with dark mat- ter, the levitation experiences contributions from damp- ing forces and fluctuations. In this paper, we study levitodynamics with multiple stochastic forces, includ- ing thermal collision, photon recoil, feedback, and oth- ers. We assume that all stochastic forces are indepen- dent of each other, following the fluctuation dissipation theorem. We observe that the fluctuation term does notnecessarily correspond one-to-one with the damping co- efficient when the feedback loop is active. Therefore, we introduce two coefficients, γTandγSto differentiate be- tween total damping and stochastic damping. The ra- tio of these coefficients distinguishes our levitodynamics from those with only single stochastic forces. Both feed- back damping and optical damping can cool down the levitated particle. We analyze the energy distribution, effective temperature, and heating and cooling behav- iors. The cooling limit differs from the result presented in Ref. [35], where the limit is determined by the balance between single fluctuation and feedback optical force. In our theory, feedback damping can dominate the cooling limit. Importantly, even in an equilibrium state with- out feedback, the effective temperature of the system is not equal to the ambient or thermal temperature. It is a complex value influenced by thermal collision, photon recoil, and possibly dark matter. We investigate the stochastic sources of forces in our study, including thermal drag, recoil, and feedback. These coefficients are derived from various theories such as fluid dynamics and quantum optics. We discover that DM collisions cannot be treated in the same way as in fluid dynamics due to the failure of linearizing the velocity distribution. It becomes necessary to extract the fundamental interaction between DM and standard model particles. The paper provides a detailed analysis of the relationship between the fundamental transfer cross section and the macroscopic transfer cross section. Al- though the dark damping coefficient is suppressed by the mass of the levitated particle, scattering can be coher- ently enhanced based on the scale of the component mi- croscopic particle, the atomic form factor, and the static structure factor. Therefore, dark damping may offer in- sights into detecting the macroscopic strength of funda- mental particles. Finally, we propose the operations of the levitation ex- periment. By precisely measuring the Lorentzian peak, a much more accurate damping coefficient can be obtained. As the thermal drag and recoil can be linearly estimated to approach zero limits, non-zero results at zero pressure and optical fluctuation can test our theory. Based on the current levitation results, we demonstrate that the fun- damental transfer cross section can be on the order of σD T∼<O(10−26)cm2. If our proposal is implemented in actual experiments, we believe that it can provide deeper constraints and reveal more details about the dark inter- action through levitation experiments. Although this paper only focuses on four types of stochastic forces, our theory can be readily extended to scenarios involving additional stochastic forces. The linear estimation can also be expanded to encompass other relationships found in the additional damping coef- ficients. Extracting the dark damping from the damping coefficient is akin to removing noise in a radio telescope in the 1960s. The cosmic microwave background (CMB) was eventually discovered in the residual noise. Simi- larly, dark damping may serve as an irremovable noise in20 levitodynamics, potentially leading to the revelation of hidden secrets in the future.ACKNOWLEDGMENTS Thanks for very useful discussion with Meng Sun and Bo-Yang Liu. 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1810.11016v2.Time_retarded_damping_and_magnetic_inertia_in_the_Landau_Lifshitz_Gilbert_equation_self_consistently_coupled_to_electronic_time_dependent_nonequilibrium_Green_functions.pdf

Time-retarded damping and magnetic inertia in the Landau-Lifshitz-Gilbert equation self-consistently coupled to electronic time-dependent nonequilibrium Green functions Utkarsh Bajpai and Branislav K. Nikoli´ c∗ Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA The conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe dy- namics of local magnetic moments, viewed as classical vectors of fixed length, with their change assumed to take place simultaneously with the cause. Here we demonstrate that recently devel- oped [M. D. Petrovi´ c et al. , Phys. Rev. Applied 10, 054038 (2018)] self-consistent coupling of the LLG equation to time-dependent quantum-mechanical description of electrons—where nonequi- librium spin density from time-dependent nonequilibrium Green function (TDNEGF) calculations is inserted within a torque term into the LLG equation while local magnetic moments evolved by the LLG equation introduce time-dependent potential in the quantum Hamiltonian of electrons— microscopically generates time-retarded damping in the LLG equation described by a memory kernel which is also spatially dependent. For sufficiently slow dynamics of local magnetic moments on the memory time scale, the kernel can be expanded into power series to extract the Gilbert damping (proportional to first time derivative of magnetization) and magnetic inertia (proportional to second time derivative of magnetization) terms whose parameters, however, are time-dependent in contrast to time-independent parameters used in the conventional LLG equation. We use examples of single or multiple local magnetic moments precessing in an external magnetic field, as well as field-driven motion of a magnetic domain wall (DW), to quantify the difference in their time evolution computed from conventional LLG equation vs. TDNEGF+LLG quantum-classical hybrid approach. The faster DW motion predicted by TDNEGF+LLG approach reveals that important quantum effects, stem- ming essentially from a finite amount of time which it takes for conduction electron spin to react to the motion of classical local magnetic moments, are missing from conventional classical micro- magnetics simulations. We also demonstrate large discrepancy between TDNEGF+LLG-computed numerically exact and, therefore, nonperturbative result for charge current pumped by a moving DW and the same quantity computed by perturbative spin motive force formula combined with the conventional LLG equation. I. INTRODUCTION The conventional Landau-Lifshitz-Gilbert (LLG) equation [1–3] is the cornerstone of numerical micro- magnetics [4] and atomistic spin dynamics [5] where one simulates the classical time evolution of many magnetic units coupled by exchange or magnetostatic interactions. The LLG equation ∂m(r,t) ∂t=−gm(r,t)×Beff(r,t)+λGm(r,t)×∂m(r,t) ∂t, (1) describes time evolution of m(r,t) as the unit vector |m|= 1 of constant length representing the direction of the local magnetization. Here gis the gyromag- netic ratio and Beffis the sum of an external mag- netic field and effective magnetic fields due to magnetic anisotropy and exchange coupling (additional stochastic magnetic field can contribute to Beffto take into ac- count finite temperature effects [6]). The second term on the right-hand side of Eq. (1) is introduced phe- nomenologically to break the time-inversion symmetry, thereby generating a damping mechanism. The conven- tional intrinsic Gilbert damping λGis assumed to be ma- terials specific and, therefore, time-independent parame- ∗bnikolic@udel.eduter. It is typically computed using the so-called breath- ing Fermi surface [7] or torque-torque correlation formu- las [8] within single-particle quantum-mechanical frame- work (additional many-body processes have to be taken into account to make λGfinite in the clean limit at low temperatures [9]). In the original form, Eq. (1) is written for a bulk material as a highly nonlinear partial differen- tial equation. It can also be re-written for a macrospin or a lattice of atomic spins leading to a system of nonlinear ordinary differential equations [5]. In the case of conducting ferromagnets, LLG equation has to be extended by including additional terms, such as: (i) spin-transfer torque [10] T∝/angbracketleftˆs/angbracketright×mdue to injected electrons generating nonequilibrium spin density /angbracketleftˆs/angbracketrightthat is noncollinear to local magnetization; ( ii) additional Gilbert damping, ( g↑↓/4π)m×∂m/∂t, due to pumping of spin currents by the dynamics of m(t) whereg↑↓is the so-called spin-mixing conductance [11]; ( iii) additional nonlocal Gilbert damping [12–18], m×(D·∂m/∂t), due to spin pumping by noncollinear magnetic textures where Dαβ=η/summationtext i(m×∂im)α(m×∂im)βis the 3×3 damping tensor,∂i=∂/∂iandα,β,i∈{x,y,z}; and ( iv) mag- netic inertia [19–25], Im×∂2m/∂t2, of relevance to ul- trafast magnetization dynamics. Like the original Gilbert damping parameter λGin Eq. (1), T,g↑↓,DandIrequire microscopic quantum-mechanical calculations which are often combined [8, 9, 26–31] with first-principles Hamil- tonians of realistic materials. Furthermore, generalizations of LLG equation havearXiv:1810.11016v2 [cond-mat.mes-hall] 6 Dec 20182 been considered to take into account the retardation ef- fects [32, 33] ∂m(r,t) ∂t=t 0dt/prime d3r/primeΓ(r,t;r/prime,t/prime)m(r/prime,t/prime) ×/bracketleftbigg −gBeff(r/prime,t/prime) +λG∂m(r/prime,t/prime) ∂t/prime/bracketrightbigg ,(2) by introducing a memory kernel Γ( r,t;r/prime,t/prime). The mem- ory kernel models space-time correlation between local magnetic moments, i.e., the fact that the cause for the change of local magnetization occurs at time t−t/primeand at position r−r/primewhile the effect at position ris vis- ible at the later time t. It has been specified phe- nomenologically, such as the sum of an instantaneous and time-dependent part which exponentially decays on a characteristic time scale defining the strength of mem- ory [32, 33]. It is also often simplified [33] by considering time-retardation only, Γ( r,t;r/prime,t/prime)→Γ(t,t/prime), so that any space-retardation effects are included only through the effective field Beff(r,t/prime). The time-retardation described by Γ( t,t/prime) is a damping mechanism in addition to well-established mechanisms— the combined effects of spin-orbit coupling and electron- phonon interaction [7, 8]—which govern λGin Eq. (1). However, the magnitude of Γ( t,t/prime) cannot be deduced from purely phenomenological considerations [32, 33]. Instead, the introduction of the memory kernel Γ( t,t/prime) can be justified microscopically [6, 22, 34–37] by us- ingquantum-classical hybrid approaches, where time- dependent quantum formalism is used to compute /angbracketleftˆs/angbracketright(t) which is then fed into the LLG equation, while in turn, lo- cal magnetization from the LLG equation generates time- dependent field in the quantum Hamiltonian of electrons. Although electron dynamics is assumed to be much faster than that of local magnetic moments, it still takes fi- nite time for electron spin to react to new position of m(r,t). This is the fundamental reason for time-retarded damping effects encoded by Eq. (2), which are present even if the intrinsic Gilbert damping in Eq. (1) is van- ishingly small due to small spin-orbit coupling (nonzero λGrequires spin-orbit coupling [7–9] and scales quadrati- cally with it [38]). Since classical micromagnetics simula- tions typically use only the conventional intrinsic Gilbert damping term in Eq. (1), while not considering explicitly the flow of conduction electrons in the presence of mag- netization dynamics, the question arises about the mag- nitude of neglected effects like time-retarded damping in standard simulations of magnetic-field- or current-driven dynamics of noncollinear magnetic textures such as mag- netic domain walls (DWs) [39–47] and skyrmions [48, 49]. Although quantum-classical approaches which auto- matically include time-retardation effects have been dis- cussed previously [6, 22, 34–37], they have been focused on the simples examples where one or two local magnetic moments (pertinent to, e.g., magnetic molecules) inter- act with either closed electronic quantum system [22, 35] (i.e., not attached to macroscopic reservoirs to allow elec- tron spin and charge currents to flow into and from an FIG. 1. Schematic view of two-terminal devices where an infi- nite 1D TB chain, describing electrons quantum-mechanically, is attached to two macroscopic reservoirs while its middle part hosts: (a) single local magnetic moment, initially oriented in the +x-direction, placed in an external magnetic field point- ing along the + z-direction; (b) 11 local magnetic moments (illustration shows 7 of them), initially oriented in the + x- direction, placed in an external magnetic field pointing along the +z-direction; (c) three-site-wide head-to-head magnetic DW whose motion is driven by an external magnetic field pointing in the + x-direction. Electrons within 1D TB chain and classical local magnetic moments interact via the s-dex- change coupling of strength Jsd, and classical local magnetic moments within the DW in (c) additionally interact with each other via the Heisenberg exchange coupling of strength J. external circuit), or open electronic quantum system but employing approximations [6, 34, 36, 37] to obtain analyt- ical solution. Thus, these approaches are not suitable for simulations of spintronic devices containing large number of noncollinear local magnetic moments. Here we employ recently developed [50] numerically exact and, therefore, nonperturbative algorithm combin- ing time-dependent nonequilibrium Green function for- malism [51, 52] with the conventional LLG Eq. (1) (TD- NEGF+LLG) to demonstrate how it effectively generates time-retardation effects, whose memory kernel can be ex- plicitly extracted in terms of TDNEGFs only in some lim- its (such as weak electron-spin/local-magnetic-moment interaction and weak coupling of the active region to macroscopic reservoirs). The paper is organized as fol- lows. Section II A introduces model quantum Hamilto- nian for electronic subsystem and classical Hamiltonian for the subsystem comprised of local magnetic moments. In Sec. II B, we show how the nonequilibrium expectation3 value of spin density /angbracketleftˆs/angbracketrighti(t) =~ 2Tr [(neq(t)−eq)|i/angbracketright/angbracketlefti|⊗], (3) inserted into the LLG Eq. (1) generates a memory ker- nel because of the structure of the nonequilibrium time- dependent density matrix neq(t) obtained from TD- NEGF calculations. Here eqis the grand canonical equi- librium density matrix; = (ˆσx,ˆσy,ˆσz) is the vector of the Pauli matrices; |i/angbracketrightelectron orbital centered on site i; and the operator |i/angbracketright/angbracketlefti|⊗acts in the composite Hilbert spaceH=Horb⊗Hspinof electronic orbital and spin degrees of freedom. In this Section, we also discuss how in the limit of slow magnetization dynamics the memory kernel can be expanded in a Taylor series in order to ex- tract conventional Gilbert damping and magnetic inertia terms, but with time- and spatially-dependent parame- tersλD i(t) andID i(t). In Secs. III A–III C we compare the dynamics of local magnetic moments driven by an exter- nal magnetic field as computed by TDNEGF+LLG vs. conventional LLG simulations for three one-dimensional (1D) examples depicted in Fig. 1(a)–(c), respectively. Sec. III C also compares pumped charge current due to the DW motion as computed by TDNEGF+LLG vs. the widely-used spin motive force (SMF) theory [12, 53] com- bined [54–56] with the conventional LLG equation. We conclude in Sec. IV. II. MODELS AND METHODS A. Coupled quantum and classical Hamiltonians The conduction electron subsystem is modeled by a quantum Hamiltonian H(t) =−γ/summationdisplay /angbracketleftij/angbracketrightˆc† iˆci−Jsd/summationdisplay iˆc† i·Mi(t)ˆci, (4) which is (assumed to be 1D for simplicity) tight-binding (TB) model where electron interacts with magnetic mo- ments localized at sites iand described by the classical vector Mi(t) of unit length. Here ˆ c† i= (ˆc† i↑,ˆc† i↓) is a row vector containing operators ˆ c† iσwhich create an electron of spinσ=↑,↓at sitei; ˆciis a column vector that con- tains the corresponding annihilation operators; γ= 1 eV is the nearest neighbor hopping; and Jsdis thes-dex- change coupling parameter between conduction electrons and local magnetic moments. The active region of de- vices depicted in Fig. 1(a)–(c) consists of 1, 11 and 21 TB sites, respectively. These are attached to the left (L) and right (R) semi-infinite ideal leads modeled by the same Hamiltonian in Eq. (4) but with Jsd= 0 eV. The leads are assumed to terminate into macroscopic reser- voirs kept at the same chemical potential since we do not apply any bias voltage to the devices in Fig. 1(a)–(c).The classical Hamiltonian describing the local mag- netic moments is given by H=−J/summationdisplay ijMi·Mj−µM/summationdisplay iMi·Bi ext−K/summationdisplay i(Mx i)2 −Jsd/summationdisplay i/angbracketleft^s/angbracketrighti·Mi,(5) whereJis the Heisenberg exchange coupling parame- ter;Bi extis the applied external magnetic field; Kis the magnetic anisotropy (in the x-direction) and/angbracketleftˆs/angbracketrightiis the nonequilibrium electronic spin density computed from Eq. (3). B. Time-retarded damping and magnetic inertia in the LLG equation self-consistently coupled to TDNEGF The quantum equation of motion for the nonequilib- rium density matrix of electrons [57, 58] i~∂neq(t) ∂t= [H(t),neq(t)] +/summationdisplay p=L,Ri[Πp(t) +Π† p(t)]. (6) is an example of a master equation for an open (i.e., con- nected to macroscopic reservoirs) quantum system [59] due to the presence of the second term on the right hand side, in addition to standard terms of the von Neumann equation. This term and the density matrix itself can be expressed using TDNEGF formalism [51, 52] as neq(t) =1 iG<(t,t/prime)|t=t/prime, (7) Πp(t/prime) =t/prime −∞dt1[G>(t/prime,t1)Σ< p(t1,t/prime)− G<(t/prime,t1)Σ> p(t1,t/prime)].(8) The central quantities of the TDNEGF formalism are the retarded Gr,σσ/prime ii/prime(t,t/prime) =−iΘ(t−t/prime)/angbracketleft{ˆciσ(t),ˆci/primeσ/prime(t)}/angbracketright and the lesser G<,σσ/prime ii/prime(t,t/prime) =i/angbracketleftˆc† i/primeσ/prime(t/prime)ˆciσ(t)/angbracketrightGreen functions (GFs) which describe the available density of states and how electrons occupy those states, respec- tively. In addition, it is also useful to introduce the greater GF, G>(t,t/prime) = [G<(t/prime,t)]†, and the advanced GF,Ga(t,t/prime) = [Gr(t,t/prime)]†. The current matrices Πp(t) make it possible to compute directly [57, 58] charge cur- rent Ip(t) =e ~Tr[Πp(t)], (9) and spin current ISα p(t) =e ~Tr[ˆσαΠp(t)], (10)4 in the L and R semi-infinite leads. The equation of mo- tion for the lesser and greater GFs is given by i~∂G>,<(t,t1) ∂t=H(t)G>,<(t,t1)+ +∞ −∞dt2/bracketleftbigg Σr tot(t,t2)G>,<(t2,t) +Σ>,< tot(t,t2)Ga(t2,t)/bracketrightbigg , (11) where Σr,>,< tot (t,t2) =/summationtext p=L,RΣr,>,< p (t,t2) and Σr,>,< p (t,t2) are the lead self-energy matrices [52, 57, 58]. The classical equation of motion for the magnetic mo- ment localized at site iis the Landau-Lifshitz equation ∂Mi(t) ∂t=−gMi(t)×Beff i(t), (12) where the effective magnetic field is Beff i(t) =−1 µM∂H/∂MiandµMis the magnitude of the magnetic moment [5]. The full TDNEGF+LLG framework [50], which we also denote as TDNEGF LLG, consists of self- consistent combination of Eq.(6) and (12) where one first solves for the nonequilibrium electronic spin density in Eq.(3), which is then fed into Eq. (12) to propagate local magnetic moments Mi(t) in the next time step. Evolving neq(t) via Eq. (6) requires time step δt= 0.1 fs for nu- merical stability, and we use the same time step to evolve LLG or Landau-Lifshitz equations for Mi(t). These up- dated local magnetic moments are fed back into the quan- tum Hamiltonian of conduction electron subsystem in Eq. (6). Thus obtained solutions for Mi(t),/angbracketleftˆs/angbracketrighti(t),Ip(t) andISαp(t) are numerically exact. For testing the im- portance of the self-consistent feedback loop, we also use TDNEGF←LLG where TDNEGF is utilized to obtain Ip(t) andISαp(t) while the local magnetic moments are evolved solely by the conventional LLG Eq. (1), i.e., by usingJsd≡0 in Eq. (5) but Jsd/negationslash= 0 is used in Eq. (4). In the weak-coupling limit [34, 60] (i.e., small Jsd) for electron-spin/local-magnetic-moment interaction it is possible to extract explicitly the generalized LLG equa- tion with a memory kernel. For this purpose we use the following expansions in the powers of small Jsd neq(t) =∞/summationdisplay n=0n(t)Jn sd, (13) Πp(t/prime) =∞/summationdisplay n=0Π(n) p(t/prime)Jn sd, (14) Gr,a,>,<(t/prime,t1) =∞/summationdisplay n=0Gr,a,>,< n (t/prime,t1)Jn sd. (15)In Appendix A, we show how to combine Eqs.(6), (11), (13), (14) and (15) to obtain the perturbative equation ∂Mi(t) ∂t=−g/bracketleftbigg Mi(t)×Beff,0 i(t)+ J2 sd µM/summationdisplay p=L,RMi(t)×+∞ −∞dt/prime/primeMi(t/prime/prime){Kp i(t/prime/prime,t)+Kp∗ i(t/prime/prime,t)}/bracketrightbigg , (16) for the dynamics of each local magnetic moment at site i, by retaining only the terms linear in Jsdin Eqs. (13)–(15). Here Beff,0 i≡−1 µM∂H0/∂Mi,H0is the classical Hamil- tonian in Eq. (5) with Jsd≡0 and Kp i(t/prime/prime,t) is defined in Appendix A. The physical origin [35] of time-retardation effects described by the second term in Eq. (16) is that, even though electron dynamics is much faster than the dynamics of local magnetic moments, the nonequilibrium spin density in Eq. (3) is always behind Mi(t) and, there- fore, never parallel to it which introduces spin torque term into the Landau-Lifshitz Eq. (12). In other words it takes finite amount of time for conduction electron spin to react to the motion of classical local magnetic mo- ments, so that nonequilibrium electrons effectively me- diate interaction of Mi(t) with the same local magnetic moment at time t/prime< t. In the full TDNEGF+LLG, such retardation effects are mediated by the nonequilib- rium electrons starting at site iat timet/primeand returning back to the same site at time t > t/prime, while in the per- turbative limit the same effect is captured by the second term in Eq. (16). The perturbative formula Eq. (16) is expected [35] to breakdown after propagation over time t∼~/Jsd. Further approximation to Eq. (16) can be made by considering sufficiently slow dynamics of local magnetic moments so that higher order terms in the Taylor series Mi(t/prime/prime)≈Mi(t)+∂Mi(t) ∂t(t/prime/prime−t)+1 2∂2Mi(t) ∂t2(t/prime/prime−t)2+..., (17) can be neglected. By defining the following quantities λD p,i(t)≡+∞ −∞dt/prime/prime(t/prime/prime−t)[Kp i(t/prime/prime,t) + K∗p i(t/prime/prime,t)],(18) and ID p,i(t)≡1 2+∞ −∞dt/prime/prime(t/prime/prime−t)2[Kp i(t/prime/prime,t) + K∗p i(t/prime/prime,t)],(19) and by retaining terms up to the second order in Eq. (17)5 we obtain the conventionally looking LLG equation ∂Mi(t) ∂t=−g/bracketleftbigg Mi(t)×Beff,0 i(t)+ J2 sd µM/braceleftbigg/summationdisplay p=L,RλD p,n(t)/bracerightbigg Mi(t)×∂Mi(t) ∂t+ J2 sd µM/braceleftbigg/summationdisplay p=L,RID p,i(t)/bracerightbigg Mi(t)×∂2Mi(t) ∂t2/bracketrightbigg .(20) However, the Gilbert damping term prefactor λD i(t) =J2 sd µM/summationdisplay p=L,RλD p,i(t), (21) and the magnetic inertia term prefactor ID i(t) =J2 sd µM/summationdisplay p=L,RID p,i(t), (22) in Eq. (20) are now time- and position-dependent. This is in sharp contrast to conventional LLG Eq. (1) em- ployed in classical micromagnetics where Gilbert damp- ing and magnetic inertia prefactors are material specific constants. III. RESULTS AND DISCUSSION A. Single local magnetic moment in an external magnetic field To compare the dynamics of local magnetic moments in full TDNEGF+LLG quantum-classical simulations vs. conventional LLG classical simulations, we first consider a well-known example [5] for which the conventional LLG equation can be analytically solved—a single local mag- netic moment which at t= 0 points along the + x- direction and then starts to precesses due to an external magnetic field pointing in the + z-direction. Its trajectory is given by [5] Mx(t) = sech/parenleftbigggλGB 1 +λGt/parenrightbigg cos/parenleftbigggB 1 +λ2 Gt/parenrightbigg ,(23a) My(t) = sech/parenleftbigggλGB 1 +λGt/parenrightbigg sin/parenleftbigggB 1 +λ2 Gt/parenrightbigg ,(23b) Mz(t) = tanh/parenleftbigggλGB 1 +λGt/parenrightbigg , (23c) where B= (0,0,B) is the applied external mag- netic field. Thus, if the conventional intrinsic Gilbert damping parameter is set to zero, λG= 0, then the local magnetic moment precesses steadily around thez-axis with Mz≡0. On the other hand, for nonzero λG>0, the local magnetic moment FIG. 2. (a) Time dependence of tanh−1(Mz) for a sin- gle local magnetic moment in Fig. 1(a) obtained from TD- NEGF+LLG simulations. Colors red to blue indicate in- creasings-dexchange coupling in steps of 0 .1 eV, ranging fromJsd= 0 eV toJsd= 1.9 eV. (b) The dynamical Gilbert damping parameter in Eq. (21) extracted from panel (a) as a function of Jsd. (c) Time dependence of Mzcomponent for a single local magnetic moment in Fig. 1(a) at large Jsd= 2.0 eV exhibits nutation as a signature of magnetic in- ertia. To generate fast magnetization dynamics and reduce simulation time, we use an unrealistically large external mag- netic field of strength B= 1000 T. The conventional intrinsic Gilbert damping parameter is set to zero, λG= 0, and the Fermi energy is EF= 0 eV. will relax towards the direction of magnetic field, i.e., lim t→∞(Mx(t),My(t),Mz(t)) = (0,0,1). Thus, such damped dynamics is signified by a linear tanh−1(Mz) vs. time dependence. Figure 2 plots results of TD- NEGF+LLG simulations for the same problem. Even though we set conventional intrinsic Gilbert damping to zero,λG= 0, Fig. 2(a) shows linear tanh−1(Mz) vs. time, independently of the strength of s-dexchange cou- pling as long as Jsd.2 eV. This means that the lo- cal magnetic moment is experiencing (time-independent) dynamical Gilbert damping λD∝J2 sd, in accord with Eq. (21) and as shown in Fig. 2(b), which is generated solely by the TDNEGF part of the self-consistent loop within the full TDNEGF+LLG scheme. ForJsd&2 eV, the dynamics of the local magnetic mo- ment also exhibits nutation [35], as shown in Fig. 2(c), which is the signature of the magnetic inertia [19–24] term∝Mi×∂2Mi/∂t2in Eq. (20). Thus, nutation be- comes conspicuous when the dynamics of the local mag- netic moments is sufficiently fast, so that ∂2Mi/∂t2is large, as well as when the interaction between the itiner- ant and localized spins is sufficiently large.6 FIG. 3. TDNEGF+LLG-computed trajectories (Mx(t),My(t),Mz(t)) on the Bloch sphere of local magnetic moment in the setup of Fig. 1(b) at: (a) site 1; and (c) site 6. The total number of local magnetic moments is N= 11, and they do not interact with each other via exchange coupling [i.e., J= 0 eV in Eq. (5)]. Panels (b) and (d) show the corresponding time dependence of Mzcomponent from panels (a) and (c), respectively. The external magnetic field isB= 1000 T, and the s-dexchange coupling strength Jsd= 0.1 eV is nonperturbative in this setup, therefore, notallowing us to extract explicitly the dynamical Gilbert damping parameter from Eq. (21). The conventional intrinsic Gilbert damping parameter is set to zero, λG= 0, and the Fermi energy is EF= 0 eV. B. Multiple exchange-uncoupled local magnetic moments in an external magnetic field In order to examine possible spatial dependence of the dynamical Gilbert damping parameter or emergence of dynamical exchange coupling [61, 62] between local mag- netic moments, we consider a chain of N= 11 magnetic moments which do not interact with each other ( J= 0) but interact with conduction electron spin ( Jsd/negationslash= 0), as illustrated in Fig. 1(b). At t= 0, all magnetic moments point in the + x-direction while the external magnetic field is in the + zdirection, and the conventional intrin- sic Gilbert damping is set to zero, λG= 0. Figures 3(a) and 3(c) show the trajectory of selected local magnetic moments ( i= 1 and 6) on the Bloch sphere forJsd= 0.1 eV. In contrast to single local mag- netic moment in Fig. 2(a), for which tanh−1(Mz) vs. time is linear using Jsd= 0.1 eV, we find that in case of multiple exchange-uncoupled magnetic moments this is no longer the case, as demonstrated by Figs. 3(b) and 3(d). Hence, the trajectory followed by these local mag- netic moments cannot be described by Eq. (23) so that FIG. 4. (a) TDNEGF+LLG-computed time dependence of Mzcomponent of local magnetic moment on sites 1, 3 and 6 in the setup of Fig. 1(b) with a total of N= 11 moments. (b) Po- sition dependence of the dynamical Gilbert damping param- eter in Eq. (21). The external magnetic field is B= 1000 T, and thes-dexchange coupling strength Jsd= 0.01 eV is per- turbative in this setup, therefore, allowing us to extract the dynamical Gilbert damping explicitly from Eq. (21). The con- ventional intrinsic Gilbert damping parameter is set to zero, λG= 0, and the Fermi energy is EF= 0 eV. the conventional-like Gilbert damping parameter cannot be extracted anymore. Thus, such a nonstandard damp- ing of the dynamics of local magnetic moments originates from time-dependence of the dynamical damping param- eterλD iin Eq. (21). Figure 4(a) shows tanh−1(Mz) vs. time for selected local magnetic moments ( i= 1,3 and 6) and smaller Jsd= 0.01 eV. Although all local magnetic moments fol- low linear tanh−1(Mz) vs. time, as predicted by the solution in Eq. (23c) of the conventional LLG equation, the dynamical Gilbert damping extracted from Eq. (23) changes from site to site as shown in Fig. 4(b). Further- more, the linear tanh−1(Mz) vs. time relation breaks down for times t&50 ps at specific sites, which then pre- vents extracting time-independent λD iat those sites. C. Magnetic field-driven motion of a domain wall composed of multiple exchange-coupled local magnetic moments In order to examine difference in predicted dynam- ics of exchange-coupled local magnetic moments by TD- NEGF+LLG framework vs. conventional LLG equa- tion, we consider the simplest example of 1D head-to- head magnetic DW depicted in Fig. 1(c). Its motion is driven by applying an external magnetic field in the + x- direction. Some type of damping mechanism is crucial for the DW to move, as demonstrated by solid lines in Fig. 5(e)–(h), obtained by solving the conventional LLG equation with λG= 0, which show how local magnetic moments precess around the magnetic field but without net displacement of the center of the DW. On the other hand, even though we set λG= 0 in TD- NEGF+LLG simulations in Fig. 5(a)–(d), the center of the DW moves to the right due to dynamically generated7 FIG. 5. (a)–(d) TDNEGF+LLG-computed snapshots of head-to-head DW in the setup of Fig. 1(c) driven by an external magnetic field of strength B= 100 T pointing in the + x-direction, in the absence ( λG= 0) or presence ( λG= 0.01) of the conventional intrinsic Gilbert damping. Panels (e)–(h) show the corresponding snapshots computed solely by the conventional LLG Eq. (1) where in the absence ( λG= 0) of the conventional intrinsic Gilbert damping the DW does not move at all. The Heisenberg exchange coupling between local magnetic moments is J= 0.01 eV;s-dexchange coupling between electrons and local magnetic moments is Jsd= 0.1 eV; magnetic anisotropy (in the x-direction) is K= 0.01 eV; and the Fermi energy of electrons is EF=−1.9 eV. The magnetic field is applied at t= 2 ps, while prior to that we evolve the conduction electron subsystem with TDNEGF until it reaches the thermodynamic equilibrium where all transient spin and charge currents have decayed to zero. time-retarded damping encoded by the memory kernel in Eq. (16). Including the conventional intrinsic Gilbert damping,λG= 0.01 as often used in micromagnetic sim- ulations of DW along magnetic nanowires [43, 44, 63], changes only slightly the result of TDNEGF+LLG sim- ulations which demonstrates that the effective dynam- ical Gilbert damping (which is also time-dependent) is about an order of magnitude larger than λG. This is also reflected in the DW velocity being much larger in TD- NEGF+LLG simulations with λG= 0 in Fig. 5(a)–(d) than in the conventional LLG equation simulations with λG= 0.01 in Fig. 5(e)–(h). It has been predicted theoretically [12, 53, 64–68] and confirmed experimentally [69] that a moving DW will pump charge current even in the absence of any applied bias voltage. The corresponding open circuit pumping voltage in the so-called spin motive force (SMF) the- ory [12, 53] is given by VSMF=1 G0 jxdx, (24a) jα(r) =Pσ0~ 2e[∂tm(r,t)×∂αm(r,t)]·m(r,t),(24b) wherejxis the pumped local charge current along the x-axis. Here σ0=σ↑+σ↓is the total conductivity; P= (σ↑−σ↓)/(σ↑+σ↓) is the spin polarization of the ferromagnet; and ∂t=∂/∂t. Equation (24) is typicallycombined [54–56] with classical micromagnetics which supplies Mi(t) that is then plugged into the discretized version [50] jx(i)∝1 a[∂tMi(t)×(Mi+1(t)−Mi(t))]·Mi(t) ∝1 a[∂Mi(t)×Mi+1(t)]·Mi(t). (25) of Eq. (24b). We denote this approach as SMF ←LLG, which is perturbative in nature [67, 70] since it considers only the lowest temporal and spatial derivatives. On the other hand, the same pumping voltage can be computed nonperturbatively VTDNEGF =Ip(t) G(t), (26) using TDNEGF expression for charge current in lead p in Eq. 9, where TDNEGF calculations are coupled to LLG calculations either self-consistently (i.e., by using TDNEGFLLG) or non-self-consistently (i.e., by us- ing TDNEGF←LLG). Here, G(t) is the conductance computed using the Landauer formula applied to two- terminal devices with a frozen at time ttexture of local magnetic moments. Figures 6(a) and 6(b) plot the pumping voltage cal- culated by TDNEGF LLG for DW motion shown in Fig. 5(a)–(d) in the absence or presence of conventional8 FIG. 6. Time dependence of pumping voltage generated by the DW motion depicted in Fig. 5(a)–(d) for: (a) λG= 0; (b)λG= 0.01. In panels (a) and (b) local magnetic mo- ments evolve in time by the full TDNEGF+LLG framework where the arrows indicate how TDNEGF sends nonequilib- rium electronic spin density into the LLG equation which, in turn, sends trajectories of local magnetic moments into TD- NEGF. Time dependence of pumping voltage generated by DW motion depicted in Fig. 5(e)–(h) for: (c) λG= 0; (d) λG= 0.01. In panels (c) and (d) local magnetic moments evolve in time using the conventional LLG equation which sends their trajectories into either TDNEGF (green) or SMF formulas (blue) in Eq. (26) or Eq. (24), respectively, to obtain the corresponding pumping voltage. Gilbert damping, respectively. The two cases are virtu- ally identical due to an order of magnitude larger dynam- ical Gilbert damping that is automatically generated by TDNEGFLLG in both Figs. 6(a) and 6(b). The nonperturbative results in Figs. 6(a) and Fig. 6(b) are quite different from SMF ←LLG predictions in Figs. 6(c) and Fig. 6(d), respectively. This is due to both failure of Eqs. (24) and (25) to describe noncoplanar and non- collinear magnetic textures with neighboring local mag- netic moments tilted by more than 10◦[50] and lack of dynamical Gilbert damping in SMF ←LLG simula- tions [54–56]. The latter effect is also emphasized by the inability of TDNEGF ←LLG in Figs. 6(c) and Fig. 6(d) to reproduce the results of self-consistent TDNEGF  LLG in Figs. 6(a) and Fig. 6(b), respectively. IV. CONCLUSIONS In conclusion, we delineated a hierarchy of theoret- ical descriptions of a nonequilibrium quantum many- body system in which conduction electron spins inter- act with local magnetic moments within a ferromagneticlayer sandwiched between normal metal electrodes. On the top of the hierarchy is a fully quantum approach, for both electrons and local magnetic moments, whose computational complexity (using either original spin op- erators [71, 72] for local magnetic moments, or their mapping to bosonic operators in order to enable ap- plication of many-body perturbation theory within the NEGF formalism [73]) makes it impractical for systems containing large number of local magnetic moments. The next approach in the hierarchy is computation- ally much less expensive quantum-classical hybrid [74] based on self-consistent coupling [50] of TDNEGF (which can be implemented using algorithms that scale linearly with both system size and simulation time [52, 58, 75]) with classical LLG equation for local magnetic moments. Such TDNEGF+LLG approach is numerically exact and, therefore, nonperturbative in the strength of electron- spin/local-magnetic-moment interaction, speed of local magnetic moment dynamics and degree of noncollinearity between them. Even though electron dynamics is much faster than localized spin dynamics, the most general sit- uation cannot be handled by integrating out [6, 34] the conduction electron degrees of freedom and by focusing only on the LLG-type equation where a much larger time step can be used to propagate spins only. Nevertheless, in the limit [34, 60] of weak electron- spin/local-magnetic-moment interaction [i.e., small Jsd in Eqs. (4) and (5)] one can derive analytically a type of generalized LLG equation [34–37] for each local mag- netic moment which is next approach in the hierarchy that sheds light onto different effects included in the nu- merically exact TDNEGF+LLG scheme. Instead of the conventional Gilbert damping term in Eq. (1), the gen- eralized LLG equation we derive as Eq. (16) contains a microscopically determined memory kernel which de- scribes time-retardation effects generated by the coupling to TDNEGF. Fundamentally, the memory kernel is due to the fact that electron spin can never follow instanta- neously change in the orientation of the local magnetic moments [35]. In the limit of slow dynamics of local magnetic moments, one can further expand the memory kernel into a Taylor series to obtain the final approach within the hierarchy whose LLG Eq. (20) is akin to the conventional one, but which contains both Gilbert damp- ing (proportional to first time derivative of local mag- netization) and magnetic inertia terms (proportional to second time derivative of local magnetization) with time- dependent parameters instead of usually assumed mate- rials specific constants. Using three simple examples—single or multiple local magnetic moments precessing in an external magnetic field or magnetic-field-driven magnetic DW motion— we demonstrate the importance of dynamically induced damping which operates even if conventional static Gilbert damping is set to zero. In the case of field- driven magnetic DW motion, we can estimate that the strength of dynamical damping is effectively an order of magnitude larger than typically assumed [43, 44, 63] con-9 ventional static Gilbert damping λG/similarequal0.01 in classical micromagnetic simulations of magnetic nanowires. In ad- dition, we show that charge pumping by the dynamics of noncoplanar and noncollinear magnetic textures, which is outside of the scope of pure micromagnetic simulations but it is often described by combining [54–56] them with the SMF theory formula [12, 53], requires to take into ac- count both the dynamical Gilbert damping and possiblylarge angle between neighboring local magnetic moments in order to reproduce numerically exact results of TD- NEGF+LLG scheme. ACKNOWLEDGMENTS This work was supported by NSF Grant No. ECCS 150909. Appendix A: Derivation of Memory Kernel in LLG equation self-consistently coupled to TDNEGF In this Appendix, we provide a detailed derivation of the memory kernel in Eq. (16). To obtain the perturbative equation of motion for local magnetic moments we start from Landau-Lifshitz Eq. (12) where the effective magnetic field can be written as Beff i(t) =Beff,0 i(t) +Jsd/angbracketleftˆs/angbracketrighti(t). (A.1) The nonequilibrium spin density is expanded up to terms linear in Jsdusing Eq. (13) /angbracketleftˆs/angbracketrighti(t) =~ 2Tr[neq(t)|i/angbracketright/angbracketlefti|⊗]−/angbracketleftˆs/angbracketrighti eq≈~ 2Tr/bracketleftbigg {0(t)+Jsd1(t)}|i/angbracketright/angbracketlefti|⊗/bracketrightbigg −/angbracketleftˆs/angbracketrighti eq=Jsd~ 2Tr[1(t)|i/angbracketright/angbracketlefti|⊗]−/angbracketleftˆs/angbracketrighti eq. (A.2) Here/angbracketleftˆs/angbracketrighti eqis the equilibrium electronic spin density i.e., /angbracketleftˆs/angbracketrighti eq= (~/2) Tr [ eq|i/angbracketright/angbracketlefti|⊗]. Furthermore, the electronic spin density in the zeroth order must vanish, i.e., Tr [ 0(t)|i/angbracketright/angbracketlefti|⊗] = 0 since for Jsd= 0 electrons are not spin- polarized. Hence, we can write Eq. (12) as ∂Mi(t) ∂t=−gMi(t)×/bracketleftbigg Beff,0 i(t) +J2 sd~ 2Tr[1(t)|i/angbracketright/angbracketlefti|⊗]−Jsd/angbracketleftˆs/angbracketrighti eq/bracketrightbigg . (A.3) To obtain analytical results, we assume that the equilibrium spin density follows the direction of local magnetic moments, so that Mi(t)×/angbracketleftˆs/angbracketrighti eq= 0. By expanding Eq. (6) we obtain i~∂0(t) ∂t= [H0(t),0(t)] +/summationdisplay p=L,Ri[Π(0) p(t) +Π(0)† p(t)], (A.4) and i~∂1(t) ∂t= [H1(t),0(t)] +i/summationdisplay p=L,R[Π(1) p(t) +Π(1)† p(t)], (A.5) where H1(t) =−/summationtext i|i/angbracketright/angbracketlefti|⊗·Mi(t). One can formally integrate Eq. (A.5) which leads to ~ 2Tr[1(t)|i/angbracketright/angbracketlefti|⊗] =/summationdisplay p=L,R1 2t −∞dt/primeTr/bracketleftbigg {Π(1) p(t/prime) +Π(1)† p(t/prime)}|i/angbracketright/angbracketlefti|⊗/bracketrightbigg . (A.6) which requires to find an expression for Π(1) p(t/prime). Using Eq. (8) and the fact that lead self-energy matrices do not depend onJsdleads to Π(1) p(t/prime) =t/prime −∞dt1[G> 1(t/prime,t1)Σ< p(t1,t/prime)−G< 1(t/prime,t1)Σ> p(t1,t/prime)]. (A.7) Equations (11) and (15) can be formally integrated to yield lesser and greater GFs in Eq. (A.7) G>,< 1(t/prime,t1) =1 i~/parenleftbiggt/prime −∞dt/prime/primeH1(t/prime/prime)G>,< 0(t/prime/prime,t1) +t/prime −∞dt/prime/prime+∞ −∞dt2/bracketleftbigg Σr tot(t/prime/prime,t2)G>,< 1(t2,t/prime/prime) +Σ>,< tot(t/prime/prime,t2)Ga 1(t2,t/prime/prime)/bracketrightbigg/parenrightbigg . (A.8)10 We further assume that the active region in Fig. 1 is weakly coupled with semi-infinite leads and, therefore, macroscopic reservoirs into which they terminate. This means that after we substitute Eq. (A.8) into Eq. (A.7) we can keep only those terms that are linear in the self-energy Π(1) p(t/prime) =1 2it/prime −∞dt/prime/primeH1(t/prime/prime)t/prime −∞dt1/bracketleftbigg G> 0(t/prime/prime,t1)Σ< p(t1,t/prime)−G< 0(t/prime/prime,t1)Σ> p(t1,t/prime)/bracketrightbigg (A.9) =i 2/summationdisplay it/prime −∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗·Mi(t/prime/prime)t/prime −∞dt1/bracketleftbigg G> 0(t/prime/prime,t1)Σ< p(t1,t/prime)−G< 0(t/prime/prime,t1)Σ> p(t1,t/prime)/bracketrightbigg (A.10) =i/summationdisplay it/prime −∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗·Mi(t/prime/prime)A0 p(t/prime/prime,t/prime), (A.11) where A0 p(t/prime/prime,t/prime) is an operator constructed out of the zeroth order terms in the expansion of GFs shown in Eq. (15) A0 p(t/prime/prime,t/prime)≡i 2t/prime −∞dt1/bracketleftbigg G> 0(t/prime/prime,t1)Σ< p(t1,t/prime)−G< 0(t/prime/prime,t1)Σ> p(t1,t/prime)/bracketrightbigg . (A.12) By plugging in Eqs. (A.11) and (A.12) into Eq. (A.6) we obtain ~ 2Tr[1(t)|i/angbracketright/angbracketlefti|⊗ˆσµ] =/summationdisplay p=L,R/summationdisplay j/summationdisplay νt −∞dt/primet/prime −∞dt/prime/primeMν j(t/prime/prime) Tr/bracketleftbigg |j/angbracketright/angbracketleftj|⊗ˆσν{A0 p(t/prime/prime,t/prime)+A0† p(t/prime/prime,t/prime)}|i/angbracketright/angbracketlefti|⊗σµ/bracketrightbigg .(A.13) Since A0 p(t/prime/prime,t/prime) is an operator constructed from the zeroth order GFs, it can be written in the followin form A0 p(t/prime/prime,t/prime) =1 2/summationdisplay mnAp mn(t/prime/prime,t/prime)|m/angbracketright/angbracketleftn|⊗12, (A.14) where 12is a 2×2 identity matrix. 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1408.6261v2.Stability_of_an_abstract_wave_equation_with_delay_and_a_Kelvin_Voigt_damping.pdf

arXiv:1408.6261v2 [math.AP] 15 May 2015Stability of an abstract–wave equation with delay and a Kelvin–Voigt damping Ka¨ ıs AMMARI∗, Serge NICAISE†and Cristina PIGNOTTI‡ Abstract. In this paper we consider a stabilization problem for an abstract wav e equation with delay and a Kelvin–Voigt damping. We prove an exponential stabilit y result for appropri- ate damping coefficients. The proofofthe main result is based on a fr equency–domainapproach. 2010 Mathematics Subject Classification : 35B35, 35B40, 93D15, 93D20. Keywords : Internal stabilization, Kelvin-Voigt damping, abstract wave equation with delay. 1 Introduction Our main goal is to study the internal stabilization of a dela yed abstract wave equation with a Kelvin–Voigt damping. More precisely, given a consta nt time delay τ >0,we consider the system given by: u′′(t)+aBB∗u′(t)+BB∗u(t−τ) = 0, in (0,+∞),(1.1) u(0) =u0, u′(0) =u1, (1.2) B∗u(t−τ) =f0(t−τ), in (0,τ), (1.3) wherea >0 is a constant, B:D(B)⊂H1→His a linear unbounded operator from a Hilbert space H1into another Hilbert space Hequipped with the respective norms || · ||H1,|| · ||Hand inner products ( ·,·)H1, (·,·)H, andB∗:D(B∗)⊂H→H1is the adjoint of B. The initial datum ( u0,u1,f0) belongs to a suitable space. We supposethat the operator B∗satisfies the following coercivity assumption: there existsC >0 such that /ba∇dblB∗v/ba∇dblH1≥C/ba∇dblv/ba∇dblH,∀v∈D(B∗). (1.4) For shortness we set V=D(B∗) and we assume that it is closed with the norm /ba∇dblv/ba∇dblV:= /ba∇dblB∗v/ba∇dblH1and that it is compactly embedded into H. ∗UR Analysis and Control of Pde, UR 13ES64, Department of Math ematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia, e-mail : kais.ammari@fsm.rnu.tn †Universit´ e de Valenciennes et du Hainaut Cambr´ esis, LAMA V, FR CNRS 2956, 59313 Valenciennes Cedex 9, France, e-mail: snicaise@univ-valenciennes.fr ‡Dipartimento di Ingegneria e Scienze dell’Informazione e M atematica, Universit` a di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila, Italy, e-mail : pigno tti@univaq.it 1Delay effects arise in many applications and practical proble ms and it is well–known that an arbitrarily small delay may destroy the well–posedn ess of the problem [16, 12, 19, 20] or destabilize a system which is uniformly asympt otically stable in absence of delay (see e.g. [9, 11], [17], [20]). Different strategies w ere recently developed to restitute either the well–posedness or the stability. In th e first case, one idea is to add a non–delay term, see [7, 19] for the heat equation. In the second case, we refer to [2, 5, 10, 17, 18] for stability results for systems with ti me delay where a standard feedback compensating the destabilizing delay effect is intr oduced. Nevertheless recent papers reveal that particular choices of the delay may resti tute exponential stability property, see [13, 4]. Note that the above system is exponentially stable in absenc e of time delay, and if a >0. On the other hand if a= 0 and −BB∗corresponds to the Laplace operator with Dirichlet boundary conditions in a bounded domain of Rn, problem (1.1)–(1.3) is not well–posed, see [16, 12, 19, 20]. Therefore in this paper in o rder to restitute the well- posedness character and its stability we propose to add the K elvin–Voigt damping term aBB∗u′. Hence the stabilization of problem (1.1)–(1.3) is perform ed using a frequency domain approach combined with a precise spectral analysis. The paper is organized as follows. The second section deals w ith the well–posedness of the problem while, in the third section, we perform the spe ctral analysis of the associated operator. In section 4, we prove the exponential stability of the system (1.1)–(1.3) if τ≤a. In the last section we give an example of an application. 2 Existence results In this section we will give a well–posedness result for prob lem (1.1)–(1.3) by using semigroup theory. Inspired from [17], we introduce the auxiliary variable z(ρ,t) =B∗u(t−τρ), ρ∈(0,1), t >0. (2.1) Then, problem (1.1)–(1.3) is equivalent to u′′(t)+aBB∗u′(t)+Bz(1,t) = 0, in (0,+∞), (2.2) τzt(ρ,t)+zρ(ρ,t) = 0 in (0 ,1)×(0,+∞),(2.3) u(0) =u0, u′(0) =u1, (2.4) z(ρ,0) =f0(−ρτ), in (0,1), (2.5) z(0,t) =B∗u(t), t > 0. (2.6) If we denote U:=/parenleftbig u,u′,z/parenrightbig⊤, then U′:=/parenleftbig u′,u′′,zt/parenrightbig⊤=/parenleftbig u′,−aBB∗u′−Bz(1,t),−τ−1zρ/parenrightbig⊤. Therefore, problem (2.2)–(2.6) can be rewritten as /braceleftbiggU′=AU, U(0) = (u0,u1,f0(−·τ))⊤,(2.7) 2where the operator Ais defined by A u v z := v −aBB∗v−Bz(·,1) −τ−1zρ , with domain D(A) :=/braceleftBig (u,v,z)⊤∈D(B∗)×D(B∗)×H1(0,1;H1) :aB∗v+z(1)∈D(B), B∗u=z(0)/bracerightBig , (2.8) in the Hilbert space H:=D(B∗)×H×L2(0,1;H1), (2.9) equipped with the standard inner product ((u,v,z),(u1,v1,z1))H= (B∗u,B∗u1)H1+(v,v1)H+ξ/integraldisplay1 0(z,z1)H1dρ, whereξ >0 is a parameter fixed later on. We will show that Agenerates a C0semigroup on Hby proving that A −cIdis maximal dissipative for an appropriate choice of cin function of ξ,τanda. Namely we prove the next result. Lemma 2.1. Ifξ >2τ a, then there exists a∗>0such that A −a−1 ∗Idis maximal dissipative in H. Proof.TakeU= (u,v,z)T∈D(A).Then we have (A(u,v,z),(u,v,z))H= (B∗v,B∗u)H1−((B(aB∗v+z(1)),v)H −ξτ−1/integraldisplay1 0(zρ,z)H1dρ. Hence, we get (A(u,v,z),(u,v,z))H= (B∗v,B∗u)H1−(aB∗v+z(1),B∗v)H1 −ξ 2τ/ba∇dblz(1)/ba∇dbl2 H1+ξ 2τ/ba∇dblz(0)/ba∇dbl2 H1. Hence reminding that z(0) =B∗uand using Young’s inequality we find that ℜ(A(u,v,z),(u,v,z))H ≤(ε−a)/ba∇dblB∗v/ba∇dbl2 H1+(1 2ε−ξ 2τ)/ba∇dblz(1)/ba∇dbl2 H1+(1 2ε+ξ 2τ)/ba∇dblB∗u/ba∇dbl2 H1. Chosing ε=a 2, we find that ℜ(A(u,v,z),(u,v,z))H≤ −a 2/ba∇dblB∗v/ba∇dbl2 H1+(1 a−ξ 2τ)/ba∇dblz(1)/ba∇dbl2 H1+(1 a+ξ 2τ)/ba∇dblB∗u/ba∇dbl2 H1. 3The choice of ξis equivalent to1 a−ξ 2τ<0, and therefore for a∗=/parenleftBig 1 a+ξ 2τ/parenrightBig−1 , ℜ(A(u,v,z),(u,v,z))H≤ −a 2/ba∇dblB∗v/ba∇dbl2 H1+(1 a−ξ 2τ)/ba∇dblz(1)/ba∇dbl2 H1+a−1 ∗/ba∇dblB∗u/ba∇dbl2 H1.(2.10) As/ba∇dblB∗u/ba∇dbl2 H1≤ /ba∇dbl(u,v,z)/ba∇dbl2 H, we get ℜ((A−a−1 ∗Id)(u,v,z),(u,v,z))H≤ −a 2/ba∇dblB∗v/ba∇dbl2 H1+(1 a−ξ 2τ)/ba∇dblz(1)/ba∇dbl2 H1≤0, which directly leads to the dissipativeness of A−a−1 ∗Id. Let us go on with the maximality, namely let us show that λI−Ais surjective for a fixedλ >0.Given (f,g,h)T∈ H,we look for a solution U= (u,v,z)T∈D(A) of (λI−A) u v z = f g h , (2.11) that is, verifying λu−v=f, λv+B(aB∗v+z(1)) =g, λz+τ−1zρ=h.(2.12) Suppose that we have found uwith the appropriate regularity. Then, v=λu−f (2.13) and we can determine z.Indeed, by (2.8), z(0) =B∗u, (2.14) and, from (2.12), λz(ρ)+τ−1zρ(ρ) =h(ρ) forρ∈(0,1). (2.15) Then, by (2.14) and (2.15), we obtain z(ρ) =B∗ue−λρτ+τe−λρτ/integraldisplayρ 0h(σ)eλστdσ. (2.16) In particular, we have z(1) =B∗ue−λτ+z0, (2.17) withz0∈H1defined by z0=τe−λτ/integraldisplay1 0h(σ)eλστdσ. (2.18) This expression in (2.12) shows that the function uverifies formally λ2u+B(aB∗(λu−f)+B∗ue−λτ+z0) =g+λf, that is, λ2u+(λa+e−λτ)BB∗u=g+λf+B(aB∗f)−Bz0. (2.19) 4Problem (2.19) can be reformulated as (λ2u+(λa+e−λτ)BB∗u,w)H= (g+λf+B(aB∗f)−Bz0,w)H,∀w∈V.(2.20) Using the definition of the adjoint of B, we get λ2(u,w)H+(λa+e−λτ)(B∗u,B∗w)H1= (g+λf,w)H+(aB∗f−z0,B∗w)H1,∀w∈V. (2.21) As the left-hand sideof (2.21) is coercive on D(B∗), the Lax–Milgram lemma guarantees the existence and uniqueness of a solution u∈Vof (2.21). Once uis obtained we define vby (2.13) that belongs to Vandzby (2.16) that belongs to H1(0,1;H1). Hence we can setr=aB∗v+z(1), it belongs to H1but owing to (2.21), it fulfils λ(v,w)H+(r,B∗w)H1= (g,w)H,∀w∈D(B∗), or equivalently (r,B∗w)H1= (g−λv,w)H,∀w∈D(B∗). Asg−λv∈H, this implies that rbelongs to D(B) with Br=g−λv. This shows that the triple U= (u,v,z) belongs to D(A) and satisfies (2.11), hence λI−Ais surjective for every λ >0. We have then the following result. Proposition 2.2. The system (1.1)–(1.3)is well–posed. More precisely, for every (u0,u1,f0)∈ H, there exists a unique solution (u,v,z)∈C(0,+∞,H)of(2.7). More- over, if(u0,u1,f0)∈D(A)then(u,v,z)∈C(0,+∞,D(A))∩C1(0,+∞,H)withv=u′ anduis indeed a solution of (1.1)–(1.3). 3 The spectral analysis AsD(B∗) is compactly embedded into H, the operator BB∗:D(BB∗)⊂H→H has a compact resolvent. Hence let ( λk)k∈N∗be the set of eigenvalues of BB∗repeated according to their multiplicity (that are positive real num bers and are such that λk→ +∞ask→+∞) and denote by ( ϕk)k∈N∗the corresponding eigenvectors that form an orthonormal basis of H(in particular for all k∈N∗,BB∗ϕk=λkϕk). 3.1 The discrete spectrum We have the following lemma. Lemma 3.1. Ifτ≤a, then any eigenvalue λofAsatisfies ℜλ <0. 5Proof.Letλ∈CandU= (u,v,z)⊤∈D(A) be such that (λI−A) u v z = 0, or equivalently v=λu, −B(aB∗v+z(·,1)) =λv, −τ−1zρ=λz.(3.1) By (2.14), we find that z(ρ) =λ−1B∗ve−λρτ. (3.2) Using this property in (3.1), we find that u∈D(B∗) is solution of λ2u+(aλ+e−λτ)BB∗u= 0. Hence a non trivial solution exists if and only if there exist sk∈N∗such that λ2 aλ+e−λτ=−λk. (3.3) This condition implies that λdoes not belong to Σ :={λ∈C:aλ+e−λτ= 0}, (3.4) and that e−λτ+λ2 λk+aλ= 0. (3.5) Writingλ=x+iy, withx,y∈R, we see that this identity is equivalent to e−τxcos(τy)+x2−y2 λk+ax= 0, (3.6) −e−τxsin(τy)+2xy λk+ay= 0. (3.7) The second equation is equivalent to eτx/parenleftBig2x λk+a/parenrightBig y= sin(τy). Hence if y/\e}atio\slash= 0, we will get eτx τ/parenleftBig2x λk+a/parenrightBig =sin(τy) τy. As the modulus of the right-hand side is ≤1, we obtain /vextendsingle/vextendsingle/vextendsingleeτx τ/parenleftBig2x λk+a/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤1, 6or equivalently/vextendsingle/vextendsingle/vextendsingle2x λk+a/vextendsingle/vextendsingle/vextendsingle≤τe−τx. Therefore if x≥0, we find that 2x λk+a≤τe−τx≤τ, which implies that 2x λk≤τ−a. Forτ < a, we arrive to a contradiction. For τ=a, the sole possibility is x= 0 and by (3.7), we find that sin(τy) =τy, which yields y= 0 and again we obtain a contradiction. Ify= 0, we see that (3.7) always holds and (3.6) is equivalent to e−τx=−x(x λk+a). This equation has no non–negative solutions xsince for x≥0, the left hand side is positive while the right–hand side is non positive, hence ag ain if a solution xexists, it has to be negative. The proof of the lemma is complete. Ifa < τ, we now show that there exist some pairs of ( a,τ) for which the system (1.1)–(1.3) becomes unstable. Hence the condition τ≤ais optimal for the stability of this system. Lemma 3.2. There exist pairs of (a,τ)such that 0< a < τand for which the associated operator Ahas a pure imaginary eigenvalue. Proof.We look for a purely imaginary eigenvalue iyofA, hence system (3.6)–(3.7) reduces to cos(τy) =y2 λk, (3.8) sin(τy) =ay. (3.9) Such a solution exists if y4 λ2 k+a2y2= 1. (3.10) One solution of this equation is yk= −a2λ2 k+/radicalBig a4λ4 k+4λ2 k 2 1 2 . 7We now take any τ∈(0,π 2yk) anda=sin(τyk) yk. Then (3.9) automatically holds, while (3.8) is valid owing to (3.10) (as cos( τyk)>0). Finally a < τbecause a τ=sin(τyk) τyk<1. Therefore with such a choice of aandτ, the operator Ahas a purely imaginary eigen- value equal to iyk. 3.2 The continuous spectrum Inspired from section 3 of [1], by using a Fredholm alternati ve technique, we perform the spectral analysis of the operator A. Recall that an operator Tfrom a Hilbert space Xinto itself is called singular if there exists a sequence un∈D(T) with no convergent subsequence such that /ba∇dblun/ba∇dblX= 1 and Tun→0 inX, see [21]. According to Theorem 1.14 of [21] Tis singular if and only if its kernel is infinite dimensional or its range is not closed. Let Σ be the set defined in (3.4). The following results hold: Theorem 3.3. 1. Ifλ∈Σ, thenλI−Ais singular. 2. Ifλ/\e}atio\slash∈Σ, thenλI−Ais a Fredholm operator of index zero. Proof.For the proof of point 1, let us fix λ∈Σ and for all k∈N∗set Uk= (uk,λuk,B∗uke−λτ·)⊤, withuk=1√λkϕk. ThenUkbelongs to D(A) and easy calculations yield (due to the assumption λ∈Σ) (λI−A)Uk=λ2(0,uk,0)⊤. Therefore we deduce that /ba∇dbl(λI−A)Uk/ba∇dblH→0,ask→ ∞. Moreover due to the property /ba∇dblB∗uk/ba∇dblH1= 1, there exist positive constants c,C,such that c≤ /ba∇dblUk/ba∇dblH≤C,∀k∈N∗. This shows that λI−Ais singular. For allλ∈C, introduce the (linear and continuous) mapping AλfromVinto its dual by /a\}b∇acketle{tAλv,w/a\}b∇acket∇i}htV′−V=λ2(v,w)H+(aλ+e−λτ)(B∗v,B∗w)H1,∀v,w∈D(B∗). Then from the proof of Lemma 2.1, we know that for λ >0,Aλis an isomorphism. Now for λ∈C\Σ, we can introduce the operator Bλ= (aλ+e−λτ)−1Aλ. 8Hence for λ∈C\Σ,Aλis a Fredholm operator of index 0 if and only if Bλis a Fredholm operator of index 0. Furthermore for λ,µ∈C\Σ asBλ−Bµis a multiple of the identity operator, due to the compact embedding of VintoV′, and asBµis an isomorphism for µ >0, we finally deduce that Aλis a Fredholm operator of index 0 for all λ∈C\Σ. Now we readily check that, for any λ∈C\Σ, we have the equivalence u∈kerAλ⇐⇒(u,λu,B∗ue−λτ·)⊤∈ker(λI−A). (3.11) This equivalence implies that dim ker( λI−A) = dim ker Aλ,∀λ∈C\Σ. (3.12) For the range property for all λ∈C\Σ introduce the inner product (u,z)λ,V:=/parenleftBig (u,λu,B∗ue−λτ·)⊤,(z,λz,B∗ze−λτ·)⊤/parenrightbig H, onVwhose associated norm is equivalent to the standard one. Denote by {y(i)}N i=1an orthonormal basis of ker Aλfor this new inner product (for shortness the dependence of λis dropped), i.e. (y(i),y(j))λ,V=δij,∀i,j= 1,...,N. Finally, for all i= 1,...,N, we set Z(i)= (y(i),λy(i),B∗y(i)e−λτ·)⊤, the element of ker( λI− A) associated with y(i)that are orthonormal with respect to the inner product of H. Let us now show that for all λ∈C\Σ, the range R(λI− A) ofλI− Ais closed. Indeed, let us consider a sequence Un= (un,vn,zn)⊤∈D(A) such that (λI−A)Un=Fn= (fn,gn,hn)⊤→F= (f,g,h)⊤inH. (3.13) Without loss of generality we can assume that (Un,Z(i))H=−αn,i,∀i= 1,...,N, (3.14) where αn,i:= ((0,fn,−τe−λτ·/integraldisplay· 0hn(σ)eλστdσ)⊤,Z(i))H. Indeed, if this is not the case, we can consider ˜Un=Un−N/summationdisplay i=1βiZ(i) that still belongs to D(A) and satisfies (λI−A)˜Un=Fn, 9as well as (˜Un,Z(i))H=−αn,i,∀i= 1,...,N, by setting βi= (Un,Z(i))H+αn,i,∀i= 1,...,N. Note that the condition (3.14) is equivalent to (un,y(i))λ,V= 0,∀i= 1,...,N. In other words, un∈(kerAλ)⊥λ,V, (3.15) where⊥λ,Vmeans that the orthogonality is taken with respect to the inn er product (·,·)λ,V. Returning to (3.13), the arguments of the proof of Lemma 2.1 i mply that Aλun=LFninV′, whereLFis defined by LF(w) := (g,w)H−τe−λτ(/integraldisplay1 0h(σ)eλστdσ,B∗w)H1+(λf+aB∗f,w)H1, whenF= (f,g,h)⊤. But it is easy to check that LFn→LFinV′. Moreover, as λ∈C\Σ,Aλis an isomorphism from (ker Aλ)⊥λ,VintoR(Aλ), hence by (3.15) we deduce that there exists a positive constant C(λ) such that /ba∇dblun−um/ba∇dblV≤C(λ)/ba∇dblLFn−LFm/ba∇dblV′,∀n,m∈N. Hence, (un)nis a Cauchy sequence in V, and therefore there exists u∈Vsuch that un→uinV, as well as Aλu=LFinV′. Then defining vby (2.13) and zby (2.16), we deduce that U:= (u,v,z)⊤belongs to D(A) and (λI−A)U=F. In other words, Fbelongs to R(λI−A). The closedness of R(λI−A) is thus proved. At this stage, for any λ∈C\Σ, we show that codimR(Aλ) = codim R(λI−A), (3.16) where for W⊂ H, codimWis the dimension of the orthogonal in HofW, while for W′⊂V′, codimW′is the dimension of the annihilator A:={v∈V:/a\}b∇acketle{tv,w/a\}b∇acket∇i}htV−V′= 0,∀w∈W′}, 10ofW′inV. Indeed, let us set N= codim R(Aλ), then there exist Nelements ϕi∈V, i= 1,...,N,such that f∈R(Aλ)⇐⇒f∈V′and/a\}b∇acketle{tf,ϕi/a\}b∇acket∇i}htV′−V= 0,∀i= 1,...,N. Consequently, for F∈ H, ifLF(that belongs to V′) satisfies LF(ϕi) = 0,∀i= 1,...,N, (3.17) then there exists a solution u∈Vof Aλu=LFin V’, and the arguments of the proof of Lemma 2.1 imply that Fis inR(λI−A). Hence, the Nconditions on F∈ Hfrom (3.17) allow to show that it belongs to R(λI− A), and therefore codimR(λI−A)≤N= codim R(Aλ). (3.18) This shows that λI−Ais a Fredholm operator. Conversely, set M= codim R(λI−A),thenthereexist MelementsΨ i= (ui,vi,zi)∈ H, i= 1,...,M, such that F∈R(λI−A)⇐⇒F∈ Hand (F,Ψi)H= 0,∀i= 1,...,M. Then, for any g∈H, if (g,vi)H= ((0,g,0)⊤,Ψi)H= 0,∀i= 1,...,M, (3.19) there exists U= (u,v,z)⊤∈D(A) such that (λI−A)U= (0,g,0), which implies that Aλu=g. This shows that R(Aλ)⊃H0, whereH0:={g∈Hsatisfying (3 .19)}. This inclusion implies that (here ⊥means the annihilator of the set in V) R(Aλ)⊥⊂H⊥ 0. Therefore R(Aλ)⊥⊂ {v∈V:/a\}b∇acketle{tv,g/a\}b∇acket∇i}htV−V′= 0,∀g∈H0} ={v∈V: (v,g)H= 0,∀g∈H0} ⊂Span{vi}M i=1∩V. Hence, codimR(Aλ)≤M= codim R(λI−A). (3.20) The inequalities (3.18) and (3.20) imply (3.16). 11Lemma 3.4. Ifτ≤a, then Σ⊂ {λ∈C:ℜλ <0}. Proof.Letλ=x+iy∈Σ, withx,y∈Rwe deduce that ax+e−τxcos(τy) = 0, ay−e−τxsin(τy) = 0. This corresponds to the system (3.6)–(3.7) with k=∞, hence the arguments as in the proof of Lemma 3.1 yield the result. Corollary 3.5. It holds σ(A) =σpp(A)∪Σ, and therefore if τ≤a σ(A)⊂ {λ∈C:ℜλ <0}. Proof.By Theorem 3.3, C\Σ⊂σpp(A)∪ρ(A). The first assertion directly follows. The second assertion follows from Lemmas 3.1 and 3.4. 4 Asymptotic behavior In this section, we show that if τ≤aandξ >2τ a, the semigroup etAdecays to the null steady state with an exponential decay rate. To obtain this, our technique is based on a frequency domain approach and combines a contradiction ar gument to carry out a special analysis of the resolvent. Theorem 4.1. Ifξ >2τ aandτ≤a, then there exist constants C,ω >0such that the semigroup etAsatisfies the following estimate /vextenddouble/vextenddoubleetA/vextenddouble/vextenddouble L(H)≤Ce−ωt,∀t >0. (4.21) Proof of theorem 4.1. We will employ the following frequency domain theorem for un i- form stability from [15, Thm 8.1.4] of a C0semigroup on a Hilbert space: Lemma 4.2. AC0semigroup etLon a Hilbert space Hsatisfies ||etL||L(H)≤Ce−ωt, for some constant C >0and forω >0if and only if ℜλ <0,∀λ∈σ(L), (4.22) and sup ℜλ≥0/ba∇dbl(λI−L)−1/ba∇dblL(H)<∞. (4.23) whereσ(L)denotes the spectrum of the operator L. 12According to Corollary 3.5 the spectrum of Ais fully included into ℜλ <0, which clearly implies (4.22). Then the proof of Theorem 4.1 is base d on the following lemma that shows that (4.23) holds with L=A. Lemma 4.3. The resolvent operator of Asatisfies condition sup ℜλ≥0/ba∇dbl(λI−A)−1/ba∇dblL(H)<∞. (4.24) Proof.Suppose that condition (4.24) is false. By the Banach-Stein haus Theorem (see [8]), there exists a sequence of complex numbers λnsuch that ℜλn≥0,|λn| →+∞ and a sequence of vectors Zn= (un,vn,zn)t∈D(A) with /ba∇dblZn/ba∇dblH= 1 (4.25) such that ||(λnI−A)Zn||H→0 asn→ ∞, (4.26) i.e., λnun−vn≡fn→0 inD(B∗), (4.27) λnvn+aB(B∗vn+zn(1))≡gn→0 inH, (4.28) λnzn+τ−1∂ρzn≡hn→0 inL2((0,1);H1). (4.29) Our goal is to derive from (4.26) that ||Zn||Hconverges to zero, that furnishes a contradiction. We notice that from (2.10) and (4.27) we have ||(λnI−A)Zn||H≥ |ℜ((λnI−A)Zn,Zn)H| ≥ ℜλn−a−1 ∗/ba∇dblB∗un/ba∇dbl2 H1+/parenleftbiggξ 2τ−1 a/parenrightbigg /ba∇dblzn(1)/ba∇dbl2 H1+a 2/ba∇dblB∗vn/ba∇dbl2 H1 =ℜλn−a−1 ∗/vextenddouble/vextenddouble/vextenddouble/vextenddoubleB∗vn+B∗fn λn/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 H1+/parenleftbiggξ 2τ−1 a/parenrightbigg /ba∇dblzn(1)/ba∇dbl2 H1+a 2/ba∇dblB∗vn/ba∇dbl2 H1. Hence using the inequality /ba∇dblB∗vn+B∗fn/ba∇dbl2 H1≤2/ba∇dblB∗vn/ba∇dbl2 H1+2/ba∇dblB∗fn/ba∇dbl2 H1, we obtain that ||(λnI−A)Zn||H≥ ℜλn−2a−1 ∗|λn|−2/ba∇dblB∗fn/ba∇dbl2 H1+/parenleftbiggξ 2τ−1 a/parenrightbigg /ba∇dblzn(1)/ba∇dbl2 H1 +(a 2−2a−1 ∗|λn|−2)/ba∇dblB∗vn/ba∇dbl2 H1. Hence for nlarge enough, say n≥n∗, we can suppose that a 2−2a−1 ∗|λn|−2≥a 4. 13and therefore for all n≥n∗, we get ||(λnI−A)Zn||H≥ ℜλn−2a−1 ∗|λn|−2/ba∇dblB∗fn/ba∇dbl2 H1+/parenleftbiggξ 2τ−1 a/parenrightbigg /ba∇dblzn(1)/ba∇dbl2 H1 +a 4/ba∇dblB∗vn/ba∇dbl2 H1. By this estimate, (4.26) and (4.27), we deduce that zn(1)→0, B∗vn→0,inH1,asn→ ∞, (4.30) and in particular, from the coercivity (1.4), that vn→0,inH,asn→ ∞. This implies according to (4.27) that un=1 λnvn+1 λnfn→0,inD(B∗),asn→ ∞, (4.31) as well as zn(0) =B∗un→0,inH1,asn→ ∞. (4.32) By integration of the identity (4.29), we have zn(ρ) =zn(0)e−τλnρ+τ/integraldisplayρ 0e−τλn(ρ−γ)hn(γ)dγ. (4.33) Hence recalling that ℜλn≥0 /integraldisplay1 0/ba∇dblzn(ρ)/ba∇dbl2 H1dρ≤2/ba∇dblzn(0)/ba∇dbl2 H1+2τ2/integraldisplay1 0/integraldisplayρ 0/ba∇dblhn(γ)/ba∇dbl2 H1dγρdρ→0,asn→ ∞. All together we have shown that /ba∇dblZn/ba∇dblHconverges to zero, that clearly contradicts /ba∇dblZn/ba∇dblH= 1. The two hypotheses of Lemma 4.2 are proved, then (4.21) holds . The proof of Theorem 4.1 is then finished. 5 Application to the stabilization of the wave equation with delay and a Kelvin–Voigt damping We study the internal stabilization of a delayed wave equati on. More precisely, we consider the system given by : utt(x,t)−a∆ut(x,t)−∆u(x,t−τ) = 0,in Ω×(0,+∞), (5.1) u= 0, on∂Ω×(0,+∞),(5.2) u(x,0) =u0(x), ut(x,0) =u1(x), in Ω, (5.3) ∇u(x,t−τ) =f0(t−τ), in Ω×(0,τ), (5.4) 14where Ω is a smooth open bounded domain of Rnanda,τ >0 are constants. Thisproblementersinourabstractframeworkwith H=L2(Ω),B=−div:D(B) = H1(Ω)n→L2(Ω),B∗=∇:D(B∗) =H1 0(Ω)→H1:=L2(Ω)n, the assumption (1.4) being satisfied owing to Poincar´ e’s inequality. The operat orAis then given by A u v z := v a∆v+ divz(·,1) −τ−1zρ , with domain D(A) :=/braceleftBig (u,v,z)⊤∈H1 0(Ω)×H1 0(Ω)×L2(Ω;H1(0,1)) :a∇v+z(·,1)∈H1(Ω), ∇u=z(·,0) in Ω/bracerightBig , (5.5) in the Hilbert space H:=H1 0(Ω)×L2(Ω)×L2(Ω×(0,1)). (5.6) According to Lemma 3.5 and Theorem 4.1 we have: Corollary 5.1. Ifτ≤a, the system (5.1)–(5.4)is exponentially stable in H, namely forξ >2τ a, the energy E(t) =1 2/parenleftbigg/integraldisplay Ω(|∇u(x,t)|2+|ut(x,t)|2)dx+ξ/integraldisplay Ω/integraldisplay1 0|∇u(x,t−τρ)|2dxdρ/parenrightbigg , satisfies E(t)≤Me−ωtE(0),∀t >0, for some positive constants Mandω. Conclusion By a careful spectral analysis combined with a frequency dom ain approach, we have shownthat thesystem (1.1)–(1.3) isexponentially stablei fτ≤aandthat thiscondition is optimal. But from the general form of (1.1), we can only con sider interior Kelvin- Voigt dampings. Hence an interesting perspective is to cons ider the wave equation with dynamical Ventcel boundary conditions with a delayed t erm and a Kelvin-Voigt damping. References [1]Z. Abbas and S. Nicaise ,The multidimensional wave equation with generalized acoustic boundary conditions I: Strong stability , SIAM J. Control Opt., 2015, to appear. 15[2]E. M. Ait Ben Hassi, K. Ammari, S. Boulite and L. Maniar ,Feedback stabilization of a class of evolution equations with delay , J. Evol. Equ., 1(2009), 103-121. [3]K. Ammari and S. Nicaise ,Stabilization of elastic systems by collocated feedback, Lecture Notes in Mathematics, Vol. 2124, Springer-Verlag, 2015. [4]K. Ammari, S. Nicaise and C. Pignotti, Stabilization by switching time-delay, Asymptotic Analysis, 83(2013), 263–283. [5]K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems Control Lett., 59(2010), 623–628. [6]K. Ammari and M. Tucsnak ,Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6(2001), 361-386. [7]A. B´atkai and S. Piazzera ,Semigroups for delay equations, Research Notes in Mathematics 10, A. K. Peters, Wellesley MA (2005). [8]H. Brezis, Analyse Fonctionnelle, Th´ eorie et Applications, Masson, Paris, 1983. [9]R. Datko, Not all feedback stabilized hyperbolic systems are robust w ith respect to small time delays in their feedbacks, SIAM J. Control Optim., 26(1988), 697-713. [10]R. Datko, J. Lagnese and P. Polis, An exemple on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1985), 152-156. [11]R. Datko, Two examples of ill-posedness with respect to time delays rev isited, IEEE Trans. Automatic Control, 42(1997), 511–515. [12]M. Dreher, R. Quintanilla and R. Racke ,Ill-posed problems in thermome- chanics, Appl. Math. Letters, 22(2009), 1374-1379. [13]M. Gugat ,Boundary feedback stabilization by time delay for one-dime nsional wave equations , IMA J. Math. Control Inform., 27(2010), 189–203. [14]F. Huang, Characteristic conditions for exponential stability of lin ear dynamical systems in Hilbert space , Ann. Differential Equations, 1(1985), 43-56. [15]B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite- dimensional Spaces , Operator Theory: Advances and Applications, 223, Birkhauser, 2012. [16]P.M. Jordan, W. Dai and R.E. Mickens ,A note on the delayed heat equation: Instability with respect to initial data , Mech. Res. Comm., 35(2008), 414-420. [17]S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks , SIAM J. Control Optim., 45(2006), 1561–1585. 16[18]S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var., 16(2010), 420– 456. [19]J. Pr¨uss,Evolutionary integral equations and applications , Monograhs Mathemat- ics,87, Birkh¨ auser Verlag, Basel, 1993. [20]R. Racke, Instability of coupled systems with delay, Commun. Pure Appl. Anal., 11(2012), 1753–1773. [21]F. Wolf, On the essential spectrum of partial differential boundary p roblems, Comm. Pure Appl. Math., 12(1959), 211–228. 17

1406.6225v2.Interface_enhancement_of_Gilbert_damping_from_first_principles.pdf

arXiv:1406.6225v2 [cond-mat.mtrl-sci] 16 Nov 2014Interface enhancement of Gilbert damping from first-princi ples Yi Liu,1,∗Zhe Yuan,1,2,†R. J. H. Wesselink,1Anton A. Starikov,1and Paul J. Kelly1 1Faculty of Science and Technology and MESA+Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands 2Institut f¨ ur Physik, Johannes Gutenberg–Universit¨ at Ma inz, Staudingerweg 7, 55128 Mainz, Germany (Dated: June 6, 2018) The enhancement of Gilbert damping observed for Ni 80Fe20(Py) films in contact with the non- magnetic metals Cu, Pd, Ta and Pt, is quantitatively reprodu ced using first-principles scattering calculations. The “spin-pumping” theory that qualitative ly explains its dependence on the Py thick- ness is generalized to include a number of extra factors know n to be important for spin transport through interfaces. Determining the parameters in this the ory from first-principles shows that inter- face spin-flipping makes an essential contribution to the da mping enhancement. Without it, a much shorter spin-flip diffusion length for Pt would be needed than the value we calculate independently. PACS numbers: 85.75.-d, 72.25.Mk, 76.50.+g, 75.70.Tj Introduction. —Magnetizationdissipation, expressedin termsofthe Gilbert dampingparameter α, is akeyfactor determining the performance of magnetic materials in a host of applications. Of particular interest for magnetic memorydevicesbasedupon ultrathin magneticlayers[ 1– 3] is the enhancement of the damping of ferromagnetic (FM) materials in contact with non-magnetic (NM) met- als [4] that can pave the way to tailoring αfor particu- lar materials and applications. A “spin pumping” theory has been developed that describes this interface enhance- ment in terms of a transverse spin current generated by the magnetization dynamics that is pumped into and ab- sorbed by the adjacent NM metal [ 5,6]. Spin pumping subsequently evolved into a technique to generate pure spin currents that is extensively applied in spintronics experiments [ 7–9]. A fundamental limitation of the spin-pumping the- ory is that it assumes spin conservation at interfaces. This limitation does not apply to a scattering theoret- ical formulation of the Gilbert damping that is based upon energy conservation, equating the energy lost by the spin system through damping to that parametrically pumped out of the scattering region by the precessing spins [10]. In this Letter, we apply a fully relativistic density functional theory implementation [ 11–13] of this scattering formalism to the Gilbert damping enhance- ment in those NM |Py|NM structures studied experimen- tally in Ref. 4. Our calculated values of αas a function of the Py thickness dare compared to the experimental results in Fig. 1. Without introducing any adjustable pa- rameters, we quantitatively reproduce the characteristic 1/ddependence aswellasthe dependenceofthe damping on the NM metal. To interpret the numerical results, we generalize the spin pumping theory to allow: (i) for interface [ 14–16] as well as bulk spin-flip scattering; (ii) the interface mix- ing conductance to be modified by spin-orbit coupling; (iii) the interface resistance to be spin-dependent. An important consequence of our analysis is that withoutinterface spin-flip scattering, the value of the spin-flip diffusion length lsfin Pt required to fit the numerical results is much shorter than a value we independently calculate for bulk Pt. A similar conclusion has recently been drawn for Co |Pt interfaces from a combination of ferromagnetic resonance, spin pumping and inverse spin Hall effect measurements [ 17]. Gilbert damping in NM |Py|NM.—We focus on the NM|Py|NM sandwiches with NM = Cu, Pd, Ta and Pt that were measured in Ref. 4. The samples were grown on insulating glass substrates, the NM layer thickness was fixed at l=5 nm, and the Py thickness dwas vari- able. To model these experiments, the conventional NM- lead|Py|NM-lead two-terminal scattering geometry with semi-infinite ballistic leads [ 10–13] has to be modified because: (i) the experiments were carried out at room 0 2 4 6 8 10 d (nm) 00.02 0.04 0.06 0.08 0.10 _Pt |Py|Pt Pd|Py|Pd Ta|Py|Ta Cu|Py|Cu Calc. Expt. NM (l)NM (l) Lead Lead Py (d) FIG. 1. (color online). Calculated (solid lines) Gilbert da mp- ing of NM |Py|NM (NM = Cu, Pd, Ta and Pt) compared to experimental measurements (dotted lines) [ 4] as a function of the Py thickness d. Inset: sketch of the structure used in the calculations. The dashed frame denotes one structural unit consisting of a Py film between two NM films.2 temperature so the 5 nm thick NM metals used in the samples were diffusive; (ii) the substrate |NM and NM |air interfaces cannot transmit charge or spin and behave ef- fectively as “mirrors”, whereas in the conventional scat- tering theory the NM leads are connected to charge and spin reservoirs. We start with the structural NM( l)|Py(d)|NM(l) unit indicated by the dashed line in the inset to Fig. 1that consists of a Py film, whose thickness dis variable, sand- wichedbetween l=5nm-thick diffusiveNM films. Several NM|Py|NM units are connected in series between semi- infinite leads to calculate the total magnetization dissi- pation of the system [ 10–13] thereby explicitly assuming a “mirror” boundary condition. By varying the number of these units, the Gilbert damping for a single unit can be extracted [ 18], that corresponds to the damping mea- sured for the experimental NM( l)|Py(d)|NM(l) system. As shown in Fig. 1, the results are in remarkably good overall agreement with experiment. For Pt and Pd, where a strong damping enhancement is observed for thin Py layers, the values that we calculate are slightly lower than the measured ones. For Ta and Cu where the enhancement is weaker, the agreement is better. In the case of Cu, neither the experimental nor the calcu- lated data shows any dependence on dindicating a van- ishinglysmalldampingenhancement. Theoffsetbetween the two horizontal lines results from a difference between the measured and calculated values of the bulk damping in Py. Acareful analysisshowsthat the calculated values ofαare inversely proportional to the Py thickness dand approach the calculated bulk damping of Py α0=0.0046 [11] in the limit of large dfor all NM metals. However, extrapolation of the experimental data yields values of α0ranging from 0.004 to 0.007 [ 19]; the spread can be partly attributed to the calibration of the Py thickness, especially when it is very thin. Generalized spin-pumping theory. —In spite of the very good agreement with experiment, our calculated re- sults cannot be interpreted satisfactorily using the spin- pumping theory [ 5] that describes the damping enhance- ment in terms of a spin current pumped through the interface by the precessing magnetization giving rise to an accumulation of spins in the diffusive NM metal, and a back-flowing spin current driven by the ensuing spin-accumulation. The pumped spin current, Ipump s= (/planckover2pi12A/2e2)Gmixm×˙m, is described using a “mixing con- ductance” Gmix[20] that is a property of the NM |FM interface [ 21,22]. Here, mis a unit vector in the di- rection of the magnetization and Ais the cross-sectional area. The theory only takes spin-orbit coupling (SOC) into account implicitly via the spin-flip diffusion length lsfof the NM metal and the pumped spin current is con- tinuous across the FM |NM interface [ 5]. With SOC included, this boundary condition does not hold. Spin-flip scattering at an interface is described by the “spin memory loss” parameter δdefined so that thespin-flip probability of a conduction electron crossing the interface is 1 −e−δ[14,15]. It alters the spin accumula- tion in the NM metal and, in turn, the backflow into the FM material. To take δand the spin-dependence of the interface resistance into account, the FM |NM interface is represented by a fictitious homogeneous ferromagnetic layerwithafinitethickness[ 15,16]. Thespincurrentand spin-resolved chemical potentials (as well as their differ- enceµs, the spin accumulation) are continuous at the boundaries of the effective “interface” layer. We impose the boundary condition that the spin current vanishes at NM|air or NM |substrate interfaces. Then the spin accu- mulation in the NM metal can be expressed as a function of the net spin-current Isflowing out of Py [ 23], which is the difference between the pumped spin current Ipump s and the backflow Iback s. The latter is determined by the spin accumulation in the NM metal close to the inter- face,Iback s[µs(Is)]. Following the original treatment by Tserkovnyak et al. [ 5],Isis determined by solving the equation Is=Ipump s−Iback s[µs(Is)] self-consistently. Fi- nally, the total damping of NM( l)|Py(d)|NM(l) can be described as α(l,d) =α0+gµB/planckover2pi1 e2MsdGmix eff=α0+gµB/planckover2pi1 e2Msd ×/bracketleftbigg1 Gmix+2ρlsfR∗ ρlsfδsinhδ+R∗coshδtanh(l/lsf)/bracketrightbigg−1 .(1) Here,R∗=R/(1−γ2 R) is an effective interface spe- cific resistance with Rthe total interface specific resis- tance between Py and NM and its spin polarization, γR= (R↓−R↑)/(R↓+R↑) is determined by the con- tributions R↑andR↓from the two spin channels [ 16].ρ is the resistivity of the NM metal. All the quantities in Eq. (1) can be experimentally measured [ 16] and calcu- lated from first-principles [ 24]. If spin-flip scattering at the interface is neglected, i.e., δ= 0, Eq. ( 1) reduces to the original spin pumping formalism [ 5]. Eq. (1) is de- rived using the Valet-Fert diffusion equation [ 25] that is still applicable when the mean free path is comparable to the spin-flip diffusion length [ 26]. Mixing conductance. —Assuming that SOC can be neglected and that the interface scattering is spin- conserving, the mixing conductance is defined as Gmix=e2 hA/summationdisplay m,n/parenleftbig δmn−r↑ mnr↓∗ mn/parenrightbig , (2) in terms of rσ mn, the probability amplitude for reflection ofaNMmetalstate nwithspin σintoaNM state mwith thesamespin. UsingEq.( 2), wecalculate GmixforPy|Pt and Py|Cu interfaces without SOC and indicate the cor- responding damping enhancement gµB/planckover2pi1Gmix/(e2MsA) on the vertical axis in Fig. 2with asterisks. When SOC is included, Eq. ( 2) is no longer applicable. Wecanneverthelessidentify aspin-pumpinginterfaceen- hancement Gmixas follows. We artificially turn off the3 0 2 4 6 8 10 d (nm)00.050.100.15αd (nm)Pt CuWithout backflow With backflow FIG. 2. (color online). Total damping calculated for Pt |Py|Pt and Cu|Py|Cu as a function of the Py thickness. The open symbols correspond to the case without backflow while the full symbols are the results shown in Fig. 1where backflow was included. The lines are linear fits to the symbols. The as- terisks on the yaxis are the values of Gmixcalculated without SOC using Eq. ( 2). backflow by connecting the FM metal to ballistic NM leads so that any spin current pumped through the in- terface propagatesawayimmediately and there is no spin accumulation in the NM metal. The Gilbert damping αd calculated without backflow (dashed lines) is linear in the Py thickness d; the intercept Γ at d= 0 represents an interface contribution. As seen in Fig. 2for Cu, Γ coincides with the orange asterisk meaning that the in- terface damping enhancement for a Py |Cu interface is, within the accuracy of the calculation, unchanged by in- cluding SOC because this is so small for Cu, Ni and Fe. By contrast, Γ and thus Gmix=e2MsAΓ/(gµB/planckover2pi1) for the Py|Pt interface is 25% larger with SOC included, con- firming the breakdown of Eq. ( 2) for interfaces involving heavy elements. The data in Fig. 1for NM=Pt and Cu are replotted as solid lines in Fig. 2for comparison. Their linearity means that we can extract an effective mixing conduc- TABLE I. Different mixing conductances calculated for Py|NM interfaces. Gmixis calculated using Eq. ( 2) without SOC.Gmixis obtainedfrom theinterceptofthetotal damping αdcalculated as a function of the Py thickness dwith SOCfor ballistic NM leads. The effective mixing conductance Gmix effis extracted from the effective αin Fig.1in the presence of 5 nm NM on either side of Py. Sharvin conductances are listed for comparison. All values are given in units of 1015Ω−1m−2. NM GSh GmixGmixGmix eff Cu 0.55 0.49 0.48 0.01 Pd 1.21 0.89 0.98 0.57 Ta 0.74 0.44 0.48 0.34 Pt 1.00 0.86 1.07 0.95tanceGmix effwith backflow in the presence of 5 nm dif- fusive NM metal attached to Py. For Py |Pt,Gmix effis only reduced slightly compared to Gmixbecause there is very little backflow. For Py |Cu, the spin current pumped into Cu is only about half that for Py |Pt. However, the spin-flipping in Cu is so weak that spin accumulation in Cu leads to a backflow that almost exactly cancels the pumped spin current and Gmix effis vanishingly small for the Py|Cu system with thin, diffusive Cu. The values of Gmix,GmixandGmix effcalculated for all four NM metals are listed in Table I. Because Gmix(Pd) andGmix(Pt) are comparable, Py pumps a similar spin current into each of these NM metals. The weaker spin- flipping and larger spin accumulation in Pd leads to a larger backflow and smaller damping enhancement. The relatively low damping enhancement in Ta |Py|Ta results from a small mixing conductance for the Ta |Py interface rather than from a large backflow. In fact, Ta behaves as a good spin sink due to its large SOC and the damp- ing enhancement in Ta |Py|Ta can not be significantly increased by suppressing the backflow. Thickness dependence of NM. —In the following we fo- cus on the Pt |Py|Pt system and examine the effect of changing the NM thickness lon the damping enhance- ment, a procedure frequently used to experimentally de- termine the NM spin-flip diffusion length [ 27–31]. The total damping calculated for Pt |Py|Pt is plotted in Fig.3as a function of the Pt thickness lfor two thick- nessesdof Py. For both d= 1 nm and d= 2 nm, αsaturates at l=1–2 nm in agreement with experiment 0 10 20 30 40 50 l (nm)0.00.51.0 0 1 2 3 4 5 l (nm)0.000.050.100.15 αd=1 nm d=2 nmPt(l)|Py(d)|Pt( l)G↑↑/G↑ G↑↓/G↑Pt@RT l↑=7.8±0.3 nm FIG. 3. αas a function of the Pt thickness lcalculated for Pt(l)|Py(d)|Pt(l). The dashed and solid lines are the curves obtained by fitting without and with interface spin memory loss, respectively. Inset: fractional spin conductances G↑↑/G↑ andG↑↓/G↑when a fully polarized up-spin current is injected into bulk Pt at room temperature. Gσσ′is (e2/htimes) the transmission probability of a spin σfrom the left hand lead into a spin σ′in the right hand lead; G↑=G↑↑+G↑↓. The value of the spin-flip diffusion length for a single spin chann el obtained by fitting is lσ= 7.8±0.3 nm.4 [17,28–31]. AfitofthecalculateddatausingEq.( 1)with δ≡0 requires just three parameters, Gmix,ρandlsf. A separate calculation gives ρ= 10.4µΩcm at T=300 K in very good agreement with the experimental bulk value of 10.8µΩcm [32]. Using the calculated Gmixfrom Table I leaves just one parameter free; from fitting, we obtain a valuelsf=0.8 nm for Pt (dashed lines) that is consis- tent with values between0.5and 1.4nm determined from spin-pumping experiments [ 28–31]. However, the dashed lines clearly do not reproduce the calculated data very well and the fit value of lsfis much shorter than that extracted from scattering calculations [ 11]. By injecting a fully spin-polarized current into diffusive Pt, we find l↑=l↓= 7.8±0.3nm, asshownin theinsettoFig. 3, and from [25,33],lsf=/bracketleftbig (l↑)−2+(l↓)−2/bracketrightbig−1/2= 5.5±0.2 nm. This value is confirmed by examining how the current polarization in Pt is distributed locally [ 34]. If we allow for a finite value of δand use the in- dependently determined Gmix,ρandlsf, the data in Fig.3(solid lines) can be fit with δ= 3.7±0.2 and R∗/δ= 9.2±1.7 fΩm2. The solid lines reproduce the calculateddatamuch better than when δ= 0 underlining the importance of including interface spin-flip scattering [17,35]. The large value of δwe find is consistent with a low spin accumulation in Pt and the corresponding very weak backflow at the Py |Pt interface seen in Fig. 2. Conductivity dependence. —Many experiments deter- mining the spin-flip diffusion length of Pt have reported Pt resistivities that range from 4.2–12 µΩcm at low tem- perature [ 35–38] and 15–73 µΩcm at room temperature [17,39–41]. The large spread in resistivity can be at- tributed to different amounts of structural disorder aris- ing during fabrication, the finite thickness of thin film samples etc. We can determine lsfandρ≡1/σfrom first principles scattering theory [ 11,12] by varying the temperature in the thermal distribution of Pt displace- ments in the range 100–500 K. The results are plot- ted (black solid circles) in Fig. 4(a).lsfshows a lin- ear dependence on the conductivity suggesting that the Elliott-Yafet mechanism [ 42,43] dominates the conduc- tion electron spin relaxation. A linear least squares fit yieldsρlsf= 0.61±0.02fΩm2that agrees very well with bulk data extracted from experiment that are either not sensitive to interface spin-flipping [ 37] or take it into ac- count [17,35,38]. For comparison, we plot values of lsf extracted from the interface-enhanced damping calcula- tions assuming δ= 0 (empty orange circles). The result- ing values of lsfare very small, between 0.5 and 2 nm, to compensate for the neglect of δ. Having determined lsf(σ), we can calculate the interface-enhanced damping for Pt |Py|Pt for different values of σPtand repeat the fitting of Fig. 3using Eq. ( 1) [44]. The parameters R∗/δandδare plotted as a func- tion of the Pt conductivity in Fig. 4(b). The spin mem- ory lossδdoes not show any significant variation about0102030R*/δ (fΩ m2) 0 0.1 0.2 0.3 σ (108Ω-1m-1)024 δ1020lsf (nm)Rojas-Sánchez Niimi Nguyen Kurt50 20 10 7 54ρ (µΩ cm) δ=0(a) (b) FIG. 4. (a) lsffor diffusive Pt as a function of its conductivity σ(solid black circles) calculated by injecting a fully polar ized current into Pt. The solid black line illustrates the linear dependence. Bulk values extracted from experiment that are eithernotsensitivetointerface spin-flipping[ 37]ortakeitinto account [ 17,35,38] are plotted (squares) for comparison. The empty circles are values of lsfdetermined from the interface- enhanced damping using Eq. ( 1) withδ= 0. (b) Fit values of R∗/δandδas a function of the conductivity of Pt obtained using Eq. ( 1). The solid red line is the average value (for different values of σ) ofδ=3.7. 3.7, i.e., it does not appear to depend on temperature- induced disorder in Pt indicating that it results mainly from scattering of the conduction electrons at the abrupt potential change of the interface. Unlike δ, the effective interfaceresistance R∗decreaseswithdecreasingdisorder in Pt and tends to saturate for sufficiently ordered Pt. It suggests that although lattice disorder at the interface does not dissipate spin angular momentum, it still con- tributestotherelaxationofthemomentumofconduction electrons at the interface. Conclusions. —We have calculated the Gilbert damp- ing for Py |NM-metal interfaces from first-principles and reproduced quantitatively the experimentally observed damping enhancement. To interpret the numerical re- sults, we generalized the spin-pumping expression for the damping to allow for interface spin-flipping, a mix- ing conductance modified by SOC, and spin dependent interface resistances. The resulting Eq. ( 1) allows the two main factors contributing to the interface-enhanced damping to be separated: the mixing conductance that determinesthespincurrentpumpedbyaprecessingmag- netization and the spin accumulation in the NM metal that induces a backflow of spin current into Py and low- ers the efficiency of the spin pumping. In particular, the latter is responsible for the low damping enhancement for Py|Cu while the weak enhancement for Py |Ta arises from the small mixing conductance. We calculate how the spin-flip diffusion length, spin5 memory loss and interface resistance depend on the con- ductivity of Pt. It is shown to be essential to take ac- count of spin memory loss to extract reasonable spin- flip diffusion lengths from interface damping. This has important consequences for using spin-pumping-related experiments to determine the Spin Hall angles that char- acterize the Spin Hall Effect [ 17]. Acknowledgments. —We are grateful to G.E.W. Bauer for a critical reading of the manuscript. Our work was financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO) through the research programme of “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) and the supercomputer facilities of NWO “Exacte Wetenschappen (Physical Sci- ences)”. It was also partly supported by the Royal Netherlands Academy of Arts and Sciences (KNAW). Z. Yuan acknowledges the financial support of the Alexan- der von Humboldt foundation. ∗Present address: Institut f¨ ur Physik, Johannes Gutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128 Mainz, Germany †zyuan@uni-mainz.de [1] See the collection of articles in Handbook of Spin Trans- port and Magnetism , edited by E. Y. 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In addition, their treatment contains an unknown empiri- cal parameter, the ratio of the spin-conserved to spin-flip relaxation times. [24] A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2006) . [25] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993) . [26] D. R. Penn and M. D. Stiles, Phys. Rev. B 72, 212410 (2005) . [27] A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez- Su´ arez, A. F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B 83, 144402 (2011) . [28] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011) . [29] K. Kondou, H. Sukegawa, S. Mitani, K. Tsukagoshi, and S. Kasai, Applied Physics Express 5, 073002 (2012) . [30] C. T. Boone, H. T. Nembach, J. M. Shaw, and T. J. Silva,J. Appl. Phys. 113, 153906 (2013) . [31] W. Zhang, V. Vlaminck, J. E. Pearson, R. Divan, S. D. Bader, and A. Hoffmann, Appl. Phys. Lett. 103, 242414 (2013) . [32] See §12: Properties of Solids - Electrical Resistivity of Pure Metals in D. R. 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1606.04483v2.Anomalous_Damping_of_a_Micro_electro_mechanical_Oscillator_in_Superfluid___3_He_B.pdf

arXiv:1606.04483v2 [cond-mat.supr-con] 7 Oct 2016Anomalous Damping of a Micro-electro-mechanical Oscillat or in Superfluid3He-B P. Zheng,1W.G. Jiang,1C.S. Barquist,1Y. Lee,1,∗and H.B. Chan2 1Department of Physics, University of Florida, Gainesville , Florida 32611-8440, USA 2Department of Physics, The Hong Kong University of Science a nd Technology, Clear Water Bay, Hong Kong (Dated: June 4, 2021) The mechanical resonanceproperties ofamicro-electro-me chanical oscillator with agapof1.25 µm was studied in superfluid3He-B at various pressures. The oscillator was driven in the l inear damping regime where the damping coefficient is independent of the osc illator velocity. The quality factor of the oscillator remains low ( Q≈80) down to 0.1 Tc, 4 orders of magnitude less than the intrinsic quality factor measured in vacuum at 4 K. In addition to the Bo ltzmann temperature dependent contribution to the damping, a damping proportional to temp erature was found to dominate at low temperatures. We propose a multiple scattering mechanism o f the surface Andreev bound states to be a possible cause for the anomalous damping. Over several decades, different families of unconven- tional superconductors have been discovered. Many of these possess high transition temperatures, which generates much interest from the community pursuing room temperature superconductors. However, the com- plete microscopic understanding of them still remains a challenge1. Superfluid3He with p-wave spin-triplet pair- ing is a prime model system to study the unconventional nature of Cooper pairs because the symmetry of the con- densate is clearly identified and the properties of the intrinsically pure bulk system are well understood to a quantitative level2. The early theoretical works3,4have revealedthe extremefragilityofCooperpairsagainstany type of impurity scattering in unconventional supercon- ductors. Interfaces and surfaces also serve as effective pair-breaking agents in these systems, which results in many intriguing surface properties5,6. The surface scat- tering in unconventional superfluids/superconductors in- duces quasiparticle mid-gap bound states spatially local- ized near the surface within the coherence length, ξ0, often called surface Andreev bound states (SABS), ac- companying selective suppression of the order param- eter components7–10. The detailed structure of SABS has been theoretically investigated for various boundary conditions9,10. In superconductors, tunneling spectroscopyhasproven to be a powerful tool for studying the pairing symmetry and surface states11. However, the detection of SABS in superfluid3He has been difficult due to the lack of an appropriate probe for the uncharged fluid. Nevertheless, various workshavesuggested the existence ofSABS12–20. Measurements of transverse acoustic impedance using quartztransducershavebeenusedtoinvestigateSABS12. The measured transverseacoustic impedance agreeswith theoretical calculations, which provides indirect confir- mation of SABS13,14. The high resolution heat capacity measurement of3He in a silver heat exchanger was able to identify the contribution from the SABS near the sil- ver surface15. In the recent experiment by the Lancaster group16, they linked the absence of the critical velocity of a wire moving without acceleration to the presence FIG. 1. A schematic side-view of the MEMS device. A mobile center plate is suspended above the bottom plate by springs (not shown). The gap Dbetween the mobile plate and the bottom plate is 1.25 µm. The thickness of each layer is shown to scale. The horizontal arrow represents the direction of t he oscillation of the shear mode. of SABS. Recent theoretical studies provide a fresh in- sight into the nature of SABS21,22. They suggest the anisotropic magnetic response of the film or surface of 3He-B with specular boundaries as a direct indicator of Majorana fermions in surface bound states. Various resonators in direct contact with liquid3He, such as torsional oscillators23, vibrating wires24,25, tun- ing forks26,27, and movingwires16, have been successfully utilized to investigatethe propertiesofits normaland su- perfluid phases. A new direction in the development of the mechanical probes is based on the nanolithography technology, such as micro- and nano-electro-mechanical system (MEMS and NEMS) devices28,29. We have devel- oped MEMS devices to study superfluid3He films30,31. Theses devices have also been successfully exploited to study the viscosityof normal liquid3He below 800mK32. In this paper, we report the measurement of the damp- ing of a MEMS device in superfluid3He-B which exhibits anomalous low temperature behavior. A plausible phys- ical mechanism involving SABS is conjectured to be re- sponsible for the observed behavior. The MEMS device used in this measurement has a mobile plate with 2 µm thickness and 200 µm lateral size. The plate is suspended above the substrate by four serpentine springs, maintaining a gap of 1.25 µm. A schematic side-view of the device is shown in Fig.1.2 When the device is submerged in the fluid, a film is formedbetween the mobile plate and the substrate, while the bulk fluid is in direct contact with the top surface of the plate. Its in-plane oscillation, called the shear mode, can be actuated and detected by the comb elec- trodes fabricated on either side of the plate. The details of the devices and the measurement scheme can be found elsewhere28,33,34. The MEMS device was studied in liquid3He at pres- sures of 9.2, 18.2, 25.2, and 28.6 bars and cooled down to a base temperature of about 250 µK by a dilution re- frigerator and a copper demagnetization stage. The res- onance spectrum of the shear mode was obtained contin- uously upon warming from the base temperature with a typical warming rate of 30 µK/hr. The temperature was determined by calibrated tuning fork thermometers26,27 below 0.6 mK and by a3He melting curve thermometer above. The PLTS-2000 was adopted as the temperature scale35. The uncertainty oftemperature measured by the tuning forks is mainly from the calibration process and is represented by error bars in Fig.2. A magnetic field of 14 mT was applied in the direction perpendicular to the plane of the film except for one of the 28.6 bar measure- ments. The full width at half maximum (FWHM), γ, and the resonance frequency, f0, were obtained by fitting the spectrum to the Lorentzian: x=Aγf0/radicalbig (f2 0−f2)2+(γf)2, (1) wherexis the vibration amplitude of the plate, A= F0/4π2mf0γis the amplitude of the Lorentzian peak, F0is the amplitude of the driving force applied on the plate,misthe effectivemassoftheplate, and fisthe fre- quency of the driving force. The FWHM is proportional to the damping coefficient in the equation of motion of a damped driven harmonic oscillator. The uncertainty from fitting is represented by error bars in Fig.2. The resonance feature of the MEMS device is sensitive to the temperature36. Forinstance, at28.6baritsqualityfactor reaches around 80 when the liquid is cooled to 300 µK, anddecreasesrapidlytoorderofunitynearthe A-Btran- sition. The mean free path, ℓ, of the3He quasiparticles is of the order of 10 µm at the transition temperature and in- creases exponentially when the temperature approaches zero due to the isotropic energy gap of3He-B2. For T<∼0.4Tc,ℓbecomes larger than any length scale of the MEMS devices, and the MEMS-superfluid system tran- sitions into the ballistic regime. This aspect is verified by the temperature independent resonance frequency ob- served in this temperature range. At low velocities, the damping has a temperature dependence solely from the density ofthe quasiparticleswhich decreasesrapidlywith temperature as exp( −∆/kBT)25. Below 0.4 Tc, the en- ergy gap ∆ develops fully to the zero-temperature value,/s48/s46/s49 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s49/s48/s50/s49/s48/s51/s49/s48/s52 /s50 /s52 /s54 /s56/s49/s48/s50/s49/s48/s51/s49/s48/s52 /s32/s32/s77/s69/s77/s83/s32 /s32/s40/s72/s122/s41 /s84/s99/s32/s47/s32/s84 /s32/s32/s57/s46/s50/s32/s98/s97/s114 /s32/s49/s56/s46/s50/s32/s98/s97/s114 /s32/s50/s53/s46/s50/s32/s98/s97/s114 /s32/s50/s56/s46/s54/s32/s98/s97/s114 /s32/s50/s56/s46/s54/s32/s98/s97/s114/s32/s40/s122/s101/s114/s111/s32/s102/s105/s101/s108/s100/s41/s77/s69/s77/s83/s32 /s32/s40/s72/s122/s41 /s84/s32/s47/s32/s84/s99/s126/s32/s84 FIG. 2. ( Color online ) The FWHM of the MEMS as a func- tion of the reduced temperature at various pressures in a log - log scale. For clarity, the error bars are only shown for the data of 9.2 and 18.2 bar. The straight line corresponds to a linear temperature dependence. ( Inset) The same data in an Arrhenius scale. A straight line in this scale represents a Boltzmann dependence. ∆0. The FWHM of the MEMS is expected to follow γ=Bexp(−∆0/kBT), (2) whereBis the damping amplitude determined by the geometry of the device and the properties of the fluid. The intrinsic FWHM measured in vacuum at 4 K, which is 0.071 Hz, is subtracted from the fitted FWHM to yield the FWHM due to the fluid only. This FWHM is plotted as a function of the reduced temperature at vari- ous pressures in Fig.2. At the lowest attainable temper- ature for 28.6 bar, 280 µK, the FWHM is around 270 Hz, which is 4 orders of magnitude larger than the intrinsic FWHM and 2 orders of magnitude larger than the TF FWHM in the same condition37. In contrast, the FWHM of MEMS in superfluid4He below 200 mK is weakly tem- perature dependent and approaches the intrinsic value38. Therefore, the anomalously large damping observed in 3He-B is believed to stem from some mechanism other than the scattering of thermal quasiparticles from the bulk. As shown in the inset of Fig.2, the FWHM does not follow Eqn.(2). Furthermore, the FWHM becomes linear in temperature below ∼0.15Tcfor the three high- est pressures. For 9.2 bar, the linear dependence is not fully developed, probably because of the relatively low Tcat this pressure. This linear temperature dependent term emerging at low temperatures keeps the FWHM of the MEMS from decreasing exponentially as expected. The coefficient of the linear temperature dependent con- tribution can be extracted from the ratio of FWHM to temperature in the low temperature limit39. The acquired linear term is then subtracted from the total FWHM. The residual FWHM is plotted against the temperature in an Arrhenius scale in Fig.3. The linear behavior of the three highest pressures demonstrates a3 /s50 /s52 /s54 /s56/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52 /s32/s32 /s32/s57/s46/s50/s32/s98/s97/s114 /s32/s49/s56/s46/s50/s32/s98/s97/s114 /s32/s50/s53/s46/s50/s32/s98/s97/s114 /s32/s50/s56/s46/s54/s32/s98/s97/s114 /s32/s50/s56/s46/s54/s32/s98/s97/s114/s32/s40/s122/s101/s114/s111/s32/s102/s105/s101/s108/s100/s41/s77/s69/s77/s83/s32 /s66/s32/s40/s72/s122/s41 /s84 /s67/s32/s47/s32/s84 FIG. 3. ( Color online ) The FWHM of the Boltzmann damp- ing,γB, against thereducedtemperature for various pressures in an Arrhenius scale. A linear fit ( straight lines ) in this scale gives the measured energy gap, ∆ m. Boltzmann exponential temperature dependence follow- ing Eqn.(2) and justifies the assumption that the damp- ing in addition to the expected thermal quasiparticles in the bulk is linear. Since its temperature dependence is not fully developed at low temperatures for 9.2 bar, the linear term is estimated by requiringthe residualFWHM to obey Eqn.(2). It was found that the residual FWHM data are sensitive to the choice of the linear coefficient. A 5% variation in the value is sufficient to skew the de- pendence of the residual FWHM. Therefore, the total FWHM can be expressed by γ=γA+γB=αT+Bexp(−∆0/kBT),(3) whereγAis the linear temperature dependent term that dominates at low temperatures, and γBis the Boltzmann exponentialtemperaturedependent termduetothe ther- mal quasiparticles in the bulk region. Hereafter, the ex- ponential term is called the Boltzmann damping and the linear term the additional damping. The coefficient of the additional damping, α, decreases by a factor of two as the pressure increases from 9.2 bar to 28.6 bar (Fig.4). The linear coefficient seems to have a linear dependence on the coherence length. For the Boltzmann damping, the data in Fig.3 can be fitted to straight lines according to Eqn.(2) to get ∆ m, the mea- suredenergygap. It wasfoundthat ∆ mismuchlessthan the known bulk value at the correspondingpressure. The pressure dependence of ∆ mis presented in terms of the BCS coherence length ξ0= ¯hvF/π∆0, wherevFis the Fermi velocity (Fig.4). The measured energy gap is sup- pressed from the bulk value and shows a strong pressure dependence. It decreases monotonically with the scaled film thickness, D/ξ0, since a larger effective thickness gives more space for the order parameter to recoverto its bulk value, hence a larger overall energy gap for the film. Also shown in the plot is a calculation which evaluates/s50/s48 /s50/s53 /s51/s48 /s51/s53/s55/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48 /s32/s32/s40/s72/s122/s47/s109/s75/s41 /s67/s111/s104/s101/s114/s101/s110/s99/s101/s32/s108/s101/s110/s103/s116/s104/s32/s40/s110/s109/s41 /s50/s48 /s52/s48 /s54/s48/s49/s46/s48/s49/s46/s53/s49/s46/s57/s32 /s32/s77/s101/s97/s115/s117/s114/s101/s100 /s32/s66/s117/s108/s107 /s32/s86/s111/s114/s111/s110/s116/s115/s111/s118/s32/s107 /s66/s84 /s67 /s68/s32/s47/s32 /s48 FIG. 4. ( Top) The coefficient of the linear temperature term, α, against the coherence length of the bulk superfluid. ( Bot- tom) The measured energy gap, ∆ m, against the scaled film thickness, D/ξ0. The error bars associated with ∆ mare from the linear fitting in Fig.3. Also plotted are the bulk en- ergy gap and the average energy gap in a film calculated by Vorontsov. The bulk gap used here is from the weak coupling plus model40. the energy gap by averaging the parallel and perpendic- ular components of the order parameter for a superfluid film with both boundaries diffusive41. Our measurement shows a much stronger suppression and pressure depen- dence of the energy gap than the theoretical estimation. One might argue the local heating from the MEMS de- vice is responsible for the additional damping. However, the film formed in our MEMS device is in good thermal contact with the surrounding bulk because of the open geometry. Furthermore, the temperature rise due to the heat dissipation is negligible. For instance, at 0.2 Tcand 9.2 bar, when the center plate is oscillating at a veloc- ity of 1.4 mm/s, the damping force on the plate is about 2 nN, which results in a dissipation powerofabout 3 pW. In addition, all the measurements were performed in the linear regime where the damping coefficient was indepen- dent of the excitation42. Any heating effect would have resulted in the increase of the FWHM at higher excita- tions. It is also unlikely that the additional damping comes from the vortices around the MEMS devices or other topological objects as suggested by Winkelmann et al.43, because multiple independent cooldowns produced con- sistentspectraatagiventemperatureandpressure. Dur- ing each thermal cycle the MEMS device was driven at high velocities beyond the linear regime where heating effect was clearly observed. The severe heating and the high velocity should have altered vortex lines or other topological objects around the device. But after a rea- sonable relaxation time, the spectrum always recovers to4 the shape right before the heating. However, it is possible that the mobile plate dissipates through the surface bound states in the vicinity of the plate, leading to the additional damping. The atomic force microscopystudy of the MEMS surfaces shows that the average height variation of the polysilicon surface is ≈10 nm, while their lateral size is ≈150 nm28. Since these length scales are much larger than the Fermi wave- length of the3He quasiparticles, the surface of the plate is diffusive. The density of states, D(E), for the surface bound states is almost independent of energy for a diffu- sive boundary9(Fig.5). It is reasonable to project that the number of quasiparticles excited in the bound states should be proportional to temperature. Therefore, the scatteringofthe quasiparticlesoff the movingplate could lead to a linear temperature dependence of the damping, if the transverse momentum transfer occurs. The perpendicular component of the order parameter is completely suppressed at either specular or diffusive boundary9. One can expect that quasiparticles will be generated with an infinitesimally small amount of energy inthesurfaceboundstates, whichareconfinedbythegap potential around the boundary within a distance char- acterized by the coherence length ξ0(Fig.5). Parallel to the plane of the plate with a specular boundary, the bound quasiparticles move with a slow velocity v/bardbl≈vL, wherevLis the Landau critical velocity. In the direction perpendicular to the plane, however, the quasiparticle moves with a fast velocity v⊥≈vF, since the energy gap is closed in this direction. The Fermi velocity of the su- perfluid varies from 60 to 35 m/s as the pressure changes from0to30bar, while thecoherencelengthchangesfrom 90 to 18 nm in the pressure range. Therefore it takes ap- proximately 1 ns for a quasiparticle to travel from the surface of the plate to the edge of the potential well, where it is then retroreflected due to the Andreev scat- tering. Thequasiparticlebecomesaquasiholeandfollows its previous path, moving towards the plate. It is scat- tered normally off the plate and Andreev scattered off the gap potential again before returning to its original position (Fig.5). This completes an entire loop involving the normal and Andreev scattering. Considering the res- onance frequency of the MEMS device ( ≈20 kHz), one estimates that about 104such scatterings occur within one cycle of the oscillation. However, for the normal scattering at the specular boundary, there is no momen- tum transfer in the parallel direction between the plate and the quasiparticles, hence no damping for the shear motion of the plate. Therefore, one expects a very small damping force for the specular boundary. This may be verified by coating the MEMS plate with a couple of lay- ers of4He atoms, since the4He atoms drastically alter the boundary conditions44,45. For a diffusive boundary, it is difficult to trace the tra- jectory of a particular quasiparticle, though the process of the multiple Andreev scattering is still valid. Those FIG. 5. (A) A schematic picture showing the surface den- sity of states of superfluid3He at a diffusive boundary9. The quasiparticles excited in the mid-gap band are promoted by the MEMS up to the edge, ∆∗. (B) The SABS confined by a potential well near the boundary at z= 0. (C) A complete scattering cycle of a quasiparticle at a specular boundary i n- volving two normal scatterings and two Andreev scatterings . havingananti-parallelgroupvelocitycomponentwithre- spect to the plate velocity vpwill have a higher chance of scattering, resulting in a net flux proportional to vp. The tiny difference between the momentum of quasiparticles andquasiholesaroundtheFermimomentumaccumulates due to the high number of scattering during one cycle of the plate motion. This multiple scattering process leads to a net momentum transfer between the plate and the bound states which are then promoted to higher energy states until the mid-gap edge ∆∗is reached9,14. We pro- posethatthisprocesscouldbetheunderlyingmechanism for the large and linear temperature dependent damping. Furthermore, our measurements in the nonlinear regime, which will be reported elsewhere, can be coherently un- derstood with this mechanism. Nonetheless, this model neither requires the presence of a film nor involves sur- facebound stateson the otherside ofthe film. Currently, we do not understand the influence of another surface in close proximity on the damping of the plate. To clarify this, we have designed MEMS devices with the substrate etched away so that both sides of the plate are exposed to bulk fluid. In conclusion, a superfluid3He film with a thickness of 1.25µm was studied by a MEMS device at various pressures. At low temperatures, an anomalously large damping on the MEMS was measured in addition to the ordinary Boltzmann damping. It was attributed to a multiple scattering picture of the interaction between the MEMS devices and the surface bound states on the film. WewouldliketoacknowledgePeterHirschfeldandAn- tonVorontsovforhelpful discussionandcalculations. We alsowanttothanktheLancasterLowTemperaturegroup5 for providing quartz tuning forks as one of the TF ther- mometers. This work was supported by the National Science Foundation, No. DMR-1205891. ∗yoonslee@phys.ufl.edu [1] M. R. Norman, Science 332, 196 (2011). [2] D. Vollhardt and P. W¨ olfle, The Superfluid Phases of Helium 3 (Taylor & Francis, 1990). [3] A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP 12, 1243 (1961). [4] A. I. Larkin, JETP Lett. 2, 130 (1965). [5] L. J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788 (1981). [6] L. J. Buchholtz, Phys. Rev. B 44, 4610 (1991). [7] V. Ambegaokar, P. G. de Gennes, and D. 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1202.5379v1.Small_data_global_existence_for_the_semilinear_wave_equation_with_space_time_dependent_damping.pdf

arXiv:1202.5379v1 [math.AP] 24 Feb 2012SMALL DATA GLOBAL EXISTENCE FOR THE SEMILINEAR WAVE EQUATION WITH SPACE-TIME DEPENDENT DAMPING YUTA WAKASUGI Abstract. In this paper we consider the critical exponent problem for t he semilinear wave equation with space-time dependent dampin g. When the damping is effective, it is expected that the critical expone nt agrees with that of only space dependent coefficient case. We shall prove that t here exists a unique global solution for small data if the power of nonline arity is larger than the expected exponent. Moreover, we do not assume that the da ta are com- pactly supported. However, it is still open whether there ex ists a blow-up solution if the power of nonlinearity is smaller than the exp ected exponent. 1.Introduction We consider the Cauchy problem for the semilinear damped wave equa tion /braceleftBigg utt−∆u+a(x)b(t)ut=f(u),(t,x)∈(0,∞)×Rn, u(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.1) where the coefficients of damping are a(x) =a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α, b(t) = (1+t)−β,witha0>0,α,β≥0,α+β <1, where/a\}b∇acketle{tx/a\}b∇acket∇i}ht= (1+|x|2)1/2. Hereuis a real-valued unknown function and ( u0,u1) is inH1(Rn)×L2(Rn). We note that u0andu1need not be compactly supported. The nonlinear term f(u) is given by f(u) =±|u|por|u|p−1u and the power psatisfies 1<p≤n n−2(n≥3),1<p<∞(n= 1,2). Our aim is to determine the critical exponent pc, which is a number defined by the following property: Ifpc<p, all small data solutions of (1.1) are global; if 1 <p≤pc, the time-local solution cannot be extended time-globally for some data. It is expected that the critical exponent of (1.1) is given by pc= 1+2 n−α. In this paper we shall prove the existence of global solutions with sm all data when p>1+2/(n−α). However, it is still open whether there exists a blow-up solution when 1<p≤1+2/(n−α). Key words and phrases. semilinear damped wave equations; critical exponent; smal l data global existence. 12 YUTA WAKASUGI When the damping term is missing and f(u) =|u|p, that is /braceleftBigg utt−∆u=|u|p,(t,x)∈(0,∞)×Rn, u(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.2) itiswellknownthatthecriticalexponent pw(n)isthepositiverootof( n−1)p2−(n+ 1)p−2 = 0 forn≥2 (pw(1) =∞). This is the famous Strauss conjecture and the proofis completed by the effort of many mathematicians (see [2,3,1 3,21,30–34,41]). For the linear wave equation with a damping term/braceleftBigg utt−∆u+c(t,x)ut= 0,(t,x)∈(0,∞)×Rn, u(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.3) there are many results about the asymptotic behavior of the solut ion. When c(t,x) =c0>0 and (u0,u1)∈(H1∩L1)×(L2∩L1), Matsumura [22] showed that the energy of solutions decays at the same rate as the corre sponding heat equation. When the space dimension is 3, using the exact expression of the solu- tion, Nishihara[24] discoveredthat the solutionof(1.3) with c(t,x) = 1is expressed asymptotically by u(t,x)∼v(t,x)+e−t/2w(t,x), wherev(t,x) is the solution of the corresponding heat equation /braceleftBigg vt−∆v= 0,(t,x)∈(0,∞)×R3, v(0,x) =u0(x)+u1(x), x∈R3 andw(t,x) is the solution of the free wave equation /braceleftBigg wtt−∆w= 0,(t,x)∈(0,∞)×R3, w(0,x) =u0(x), wt(0,x) =u1(x),∈R3. These results indicate a diffusive structure of damped waveequatio ns. On the other hand, Mochizuki [23] showed that if 0 ≤c(t,x)≤C(1+|x|)−1−δ, whereδ>0, then the energy of solutions of (1.3) does not decay to 0 for nonzero da ta and the solu- tion is asymptotically free. We can interpret this result as (1.3) loses its “parabolic- ity”and recoverits “hyperbolicity”. Wirth [38,39] treated time-d ependent damping case, that is c(t,x) =b(t) in (1.3). By the Fourier transform method, he got sev- eral sharpLp−Lqestimates of the solution and showed that there exists diffusive structure for general b(t) including b(t) =b0(1+t)−β(−1<β <1). Todorova and Yordanov [37] considered the case c(t,x) =a(x) =a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αwithα∈[0,1) and J. S. Kenigson and J. J. Kenigson [16] considered space-time depen dent coefficient casec(t,x) =a(x)b(t),a(x) =a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α,b(t) = (1 +t)−β,(0≤α+β <1). They established the energy decay estimate that also implies diffusive stru cture even in the decaying coefficient cases. From these results, the decay rat e−1 of the coeffi- cient of the damping term is the threshold of parabolicity. This is the r eason why we assume α+β <1 for (1.1). We mention that recently, Ikehata, Todorova and Yordanov[12] treated the case c(t,x) =a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1and obtained almost optimal decay estimates. There are also many results for the semilinear damped wave equation with ab- sorbing semilinear term:/braceleftBigg utt−∆u+a(x)b(t)ut+|u|p−1u= 0,(t,x)∈(0,∞)×Rn, u(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.4)SEMILINEAR DAMPED WAVE EQUATIONS 3 It is well known that there exists a unique global solution even for lar ge initial data. Whena(x)b(t) = 1, that is constant coefficient case, Kawashima, Nakao and Ono [15], Karch[14], Hayashi, KaikinaandNaumkin [7], Ikehata, Nishiharaan dZhao[9] and Nishihara [25] showed global existence of solutions and that the ir asymptotic profile is given by a constant multiple of the Gauss kernel for 1+2 /n<pandn≤4. For 1< p≤1 + 2/n, Nishihara and Zhao [28], Ikehata, Nishihara and Zhao [9], Nishihara [25] proved that the decay rate of the solution agrees wit h that of a self- similarsolutionofthe correspondingheatequation. Hayashi,Kaikina andNaumkin [4–7] proved the large time asymptotic formulas in terms of the weigh ted Sobolev spaces. These results indicate the critical exponent for (1.4) with a(x)b(t) = 1 is given bypc= 1+2 n. In this case the critical exponent means the turning point of the asymptotic behavior of the solution. When b(t) = 1,a(x) =/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α(0≤α<1), namely space-dependent damping case, Nishihara [26] established d ecay estimates of solutions and conjectured the critical exponent is given by pc= 1+2/(n−α). Whena(x) = 1,b(t) = (1 +t)−β(−1< β <1), Nishihara and Zhai [29] proved decay estimates of solutions and conjectured the critical expone nt ispc= 1+2/n. Finally in the case a(x) =/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α,b(t) = (1+t)−β(0≤α+β <1), Lin, Nishihara and Zhai [19,20] showed decay estimates of the solution and conje ctured the critical exponent is pc= 1 + 2/(n−α). They used a weighted energy method, which is originally developed by Todorova and Yordanov [35,36]. In this paper we shall essentially use the techniques and method that they used. Li and Zhou [17] considered the semilinear damped wave equation utt−∆u+ut=|u|p. (1.5) They proved that if n≤2,1<p≤1+2 nand the data are positive on average, then the local solution of (1.5) must blow up in a finite time. Todorova and Yo rdanov [35,36] developed a weighted energy method using the function whic h has the form e2ψand determined that the critical exponent of (1.5) is pc= 1+2 n, which is well known as Fujita’s critical exponent for the heat equatio nut−∆u= up(see [1]). More precisely, they proved small data global existence in the case p>1+2/nand blow-up for all solutions of (1.5) with positive on average data in the case 1< p <1+2/n. Later on Zhang [40] showed that the critical exponent p= 1+2/nbelongstotheblow-upregion. WementionthatTodorovaandYorda nov [35,36] assumed data have compact support and essentially used t his property. However, Ikehata and Tanizawa [10] removed this assumption. Ik ehata, Todorova and Yordanov [11] investigated the space-dependent coefficient c ase: utt−∆u+a(x)ut=|u|p, (1.6) where a(x)∼a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α,|x| → ∞,radially symmetric and 0 ≤α<1. They proved that the critical exponent of (1.5) is given by pc= 1+2 n−α by using a refined multiplier method. Their method also depends on the finite propagation speed property. Recently, Nishihara [27] and Lin, Nis hihara and Zhai4 YUTA WAKASUGI [20] considered the semilinear wave equation with time-dependent da mping utt−∆u+b(t)ut=|u|p, (1.7) where b(t) =b0(1+t)−β, β∈(−1,1). They proved that the critical exponent of (1.7) is pc= 1+2 n. This shows that, roughly speaking, time-dependent coefficients of damping term do not influence the critical exponent. Therefore we expect that th e critical exponent of the semilinear wave equation (1.1) is pc= 1+2 n−α. To state our results, we introduce an auxiliary function ψ(t,x) :=A/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α (1+t)1+β(1.8) with A=(1+β)a0 (2−α)2(2+δ), δ>0 (1.9) This type of weight function is first introduced by Ikehata and Taniz awa [10]. We have the following result: Theorem 1.1. If p>1+2 n−α, then there exists a small positive number δ0>0such that for any 0< δ≤δ0the following holds: If I2 0:=/integraldisplay Rne2ψ(0,x)(u2 1+|∇u0|2+|u0|2)dx is sufficiently small, then there exists a unique solution u∈C([0,∞);H1(Rn))∩ C1([0,∞);L2(Rn))to(1.1)satisfying/integraldisplay Rne2ψ(t,x)|u(t,x)|2dx≤Cδ(1+t)−(1+β)n−2α 2−α+ε,(1.10) /integraldisplay Rne2ψ(t,x)(|ut(t,x)|2+|∇u(t,x)|2)dx≤Cδ(1+t)−(1+β)(n−α 2−α+1)+ε, where ε=ε(δ) :=3(1+β)(n−α) 2(2−α)(2+δ)δ (1.11) andCδis a constant depending on δ. Remark 1.2. When1<p≤1+2/(n−α), it is expected that no matter how small the data are, if the data have some shape, then the correspond ing local solution blows up in finite time. However, we have no result. Remark 1.3. We do not assume that the data are compactly supported. Hence our result is an extension of the results of Ikehata, Todorov a and Yordanov [11] to noncompactly supported data cases. However, we prove onl y the case a(x) = a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α.SEMILINEAR DAMPED WAVE EQUATIONS 5 As a consequence of the main theorem, we have an exponential dec ay estimate outside a parabolic region. Corollary 1.4. If p>1+2 n−α, then there exists a small positive number δ0>0such that for any 0< δ≤δ0the following holds: Take ρandµso small that 0<ρ<1−α−β,and0<µ<2A, and put Ωρ(t) :={x∈Rn;/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α≥(1+t)1+β+ρ}. Then, for the global solution uin Theorem 1.1, we have the following estimate/integraldisplay Ωρ(t)(u2 t+|∇u|2+u2)dx≤Cδ,ρ,µ(1+t)−(1+β)(n−2α) 2−α+εe−(2A−µ)(1+t)ρ,(1.12) hereεis defined by (1.11)andCδ,ρ,µis a constant depending on δ,ρandµ. Namely, the decay rate of solution in the region Ω ρ(t) is exponential. We note that the support of u(t) and the region Ω ρ(t) can intersect even if the data are compactly supported. This phenomenon was first discovered by To dorova and Yordanov [36]. We can interpret this result as follows: The support o f the solution is strongly suppressed by damping, so that the solution is concentr ated in the parabolic region much smaller than the light cone. 2.Proof of Theorem 1.1 In this section we prove our main result. At first we prepare some no tation and terminology. We put /ba∇dblf/ba∇dblLp(Rn):=/parenleftbigg/integraldisplay Rn|f(x)|pdx/parenrightbigg1/p ,/ba∇dblu/ba∇dbl:=/ba∇dblu/ba∇dblL2(Rn). ByH1(Rn) we denote the usual Sobolev space. For an interval Iand a Banach spaceX, we define Cr(I;X) as the Banach space whose element is an r-times continuously differentiable mapping from ItoXwith respect to the topology in X. The letter Cindicates the generic constant, which may change from line to the next line. To prove Theorem 1.1, we use a weighted energy method which was or iginally developed by Todorova and Yordanov [35,36]. We first describe th e local existence: Proposition 2.1. For anyδ >0, there exists Tm∈(0,+∞]depending on I2 0 such that the Cauchy problem (1.1)has a unique solution u∈C([0,Tm);H1(Rn))∩ C1([0,Tm);L2(Rn)), and ifTm<+∞then we have liminf t→Tm/integraldisplay Rneψ(t,x)(u2 t+|∇u|2+u2)dx= +∞. We can prove this proposition by standard arguments (see [10]). W e prove a priori estimate for the following functional: M(t) := sup 0≤τ<t/braceleftbigg (1+τ)B+1−ε/integraldisplay Rne2ψ(u2 t+|∇u|2)dx +(1+τ)B−ε/integraldisplay Rne2ψa(x)b(t)u2dx/bracerightbigg , (2.1)6 YUTA WAKASUGI where B:=(1+β)(n−α) 2−α+β andεis given by (1.11). From (1.8), (1.9), it is easy to see that −ψt=1+β 1+tψ, (2.2) ∇ψ=A(2−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αx (1+t)1+β, (2.3) ∆ψ=A(2−α)(n−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α (1+t)1+β+A(2−α)α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−2−α (1+t)1+β ≥(1+β)(n−α) (2−α)(2+δ)a(x)b(t) 1+t =:/parenleftbigg(1+β)(n−α) 2(2−α)−δ1/parenrightbigga(x)b(t) 1+t. (2.4) Here and after, δi(i= 1,2,...) is a positive constant depending only on δsuch that δi→0+asδ→0+. We also have (−ψt)a(x)b(t) =Aa0(1+β)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−2α (1+t)2+2β ≥a0(1+β) (2−α)2AA2(2−α)2/a\}b∇acketle{tx/a\}b∇acket∇i}ht−2α|x|2 (1+t)2+2β = (2+δ)|∇ψ|2. (2.5) By multiplying (1.1) by e2ψut, it follows that ∂ ∂t/bracketleftbigge2ψ 2(u2 t+|∇u|2)/bracketrightbigg −∇·(e2ψut∇u) +e2ψ/parenleftbigg a(x)b(t)−|∇ψ|2 −ψt−ψt/parenrightbigg u2 t+e2ψ −ψt|ψt∇u−ut∇ψ|2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T1 =∂ ∂t/bracketleftbig e2ψF(u)/bracketrightbig +2e2ψ(−ψt)F(u), (2.6) whereFis the primitive of fsatisfyingF(0) = 0, namely F′(u) =f(u). Using the Schwarz inequality and (2.5), we can calculate T1=e2ψ −ψt(ψ2 t|∇u|2−2ψtut∇u·∇ψ+u2 t|∇ψ|2) ≥e2ψ −ψt/parenleftbigg1 5ψ2 t|∇u|2−1 4u2 t|∇ψ|2/parenrightbigg ≥e2ψ/parenleftbigg1 5(−ψt)|∇u|2−a(x)b(t) 4(2+δ)u2 t/parenrightbigg .SEMILINEAR DAMPED WAVE EQUATIONS 7 From this and (2.5), we obtain ∂ ∂t/bracketleftbigge2ψ 2(u2 t+|∇u|2)/bracketrightbigg −∇·(e2ψut∇u) +e2ψ/braceleftbigg/parenleftbigg1 4a(x)b(t)−ψt/parenrightbigg u2 t+−ψt 5|∇u|2/bracerightbigg ≤∂ ∂t/bracketleftbig e2ψF(u)/bracketrightbig +2e2ψ(−ψt)F(u). (2.7) By multiplying (2.7) by ( t0+t)B+1−ε, heret0≥1 is determined later, it follows that ∂ ∂t/bracketleftbigg (t0+t)B+1−εe2ψ 2(u2 t+|∇u|2)/bracketrightbigg −(B+1−ε)(t0+t)B−εe2ψ 2(u2 t+|∇u|2) −∇·((t0+t)B+1−εe2ψut∇u) +e2ψ(t0+t)B+1−ε/braceleftbigg/parenleftbigg1 4a(x)b(t)−ψt/parenrightbigg u2 t+−ψt 5|∇u|2/bracerightbigg ≤∂ ∂t/bracketleftbig (t0+t)B+1−εe2ψF(u)/bracketrightbig −(B+1−ε)(t0+t)B−εe2ψF(u) +2(t0+t)B+1−εe2ψ(−ψt)F(u). (2.8) We put E(t) :=/integraldisplay Rne2ψ(u2 t+|∇u|2)dx, Eψ(t) :=/integraldisplay Rne2ψ(−ψt)(u2 t+|∇u|2)dx, J(t;g) :=/integraldisplay Rne2ψgdx, J ψ(t;g) :=/integraldisplay Rne2ψ(−ψt)gdx. Integrating (2.8) over the whole space, we have 1 2d dt/bracketleftbig (t0+t)B+1−εE(t)/bracketrightbig −1 2(B+1−ε)(t0+t)B−εE(t) +1 4(t0+t)B+1−εJ(t,a(x)b(t)u2 t)+1 5(t0+t)B+1−εEψ(t) ≤d dt/bracketleftbigg (t0+t)B+1−ε/integraldisplay e2ψF(u)dx/bracketrightbigg +C(t0+t)B+1−εJψ(t;|u|p+1)+C(t0+t)B−εJ(t;|u|p+1) (2.9)8 YUTA WAKASUGI Therefore, we integrate (2.9) on the interval [0 ,t] and obtain the estimate for ( t0+ t)B+1−εE(t), which is the first term of M(t): (t0+t)B+1−εE(t)−C/integraldisplayt 0(t0+τ)B−εE(τ)dτ +/integraldisplayt 0(t0+τ)B+1−εJ(τ;a(x)b(t)u2 t)+(t0+τ)B+1−εEψ(τ)dτ ≤CI2 0+C(t0+t)B+1−εJ(t;|u|p+1) +C/integraldisplayt 0(t0+τ)B+1−εJψ(τ;|u|p+1)dτ +C/integraldisplayt 0(t0+t)B−εJ(τ;|u|p+1)dτ. (2.10) In orderto complete a priori estimate, however, we have to manag e the second term of the inequality above whose sign is negative, and we also have to est imate the second term of M(t). The following argument, which is little more complicated, can settle both these problems. At first, we multiply (1.1) by e2ψuand have ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg uut+a(x)b(t) 2u2/parenrightbigg/bracketrightbigg −∇·(e2ψu∇u) +e2ψ |∇u|2+/parenleftbigg −ψt+β 2(1+t)/parenrightbigg a(x)b(t)u2+2u∇ψ·∇u/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T2−2ψtuut−u2 t =e2ψuf(u). (2.11) We calculate e2ψT2= 4e2ψu∇ψ·∇u−2e2ψu∇ψ·∇u = 4e2ψu∇ψ·∇u−∇·(e2ψu2∇ψ)+2e2ψu2|∇ψ|2+e2ψ(∆ψ)u2 and by (2.4) we can rewrite (2.11) to ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg uut+a(x)b(t) 2u2/parenrightbigg/bracketrightbigg −∇·(e2ψ(u∇u+u2∇ψ)) +e2ψ/braceleftBig |∇u|2+4u∇u·∇ψ+(−ψt)a(x)b(t)+2|∇ψ|2)u2 /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright T3 +(B−2δ1)a(x)b(t) 2(1+t)u2−2ψtuut−u2 t/bracerightBig ≤e2ψuf(u). (2.12)SEMILINEAR DAMPED WAVE EQUATIONS 9 It follows from (2.5) that T3=|∇u|2+4u∇u·∇ψ +/braceleftbigg/parenleftbigg 1−δ 3/parenrightbigg (−ψt)a(x)b(t)+2|∇ψ|2/bracerightbigg u2+δ 3(−ψt)a(x)b(t)u2 ≥ |∇u|2+4u∇u·∇ψ +/parenleftbigg 4+δ 3−δ2 3/parenrightbigg |∇ψ|2u2+δ 3(−ψt)a(x)b(t)u2 =/parenleftbigg 1−4 4+δ2/parenrightbigg |∇u|2+δ2|∇ψ|2u2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle2√4+δ2∇u+/radicalbig 4+δ2u∇ψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +δ 3(−ψt)a(x)b(t)u2 ≥δ3(|∇u|2+|∇ψ|2u2)+δ 3(−ψt)a(x)b(t)u2, where δ2:=δ 6−δ2 6, δ3:= min(1 −4 4+δ2,δ2). Thus, we obtain ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg uut+a(x)b(t) 2u2/parenrightbigg/bracketrightbigg −∇·(e2ψ(u∇u+u2∇ψ)) +e2ψδ3|∇u|2 +e2ψ/parenleftbigg δ3|∇ψ|2+δ 3(−ψt)a(x)b(t)+(B−2δ1)a(x)b(t) 2(1+t)/parenrightbigg u2 +e2ψ(−2ψtuut−u2 t) ≤e2ψuf(u). (2.13) Following Nishihara [19], related to the size of 1+ |x|2and the size of (1+ t)2, we divide the space Rninto two different zones Ω( t;K,t0) and Ωc(t;K,t0), where Ω = Ω(t;K,t0) :={x∈Rn;(t0+t)2≥K+|x|2}, and Ωc= Ωc(t;K,t0) :=Rn\Ω(t;K,t0) withK≥1 determined later. Since a(x)b(t)≥a0(t+t0)−(α+β)in the domain Ω, we multiply (2.7) by ( t0+t)α+βand obtain ∂ ∂t/bracketleftbigge2ψ 2(t0+t)α+β(u2 t+|∇u|2)/bracketrightbigg −∇·(e2ψ(t0+t)α+βut∇u) +e2ψ/bracketleftbigg/parenleftbigga0 4−α+β 2(t0+t)1−α−β/parenrightbigg +(t0+t)α+β(−ψt)/bracketrightbigg u2 t +e2ψ/bracketleftbigg−ψt 5(t0+t)α+β−α+β 2(t0+t)1−α−β/bracketrightbigg |∇u|2 ≤∂ ∂t[(t0+t)α+βe2ψF(u)]−α+β (t0+t)1−α−βe2ψF(u) +2(t0+t)α+βe2ψ(−ψt)F(u). (2.14)10 YUTA WAKASUGI Letνbe a small positive number depends on δ, which will be chosen later. By (2.14)+ν(2.13), we have ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg(t0+t)α+β 2u2 t+νuut+νa(x)b(t) 2u2+(t0+t)α+β 2|∇u|2/parenrightbigg/bracketrightbigg −∇·(e2ψ(t0+t)α+βut∇u+νe2ψ(u∇u+u2∇ψ)) +e2ψ/bracketleftbigg/parenleftbigga0 4−α+β 2(t0+t)1−α−β−ν/parenrightbigg +(t0+t)α+β(−ψt)/bracketrightbigg u2 t +e2ψ/bracketleftbigg νδ3−α+β 2(t0+t)1−α−β+−ψt 5(t0+t)α+β/bracketrightbigg |∇u|2 +e2ψ/bracketleftbigg ν(δ3|∇ψ|2+δ 3(−ψt)a(x)b(t)+(B−2δ1)a(x)b(t) 2(1+t)/bracketrightbigg u2 +2νe2ψ(−ψt)uut ≤∂ ∂t[(t0+t)α+βe2ψF(u)]−α+β (t0+t)1−α−βe2ψF(u) +2(t0+t)α+βe2ψ(−ψt)F(u)+νe2ψuf(u). (2.15) Bythe Schwarzinequality, the last term ofthe left hand side in the ab oveinequality can be estimated as |2ν(−ψt)uut| ≤νδ 3(−ψt)a(x)b(t)u2+3ν a0δ(−ψt)(t0+t)α+βu2 t. Thus, we have ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg(t0+t)α+β 2u2 t+νuut+νa(x)b(t) 2u2+(t0+t)α+β 2|∇u|2/parenrightbigg/bracketrightbigg −∇·(e2ψ(t0+t)α+βut∇u+νe2ψ(u∇u+u2∇ψ)) +e2ψ/bracketleftbigg/parenleftbigga0 4−α+β 2(t0+t)1−α−β−ν/parenrightbigg +/parenleftbigg 1−3ν a0δ/parenrightbigg (t0+t)α+β(−ψt)/bracketrightbigg u2 t +e2ψ/bracketleftbigg νδ3−α+β 2(t0+t)1−α−β+−ψt 5(t0+t)α+β/bracketrightbigg |∇u|2 +e2ψ/bracketleftbigg ν/parenleftbigg δ3|∇ψ|2+(B−2δ1)a(x)b(t) 2(1+t)/parenrightbigg/bracketrightbigg u2 ≤∂ ∂t[(t0+t)α+βe2ψF(u)]−α+β (t0+t)1−α−βe2ψF(u) +2(t0+t)α+βe2ψ(−ψt)F(u)+νe2ψuf(u). (2.16) Now we choose the parameters νandt0such that a0 4−α+β 2(t0+t)1−α−β−ν≥c0,1−3ν a0δ≥c0, νδ3−α+β 2(t0+t)1−α−β≥c0, νδ3≥c0,1 5≥c0, hold for some constant c0>0. This is possible because we first determine ν sufficiently small depending on δand then we choose t0sufficiently large depending onν. Therefore, integrating (2.16) on Ω, we obtain the following energy inequality: d dtEψ(t;Ω(t;K,t0))−N1(t)−M1(t)+Hψ(t;Ω(t;K,t0))≤P1,(2.17)SEMILINEAR DAMPED WAVE EQUATIONS 11 where Eψ(t;Ω) =Eψ(t;Ω(t;K,t0)) :=/integraldisplay Ωe2ψ/parenleftbigg(t0+t)α+β 2u2 t+νuut+νa(x)b(t) 2u2+(t0+t)α+β 2|∇u|2/parenrightbigg dx, N1(t) :=/integraldisplay Sn−1e2ψ/parenleftbigg(t0+t)α+β 2u2 t+νuut+νa(x)b(t) 2u2 +(t0+t)α+β 2|∇u|2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle |x|=√ (t0+t)2−K ×[(t0+t)2−K](n−1)/2dθ·d dt/radicalbig (t0+t)2−K, M1(t) :=/integraldisplay ∂Ω(e2ψ(t0+t)α+βut∇u+νe2ψ(u∇u+u2∇ψ))·/vector ndS, Hψ(t;Ω) =Hψ(t;Ω(t;K,t0)) :=c0/integraldisplay Ωe2ψ(1+(t0+t)α+β(−ψt))(u2 t+|∇u|2)dx +ν(B−2δ1)/integraldisplay Ωe2ψa(x)b(t) 2(1+t)u2dx, P1:=d dt/bracketleftbigg (t0+t)α+β/integraldisplay Ωe2ψF(u)dx/bracketrightbigg −/integraldisplay Sn−1(t0+t)α+βe2ψF(u)/vextendsingle/vextendsingle/vextendsingle |x|=√ (t0+t)2−K ×[(t0+t)2−K](n−1)/2dθ·d dt/radicalbig (t0+t)2−K +C/integraldisplay Ωe2ψ(1+(t0+t)α+β(−ψt))|u|p+1dx. Here/vector ndenotes the unit outer normal vector of ∂Ω. We note that by ν≤a0/4 and |νuut| ≤νa(x)b(t) 4u2+ν(t0+t)α+β a0u2 t, it follows that c/integraldisplay Ωe2ψ(t0+t)α+β(u2 t+|∇u|2)dx+c/integraldisplay Ωe2ψa(x)b(t)u2dx ≤Eψ(t;Ω(t;K,t0)) ≤C/integraldisplay Ωe2ψ(t0+t)α+β(u2 t+|∇u|2)dx+C/integraldisplay Ωe2ψa(x)b(t)u2dx for some constants c>0 andC >0. Next, we derive an energy inequality in the domain Ωc. We use the notation /a\}b∇acketle{tx/a\}b∇acket∇i}htK:= (K+|x|2)1/2.12 YUTA WAKASUGI Sincea(x)b(t)≥a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−(α+β) Kin Ωc(t,;K,t0), we multiply (2.7) by /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Kand obtain ∂ ∂t/bracketleftbigge2ψ 2/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(u2 t+|∇u|2)/bracketrightbigg −∇·(e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Kut∇u) +e2ψ/parenleftBiga0 4+(−ψt)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K/parenrightBig u2 t+1 5e2ψ(−ψt)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K|∇u|2 +(α+β)e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β−2 Kx·ut∇u ≤∂ ∂t[e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β KF(u)]+2e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt)F(u). (2.18) By (2.18) + ˆν×(2.13), here ˆνis a small positive parameter determined later, it follows that ∂ ∂t/bracketleftBigg e2ψ/parenleftBigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2u2 t+ ˆνuut+ˆνa(x)b(t) 2u2+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2|∇u|2/parenrightBigg/bracketrightBigg −∇·(e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Kut∇u+ ˆνe2ψ(u∇u+u2∇ψ)) +e2ψ/bracketleftBiga0 4−ˆν+(−ψt)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K/bracketrightBig u2 t+e2ψ/bracketleftbigg ˆνδ3+−ψt 5/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K/bracketrightbigg |∇u|2 +e2ψ/bracketleftbigg ˆν/parenleftbigg δ3|∇ψ|2+δ 3(−ψt)a(x)b(t)+(B−2δ1)a(x)b(t) 2(1+t)/parenrightbigg/bracketrightbigg u2 +e2ψ[(α+β)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β−2 Kx·ut∇u−2ˆνψtuut/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright T4] ≤∂ ∂t/bracketleftBig e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β KF(u)/bracketrightBig +2e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt)F(u)+ ˆνe2ψuf(u).(2.19) The termsT4can be estimated as |(α+β)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β−2 Kx·ut∇u| ≤ˆνδ3 2|∇u|2+(α+β)2 2ˆνδ3K2(1−α−β)u2 t, |2ˆν(−ψt)uut| ≤ˆνδ 3(−ψt)a(x)b(t)u2+3ˆν a0δ(−ψt)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Ku2 t. From this we can rewrite (2.19) as ∂ ∂t/bracketleftBigg e2ψ/parenleftBigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2u2 t+ ˆνuut+ˆνa(x)b(t) 2u2+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2|∇u|2/parenrightBigg/bracketrightBigg −∇·(e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Kut∇u+ ˆνe2ψ(u∇u+u2∇ψ)) +e2ψ/bracketleftbigg/parenleftbigga0 4−ˆν−(α+β)2 2ˆνδ3K2(1−α−β)/parenrightbigg +/parenleftbigg 1−3ˆν a0δ/parenrightbigg (−ψt)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K/bracketrightbigg u2 t +e2ψ/bracketleftbiggˆνδ3 2+−ψt 5/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K/bracketrightbigg |∇u|2 +e2ψ/bracketleftbigg ˆν/parenleftbigg δ3|∇ψ|2+(B−2δ1)a(x)b(t) 2(1+t)/parenrightbigg/bracketrightbigg u2 ≤∂ ∂t/bracketleftBig e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β KF(u)/bracketrightBig +2e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt)F(u)+ ˆνe2ψuf(u).(2.20) Now we choose the parameters ˆ νandKin the same manner as before. Indeed taking ˆνsufficiently small depending on δand then choosing Ksufficiently largeSEMILINEAR DAMPED WAVE EQUATIONS 13 depending on ˆ ν, we can obtain a0 4−ˆν−(α+β)2 2ˆνδ3K2(1−α−β)≥c1,1−3ˆν a0δ≥c1, νδ3≥c1,1 5≥c1 for some constant c1>0. Consequently, By integrating (2.20) on Ωc, the energy inequality on Ωcfollows: d dtEψ(t;Ωc(t;K,t0))+N2(t)+M2(t)+Hψ(t;Ωc(t;K,t0))≤P2,(2.21) where Eψ(t;Ωc) =Eψ(t;Ωc(t;K,t0)) :=/integraldisplay Ωce2ψ/parenleftBigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2u2 t+ ˆνuut+ˆνa(x)b(t) 2u2+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2|∇u|2/parenrightBigg dx, N2(t) :=/integraldisplay Sn−1e2ψ/parenleftBigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2u2 t+ ˆνuut+ˆνa(x)b(t) 2u2 +/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K 2|∇u|2)/vextendsingle/vextendsingle/vextendsingle |x|=√ (t0+t)2−K ×[(t0+t)2−K](n−1)/2dθ·d dt/radicalbig (t0+t)2−K, M2(t) :=/integraldisplay ∂Ωc(e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Kut∇u+ ˆνe2ψ(u∇u+u2∇ψ))·/vector ndS, Hψ(t;Ωc) =Hψ(t;Ωc(t;K,t0)) :=c1/integraldisplay Ωe2ψ(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))(u2 t+|∇u|2)dx +ˆν(B−2δ1)/integraldisplay Ωce2ψa(x)b(t) 2(1+t)u2dx, P2:=d dt/bracketleftbigg/integraldisplay Ωce2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β KF(u)dx/bracketrightbigg +/integraldisplay Sn−1/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Ke2ψF(u)/vextendsingle/vextendsingle/vextendsingle |x|=√ (t0+t)2−K ×[(t0+t)2−K](n−1)/2dθ·d dt/radicalbig (t0+t)2−K +C/integraldisplay Ωce2ψ(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))|u|p+1dx. In a similar way as the case in Ω, we note that c/integraldisplay Ωce2ψ(t0+t)α+β(u2 t+|∇u|2)dx+c/integraldisplay Ωce2ψa(x)b(t)u2dx ≤Eψ(t;Ωc(t;K,t0)) ≤C/integraldisplay Ωce2ψ(t0+t)α+β(u2 t+|∇u|2)dx+C/integraldisplay Ωce2ψa(x)b(t)u2dx for some constants c>0 andC >0.14 YUTA WAKASUGI We add the energy inequalities on Ω and Ωc. We note that replacing νand ˆνby ν0:= min{ν,ˆν}, we can still have the inequalities (2.17) and (2.21), provided that we retaket0andKlarger. By((2.17)+(2.21))×(t0+t)B−ε, we have d dt[(t0+t)B−ε(Eψ(t;Ω)+Eψ(t;Ωc))] −(B−ε)(t0+t)B−1−ε(Eψ(t;Ω)+Eψ(t;Ωc))/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright T5 + (t0+t)B−ε(Hψ(t;Ω)+Hψ(t;Ωc))/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright T6 ≤(t0+t)B−ε(P1+P2), (2.22) here we note that N1(t) =N2(t), M1(t) =M2(t) on∂Ω. Since |ν0uut| ≤ν0δ4 2a(x)b(t)u2+ν0 2δ4a0(t0+t)α+βu2 t on Ω and |ν0uut| ≤ν0δ4 2a(x)b(t)u2+ν0 2δ4a0/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β Ku2 t on Ωc, we have −T5+T6≥(t0+t)B−εI1+(t0+t)B−εI2, (2.23) where I1:=/integraldisplay Ωe2ψ/braceleftBigc0 2(1+(t0+t)α+β(−ψt))−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg (t0+t)α+β/bracerightbigg u2 t +e2ψ/braceleftbiggc0 2(1+(t0+t)α+β(−ψt))−B−ε 2(t0+t)(t0+t)α+β/bracerightbigg |∇u|2dx +/integraldisplay Ωce2ψ/braceleftbiggc1 2(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K/bracerightbigg u2 t +e2ψ/braceleftbiggc1 2(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))−B−ε 2(t0+t)/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K/bracerightbigg |∇u|2dx =:I11+I12, I2:=ν0(B−2δ1−(1+δ4)(B−ε))/parenleftbigg/integraldisplay Ω+/integraldisplay Ωc/parenrightbigg e2ψa(x)b(t) 2(1+t)u2dx +c2 2/integraldisplay Rne2ψ(u2 t+|∇u|2)dx, wherec2:= min(c0,c1). Recall the definition of εandδ1(i.e. (1.11) and (2.4)). A simple calculation shows ε= 3δ1. Choosing δ4sufficiently small depending on ε, we have (t0+t)B−εI2≥c3(t0+t)B−1−ε/integraldisplay Rne2ψa(x)b(t)u2dx+c2 2(t0+t)B−εE(t) for some constant c3>0. Next, we prove that I1≥0. By noting that α+β <1, it is easy to see that I11≥0 if we retake t0larger depending on c0,ν0andδ4. To estimateI12, we further divide the region Ωcinto Ωc(t;K,t0) = (Ωc(t;K,t0)∩ΣL)∪(Ωc(t;K,t0)∩Σc L),SEMILINEAR DAMPED WAVE EQUATIONS 15 where ΣL:={x∈Rn;/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α≤L(1+t)1+β},Σc L:=Rn\ΣL withL≫1 determined later. First, since K+|x|2≤K(1+|x|2)≤KL2/(2−α)(1+ t)2(1+β)/(2−α)on Ωc∩ΣL, we have c1 2(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K ≥c1 2−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg K(α+β)/2L(α+β)/(2−α)(1+t)(1+β)(α+β) 2−α. We note that −1+(1+β)(α+β) 2−α<0 byα+β <1. Thus, we obtain c1 2−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg K(α+β)/2L(α+β)/(2−α)(1+t)(1+β)(α+β) 2−α≥0 for larget0depending on LandK. Secondly, on Ωc∩Σc L, we have c1 2(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K ≥/braceleftbiggc1 2(1+β)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α (1+t)2+β−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg/bracerightbigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K ≥/braceleftbiggc1 2(1+β)L 1+t−B−ε 2(t0+t)/parenleftbigg 1+2ν0 δ4a0/parenrightbigg/bracerightbigg /a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K. Therefore one can obtain I12≥0, provided that L≥B−ε c1(1+β)(1 +2ν0 δ4a0). Conse- quently, we have I1≥0. By (2.23) and that we mentioned above, it follows that −T5+T6≥c3(t0+t)B−1−ε/integraldisplay Rne2ψa(x)b(t)u2dx+c2 2(t0+t)B−εE(t). Therefore, we have d dt[(t0+t)B−ε(Eψ(t;Ω)+Eψ(t;Ωc)]+c2 2(t0+t)B−εE(t) +c3(t0+t)B−1−εJ(t;a(x)b(t)u2) ≤(t0+t)B−ε(P1+P2). (2.24) Integrating (2.24) on the interval [0 ,t], one can obtain the energy inequality on the whole space: (t0+t)B−ε(Eψ(t;Ω)+Eψ(t;Ωc))+c2 2/integraldisplayt 0(t0+τ)B−εE(τ)dτ +c3/integraldisplayt 0(t0+τ)B−1−εJ(τ;a(x)b(τ)u2)dτ ≤CI2 0+/integraldisplayt 0(t0+τ)B−ε(P1+P2)dτ. (2.25)16 YUTA WAKASUGI By (2.25) +µ×(2.10), hereµis a small positive parameter determined later, it follows that (t0+t)B−εEψ(t;Ω)+(t0+t)B−εEψ(t;Ωc) +/integraldisplayt 0c2 2(t0+τ)B−εE(τ)−µC(t0+τ)B−εE(τ)dτ +c3/integraldisplayt 0(t0+τ)B−1−εJ(τ;a(x)b(τ)u2)dτ+µ(t0+t)B+1−εE(t) +µ/integraldisplayt 0(t0+τ)B+1−εJ(τ;a(x)b(τ)u2 t)+(t0+τ)B+1−εEψ(τ)dτ ≤CI2 0+P +C(t0+t)B+1−εJ(t;|u|p+1) +C/integraldisplayt 0(t0+τ)B+1−εJψ(τ;|u|p+1)dτ +C/integraldisplayt 0(t0+τ)B−εJ(τ;|u|p+1)dτ, (2.26) where P=/integraldisplayt 0(t0+τ)B−ε(P1+P2)dτ. Now we choose µsufficiently small, then we can rewrite (2.26) as (t0+t)B+1−εE(t)+(t0+t)B−εJ(t;a(x)b(t)u2) ≤CI2 0+P+C(t0+t)B+1−εJ(t;|u|p+1) +C/integraldisplayt 0(t0+τ)B+1−εJψ(τ;|u|p+1)dτ +C/integraldisplayt 0(t0+τ)B−εJ(τ;|u|p+1)dτ. (2.27) We shall estimate the right hand side of (2.27). We need the following le mma. Lemma 2.2 (Gagliardo-Nirenberg) .Letp,q,r(1≤p,q,r≤ ∞)andσ∈[0,1] satisfy 1 p=σ/parenleftbigg1 r−1 n/parenrightbigg +(1−σ)1 q except forp=∞orr=nwhenn≥2. Then for some constant C=C(p,q,r,n)> 0, the inequality /ba∇dblh/ba∇dblLp≤C/ba∇dblh/ba∇dbl1−σ Lq/ba∇dbl∇h/ba∇dblσ Lr,for anyh∈C1 0(Rn) holds. We first estimate ( t0+t)B+1−εJ(t;|u|p+1). From the above lemma, we have J(t;|u|p+1)≤C/parenleftbigg/integraldisplay Rne4 p+1ψu2dx/parenrightbigg(1−σ)(p+1)/2 ×/parenleftbigg/integraldisplay Rne4 p+1ψ|∇ψ|2u2dx +/integraldisplay Rne4 p+1ψ|∇u|2dx/parenrightbiggσ(p+1)/2 (2.28)SEMILINEAR DAMPED WAVE EQUATIONS 17 withσ=n(p−1) 2(p+1). Since e4 p+1ψu2= (e2ψa(x)b(t)u2)a(x)−1b(t)−1e(4 p+1−2)ψ ≤C(e2ψa(x)b(t)u2)/bracketleftBigg/parenleftbigg/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α (1+t)1+β/parenrightbiggα 2−α e(4 p+1−2)ψ/bracketrightBigg ×(1+t)β+(1+β)α/(2−α) ≤C(1+t)β+(1+β)α/(2−α)e2ψa(x)b(t)u2 and e4 (p+1)ψ|∇ψ|2u2≤C/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−2α (1+t)2+2βe1 2(4 p+1−2)ψe1 2(4 p+1−2)ψe2ψu2 ≤Ce1 2(4 p+1−2)ψe2ψ/bracketleftBigg/parenleftbigg/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α (1+t)1+β/parenrightbigg2−2α 2−α e1 2(4 p+1−2)ψ/bracketrightBigg ×(1+t)−2(1+β)+(1+β)(2−2α)/(2−α)u2 ≤C(1+t)−2(1+β)/(2−α)e1 2(4 p+1−2)ψe2ψu2 ≤C(1+t)−2(1+β)/(2−α)(1+t)β+(1+β)α/(2−α)e2ψa(x)b(t)u2, we can estimate (2.28) as J(t;|u|p+1)≤C(1+t)[β+(1+β)α/(2−α)](1−σ)(p+1)/2J(t;a(x)b(t)u2)(1−σ)(p+1)/2 ×[(1+t)−1J(t;a(x)b(t)u2)+E(t)]σ(p+1)/2 and hence (t0+t)B+1−εJ(t;|u|p+1)≤C/parenleftBig (t0+t)γ1M(t)(p+1)/2+(t0+t)γ2M(t)(p+1)/2/parenrightBig , where γ1=B+1−ε+/bracketleftbigg β+(1+β)α 2−α/bracketrightbigg1−σ 2(p+1)−σ 2(p+1) −(B−ε)p+1 2, γ2=B+1−ε+/bracketleftbigg β+(1+β)α 2−α/bracketrightbigg1−σ 2(p+1)−(B−ε)1−σ 2(p+1) −(B+1−ε)σ 2(p+1). By a simple calculation it follows that if p>1+2 n−α, then by taking εsufficiently small (i.e. δsufficiently small) both γ1andγ2are negative. We note that Jψ(t;|u|p+1) =/integraldisplay Rne2ψ(−ψt)|u|p+1dx ≤C 1+t/integraldisplay Rne(2+ρ)ψ|u|p+1dx,18 YUTA WAKASUGI whereρis a sufficiently small positive number. Therefore, we can estimate th e terms/integraldisplayt 0(t0+τ)B+1−εJψ(τ;|u|p+1)dτand/integraldisplayt 0(t0+τ)B−εJ(τ;|u|p+1)dτ in the same manner as before. Noting that P1+P2=d dt/bracketleftbigg (t0+t)α+β/integraldisplay Ωe2ψF(u)dx+/integraldisplay Ωce2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β KF(u)dx/bracketrightbigg +C/integraldisplay Ωe2ψ(1+(t0+t)α+β(−ψt))|u|p+1dx +C/integraldisplay Ωce2ψ(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))|u|p+1dx, we have P=/integraldisplayt 0(t0+τ)B−ε(P1+P2)dτ ≤CI2 0+C(t0+t)B−ε/integraldisplay Ωe2ψ(t0+t)α+βF(u)dx +C(t0+t)B−ε/integraldisplay Ωce2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β KF(u)dx +C/integraldisplayt 0(t0+τ)B−1−ε/integraldisplay Ωe2ψ(t0+τ)α+βF(u)dxdτ +C/integraldisplayt 0(t0+τ)B−1−ε/integraldisplay Ωce2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β KF(u)dxdτ +C/integraldisplayt 0(t0+τ)B−ε/integraldisplay Ωe2ψ(1+(t0+τ)α+β(−ψt))|u|p+1dxdτ +C/integraldisplayt 0(t0+τ)B−ε/integraldisplay Ωce2ψ(1+/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K(−ψt))|u|p+1dxdτ. We calculate e2ψ/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K=e2A/angbracketleftx/angbracketright2−α (1+t)1+β/a\}b∇acketle{tx/a\}b∇acket∇i}htα+β K ≤Ce2A/angbracketleftx/angbracketright2−α (1+t)1+β/parenleftbigg/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α (1+t)1+β/parenrightbiggα+β 2−α (1+t)(α+β)(1+β) 2−α ≤Ce(2+ρ)ψ(1+t)(α+β)(1+β) 2−α for smallρ>0. Noting that(α+β)(1+β) 2−α<1 and taking ρsufficiently small, we can estimate the terms Pin the same manner as estimating ( t0+t)B+1−εJ(t;|u|p+1). Consequently, we have a priori estimate for M(t): M(t)≤CI2 0+CM(t)(p+1)/2. This shows that the local solution of (1.1) can be extended globally. W e note that e2ψa(x)b(t)≥c(1+t)−(1+β)α 2−α−β with some constant c>0. Then we have/integraldisplay Rne2ψa(x)b(t)u2dx≥c(1+t)−(1+β)α 2−α−β/integraldisplay Rnu2dx. (2.29)SEMILINEAR DAMPED WAVE EQUATIONS 19 This implies the decay estimate of global solution (1.10) and completes the proof of Theorem 1.1. Proof of Corollary 1.4. In a similar way to derive (2.29), we have /integraldisplay Rne2ψa(x)b(t)u2dx≥c(1+t)−(1+β)α 2−α−β/integraldisplay Rne(2A−µ)/angbracketleftx/angbracketright2−α (1+t)βu2dx. By noting that /a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α (1+t)1+β≥(1+t)ρ on Ωρ(t) and Theorem 1.1, it follows that (1+t)−(1+β)α 2−α−β/integraldisplay Ωρ(t)e(2A−µ)(1+t)ρ(u2 t+|∇u|2+u2)dx ≤C(1+t)−(1+β)α 2−α−β/integraldisplay Ωρ(t)e(2A−µ)/angbracketleftx/angbracketright2−α (1+t)β(u2 t+|∇u|2+u2)dx ≤C/integraldisplay Rne2ψ(u2 t+|∇u|2+a(x)b(t)u2)dx ≤C(1+t)−B+ε. Thus, we obtain /integraldisplay Ωρ(t)(u2 t+|∇u|2+u2)dx≤C(1+t)−(1+β)(n−2α) 2−α+εe−(2A−µ)(1+t)ρ. This proves Corollary 1.4. /square Acknowledgement The author is deeply grateful to Professors Ryo Ikehata, Kenji Nishihara, Tatsuo Nishitani, Akitaka Matsumura and Michael Reissig. They gave me cons tructive comments and warm encouragement again and again. References [1]H. Fujita ,On the blowing up of solutions of the Cauchy problem for ut= ∆u+u1+α, J. Fac. Sci. Univ. Tokyo Sec. I, 13(1996), 109-124. [2]V. Georgiev, H. Lindblad, C. Sogge ,Weighted Strichartz estimates and global existence for semilinear wave equation , Amer. J. Math., 119(1997), 1291-1319. [3]R. Glassey ,Existence in the large /squareu=F(u)in two dimensions , Math. Z., 178(1981), 233-261. [4]N. Hayashi, E.I. Kaikina, P.I. Naumkin ,Damped wave equation in the subcritical case , J. Differential Equations 207(2004), 161-194. [5]N. Hayashi, E.I.Kaikina, P.I. Naumkin ,Damped wave equation with a critical nonlinearity , Trans. Smer. Math. Soc., 358(2006), 1165-1185. [6]N. Hayashi, E.I. Kaikina, P.I. Naumkin ,On the critical nonlinear damped wave equation with large initial data , J. Math. Anal. Appl., 334(2007), 1400- 1425. [7]N. Hayashi, E.I.Kaikina, P.I. Naumkin ,Asymptotics for nonlinear damped wave equations with large initial data , Sib.´Elektron. Mat. Izv. 4(2007), 249-277. [8]Y. Han, A. Milani ,On the diffusion phenomenon of quasilinear hyperbolic waves , Bull. Sci. Math.124(5) (2000) 415-433. [9]R. Ikehata, K. Nishihara, H. Zhao ,Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption , J. Differential Equations 226(2006), 1-29. [10]R. Ikehata, K. Tanizawa ,Global existence of solutions for semilinear damped wave eq ua- tions in RNwith noncompactly supported initial data , Nonlinear Anal., 61(2005), 1189-1208.20 YUTA WAKASUGI [11]R. Ikehata, G. Todorova, B. Yordanov ,Critical exponent for semilinear wave equations with space-dependent potential , Funkcialaj Ekvacioj, 52(2009) 411-435. [12]R. Ikehata, G. Todorova, B. Yordanov ,Optimal Decay Rate of the Energy for Wave Equations with Critical Potential J. Math. Soc. Japan (to appear). [13]F. John ,Blow-up of solutions of nonlinear wave equations in three sp ace dimensions , Manuscripta Math., 28(1979), 235-268. [14]R.Karch , Selfsimilar profiles in large time asymptotics of solutions to damped wave equations , Studia Math.143(2000), 175-197. [15]S. Kawashima, M. Nakao, K. Ono ,On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative t erm, J. Math. Soc. Japan, 47 (1995), 617-653. [16]J. S. Kenigson, J. J. Kenigson ,Energy decay estimates for the dissipative wave equation with space-time dependent potential , Math. Methods Appl. Sci. 34 (2011), no. 1, 48-62. [17]T. T. Li, Y. Zhou ,Breakdown to solutions to /squareu+ut=|u|1+α, Discrete Cont. Dynam. Syst.1(1995), 503-520. [18]J. Lin, K. Nishihara, J. Zhai ,L2-estimates of solutions for damped wave equations with space-time dependent damping term , J. Differential Equations, 248(2010), 403-422. [19]J. Lin, K. Nishihara, J. Zhai ,Decay property of solutions for damped wave equations with space-time dependent damping term , J. Math. Anal. Appl. 374(2011) 602-614 [20]J. Lin, K. Nishihara, J. Zhai ,Critical exponent for the semilinear wave equation with time-dependent damping , preprint. [21]H. Lindblad, C. Sogge ,Long-time existence for small amplitude semilinear wave eq uations, Amer. J. Math., 118(1996), 1047-1135. [22]A. Matsumura ,On the asymptotic behavior of solutions of semi-linear wave equations , Publ. Res. Inst. Math. Sci., 12(1976), 169-189. [23]K. Mochizuki Scattering theory for wave equations with dissipative term s, Publ. Res. Inst. Math.Sci., 12(1976/77), 383-390. [24]K. Nishihara ,Lp−Lqestimates to the damped wave equation in 3-dimensional spac e and their application , Math. Z., 244(2003), 631-649. [25]K. Nishihara ,Global asymptotics for the damped wave equation with absorp tion in higher dimensional space , J. Math. Soc. Japan, 58(2006), 805-836. [26]K. Nishihara ,Decay properties for the damped wave equation with space dep endent potential and absorbed semilinear term , Comm. Partial Differential Equations, 35(2010), 1402-1418. [27]K. Nishihara ,Asyptotic behavior of solutions to the semilinear wave equa tion with time- dependent damping , Tokyo J. Math. (to appear). [28]K. Nishihara, H. Zhao ,Decay properties of solutions to the Cauchy problem for the d amped wave equation with absorption , J. Math. Anal. Appl., 313(2006), 598-610. [29]K. Nishihara, J. Zhai ,Asymptotic behavior of time dependent damped wave equation s, J. Math. Anal. Appl., 360(2009), 412-421. [30]M. Rammaha ,Nonlinear wave equations in high dimensions , Differential equations and ap- plications, 322-326, Ohio University (1998). [31]T. Sideris ,Nonexistence of global solutions to semilinear wave equati ons in high dimensions , J. Differential Equations, 52(1984), 378-406. [32]W. Strauss ,Nonlinear scattering theory at low energy , J. Funct. Anal., 41(1981), 110-133. [33]W. Strauss ,Nonlinear wave equations , C.B.M.S. Lecture Notes 73, Amer. Math. Soc., Prov- idence, RI, 1989. [34]D. Tataru ,Strichartz estimates in the hyperbolic space and global exi stence for the semilinear wave equation , Trans. Amer. Math. Sci., 353(2000), 795-807. [35]G. Todorova, B. Yordanov ,Critical exponent for a nonlinear wave equation with dampin g, C. R. Acad. Sci. Paris S´ er. I Math., 330(2000), 557-562. [36]G. Todorova, B. Yordanov ,Critical exponent for a nonlinear wave equation with dampin g, J. Differential Equations, 174(2001), 464-489. [37]G. Todorova, B. Yordanov ,Weighted L2-estimates for dissipative wave equations with variable coefficients , J. Differential Equations, 246(2009), 4497-4518. [38]J. Wirth ,Wave equations with time-dependent dissipation. I. Non-eff ective dissipation , J. Differential Equations, 222(2006), 487-514.SEMILINEAR DAMPED WAVE EQUATIONS 21 [39]J. Wirth ,Wave equations with time-dependent dissipation. II. Effect ive dissipation , J. Dif- ferential Equations 232(2007), 74-103. [40]Qi S. Zhang ,A blow-up result for a nonlinear wave equation with damping: the critical case , C. R. Acad. Sci. Paris S´ er. I Math., 333(2001), 109-114. [41]Y. Zhou ,Cauchy problem for semilinear wave equations in four space d imensions with small initial data , J. Partial Defferential Equations, 8(1995), 135-144. Department of Mathematics, Graduate School of Science, Osa ka University, Osaka, Toyonaka, 560-0043, Japan E-mail address :y-wakasugi@cr.math.sci.osaka-u.ac.jp

1911.09400v1.Low_damping_and_microstructural_perfection_of_sub_40nm_thin_yttrium_iron_garnet_films_grown_by_liquid_phase_epitaxy.pdf

1 Low damping and microstructural perfection of sub-4 0nm-thin yttrium iron garnet films grown by liquid phase epi taxy Carsten Dubs, 1* Oleksii Surzhenko, 1 Ronny Thomas, 2 Julia Osten, 2 Tobias Schneider, 2 Kilian Lenz, 2 Jörg Grenzer, 2 René Hübner, 2 Elke Wendler 3 1 INNOVENT e.V. Technologieentwicklung, Prüssingstr. 27B, 07745 Jena, Germany 2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany 3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, Germany * Correspondence: cd@innovent-jena.de The field of magnon spintronics is experiencing an increasing interest in the development of solutions for spin-wave-based data transport and pr ocessing technologies that are complementary or alternative to modern CMOS architectures. Nanometer -thin yttrium iron garnet (YIG) films have been the gold standard for insulator-based spintron ics to date, but a potential process technology tha t can deliver perfect, homogeneous large-diameter fil ms is still lacking. We report that liquid phase epitaxy (LPE) enables the deposition of nanometer-t hin YIG films with low ferromagnetic resonance losses and consistently high magnetic qua lity down to a thickness of 20 nm. The obtained epitaxial films are characterized by an ideal stoic hiometry and perfect film lattices, which show neither significant compositional strain nor geomet ric mosaicity, but sharp interfaces. Their magneto-static and dynamic behavior is similar to t hat of single crystalline bulk YIG. We found, that the Gilbert damping coefficient α is independent of the film thickness and close to 1 × 10 -4, and that together with an inhomogeneous peak-to-peak li newidth broadening of ∆H0|| = 0.4 G, these values are among the lowest ever reported for YIG f ilms with a thickness smaller than 40 nm. These results suggest, that nanometer-thin LPE films can be used to fabricate nano- and micro-scaled circuits with the required quality for magnonic dev ices. The LPE technique is easily scalable to YIG sample diameters of several inches. I. INTRODUCTION Yttrium iron garnet (Y 3Fe 5O12 ; YIG) in the micrometer thickness range is the mat erial of choice in radio-frequency (RF) engineering for decades (see, e.g., Refs. [1-5]). Especially the lowest spin wave loss of all known magnetic materials and the f act, that it is a dielectric are of decisive importance. Since one has learned how to grow YIG f ilms in the nanometer thickness range, there has been a renaissance of this material, as its mag netic and microwave properties are in particular demand in many areas of modern physics. A growing field of application for magnetic garnets is (i) magnonics, which deals with future potential devices for data transfer and processing using spin waves [1,6-9]. The significant thickness reduction achieved today allows reducing the circui t sizes from classical millimeter dimensions [1] down to 50 nm [10-12] . Another important field is (ii) spintronics: By in creasing the YIG surface- to-volume ratio as much as possible (while keeping its magnetic properties), physical phenomena, such as the inverse spin Hall effect [13], spin-tra nsfer torque [14], and the spin Seebeck effect [15] (generated by a spin angular momentum transfer at t he interfaces between YIG and a nonmagnetic metallic conductor layer) become much more efficien t [7,16-29]. Also (iii) the field of terahertz physics, which uses ultrafast spin dynamics to cont rol ultrafast magnetism, for example for potential 2 terahertz spintronic devices [30,31,32], and (iv) t he field of low-temperature physics, which deals with magnetization dynamics at cryogenic temperatur es [33 ] for prospective quantum computer systems, are possible fields of applications for na nometer-thin iron garnet films. There are several different techniques to grow YIG on different substrates. (i) Pulsed laser deposition (PLD) is an excellent technique for fabr icating small samples of nanometer-thin YIG films with narrow ferromagnetic resonance (FMR) lin ewidths [17,19,21,22,28,34-36] whereas its up-scaling to larger sample dimensions of several i nches is challenging. (ii) Magnetron sputtered YIG usually yields wider FMR linewidths, and inhomo geneous line broadening is frequently observed [37-40]. (iii) For large-scale, low-cost c hemical solution techniques, such as spin coating, strongly broadened FMR linewidths and increased Gil bert damping parameters were reported [41,42]. (iv) Liquid phase epitaxy (LPE) from high- temperature solutions (flux melts), is a well- established technique. Since nucleation and crystal growth take place under almost thermodynamic equilibrium conditions, this guarantees high qualit y with respect to narrow absolute FMR linewidths and a small Gilbert damping coefficient [43-45] at the same time, making LPE comparable or superior to the other growth techniques. In additio n, LPE allows YIG to be deposited in the required quality on 3- or 4-inch wafers [46]. This is import ant for possible applications mentioned above. So far, classical LPE was applied to grow micromete r-thick samples used for magneto-static microwave devices [47,48] or for magneto-optical im aging systems [49]. The typical shortcomings of the LPE technology making thin-film growth so di fficult lie in the fact, that, due to high growth rates, nanometer-thin films were technologically di fficult to access. The etch-back processes in high- temperature solutions or interdiffusion processes a t the substrate/film interface at high temperatures usually prevent sharp interfaces. In addition, film contamination by flux melt constituents (if it is not a self-flux without foreign components) is unavoida ble in most cases. Nevertheless, it was recently demonstrated, that epitaxial films of 100 nm or thi nner are also accessible with this technique [50,51]. In this study, we will show that we are able to dep osit nanometer-thin YIG LPE films with low FMR losses and consistently high magnetic quality down to a thickness of 20 nm. There is no thinnest "ultimate" thickness for iron garnet LPE films, as it is sometimes claimed. It should be pointed out, that, in addition to the damping properties, magnetic anisotropy contributions as a function of the sample stoichiom etry and film/substrate pairing are also of great importance, since they determine the static and dyn amic magnetization of the epitaxial iron garnet films and thus their possible applications. For exa mple, large negative uniaxial anisotropy fields were usually observed for garnet films under compre ssion, such as for YIG on gadolinium gallium garnet (Gd 3Ga 5O12 ; GGG) or other suitable substrates with smaller latt ice parameters grown by gas phase deposition techniques (see e.g. Refs. [35,36, 52-57]), which favors in-plane magnetization. Large perpendicular magnetic anisotropies, on the o ther hand, can be found for films under tensile strain, e.g. on substrates with larger lattice para meter or for rare earth iron garnet films with smal ler lattice parameter than GGG (see e.g. Refs. [58-62] ). Between these two extremes are YIG LPE films, which are usually grown on standard GGG subs trates and exhibit small tensile strain if no lattice misfit compensation, e.g. by La ion substit ution [50,63], has been performed. Such films are characterized by a small uniaxial magnetic anisotropy and dominan t shape anisotropy when no larger growth–induced anisotropy contributions due to Pb or Bi substitution occurs [64]. However, only little information about the structur al properties and the thickness-dependent magnetic anisotropy contributions of nanometer-thin LPE films has been published so far, which is why we are concentrating on these properties for YI G films with thicknesses down to 10 nm. This allowed us to describe the intrinsic damping behavi or over a wide frequency range and to determine a set of magnetic anisotropy parameters for all inv estigated films. 3 II. EXPERIMENTAL DETAILS Nanometer-thin YIG films were deposited on 1-inch ( 111) GGG substrates by LPE from PbO-B 2O3- based high-temperature solutions (HTL) at about 865 °C using the isothermal dipping method (see e.g. [65]). Nominally pure Y 3Fe 5O12 films with smooth surfaces were obtained within on e minute deposition time on horizontally rotated substrates with rotation rates of 100 rpm. The only variable growth parameter for all samples in this study was the degree of undercooling ( ∆T = TL-Tepitaxy ) that was restricted to ∆T ≤ 5 K to obtain films with thicknesses between 10 an d 110 nm. Here TL is the liquidus temperature of the high-temperature soluti on and Tepitaxy is the deposition temperature. After deposition, the samples were pulled out of the solu tion followed by a spin-off of most of the liquid melt remnants at 1000 rpm, pulled out of the furnac e and cooled down to room temperature. Subsequently, the sample holder had to be stored wi th the sample in a diluted, hot nitric-acetic-acid solution to remove the rest of the solidified solut ion residues. Finally, the reverse side YIG film of the doubled-sided grown samples was removed by mech anical polishing and samples were cut into chips of different sizes by a diamond wire saw. The film thicknesses were determined by X-ray reflectometry (XRR) and by high-resolution X-ray di ffraction (HR-XRD) analysis, and the latter data were used to calculate anisotropy and magnetiz ation values. Atomic force microscopy (AFM) using a Park Scientif ic M5 instrument was carried out for each sample at three different regions over 400 µm2 ranges to determine the root-mean-square (RMS) surface roughness. The XRR measurements were carried out using a PANan alytical/X-Pert Pro system. For the HR- XRD investigations, a Seifert-GE XRD3003HR diffract ometer using a point focus was equipped with a spherical 2D Göbel mirror and a Bartels mono chromator on the source side. Both systems use Cu Kα1 radiation. Reciprocal space maps (RSMs) were measu red with the help of a position-sensitive detector (Mythen 1k) at the symmetric (444) and (88 8) as well as the asymmetric (088), (624), and (880) reflections. To obtain the highest possible a ngular resolution for symmetric θ−2 θ line scans, a triple-axis analyzer in front of a scintillation co unter was installed on the detector. Using a recurs ive dynamical algorithm implemented in the commercial p rogram RC_REF_Sim_Win [66], the vertical lattice misfits were calculated. Rutherford backscattering spectrometry (RBS) was ap plied to investigate the composition of the grown YIG films using 1.8 MeV He ions and a backsca ttering angle of 168°. Backscattering events were registered with a common Si detector. The ener gy calibration of the multichannel analyzer revealed 3.61 keV per channel. A thin carbon layer was deposited on top of the samples to avoid charging during analysis. The samples were tilted b y 5° with respect to the incoming He ion beam and rotated around the axis perpendicular to the sa mple surface in order to obtain reliable random spectra. The analysis of the measured spectra was p erformed by a home-made software [67] based on the computer code NDF [68] and then enables the calculation of the RBS spectra. The measured data were fitted by calculated spectra to extract t he film composition. In this way, the Fe-to-Y ratio of the films was determined. Because of the low mas s of oxygen, the O signal of the deposited films is too low for quantitative analysis. High-resolution transmission electron microscopy (H R-TEM) investigations were performed with an image C s-corrected Titan 80-300 microscope (FEI) operated a t an accelerating voltage of 300 kV. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) imaging and spectrum imaging analysis based on energy-dispe rsive X-ray spectroscopy (EDXS) were done at 200 kV with a Talos F200X microscope equipped wi th a Super-X EDXS detector system (FEI). Prior to TEM analysis, the specimen mounted in a hi gh-visibility low-background holder was placed for 10 s into a Model 1020 Plasma Cleaner (Fischion e) to remove possible contaminations. Classical cross-sectional TEM-lamella preparation was done by sawing, grinding, polishing, dimpling, and 4 final Ar-ion milling. Quantification of the element maps including Bremsstrahlung background correction based on the physical TEM model, series fit peak deconvolution, and application of tabulated theoretical Cliff-Lorimer factors as well as absorption correction was done for the elements Y (K α line), Fe (K α line), Gd (L α line), Ga (K α line), O (K line), and C (K line) using the ESPRIT software version 1.9 (Bruker). The ferromagnetic resonance (FMR) absorption spectr a were taken on two different setups. The frequency-swept measurements were recorded on a Roh de & Schwarz ZVA 67 vector network analyzer attached to a broadband stripline. The YIG /GGG sample was mounted face-down on the stripline, and the transmission signals S21 and S12 were recorded using a source power of -10 dBm (= 0.1 mW). The microwave frequency was swept across t he resonance frequency fres , while the in- plane magnetic field H remained constant. Each recorded frequency spectru m was fitted by a Lorentz function and allowed us to define the reson ance frequency fres and the frequency linewidth ΔfFWHM corresponding to the applied field H = Hres . In addition, field-swept measurements were carried out with another setup using an Agilent E8364B vector network analyzer and an 80-µm-wide coplanar waveguide. Again, the microwave transmission parameter S21 was recorded as the FMR signal. This time, the mic rowave frequency was kept constant and the external magnetic field w as swept through resonance. This facilitates tracking the FMR signals over large frequency range s. The microwave power was set to 0 dBm (= 1 mW). In addition, this setup allowed for azimutha l and polar angle-dependent measurements to determine the anisotropy and damping contributions in detail. The FMR spectra were fitted by a complex Lorentz function to retrieve the resonance field Hres and field-swept peak-to-peak linewidth ΔHpp . By fitting the four sets of resonance field data, i.e. (i) the in-plane and (ii) the perpendicular- to-plane frequency dependence as well as (iii) the azimuthal and (iv) polar angular dependences at f = 10 GHz, with the resonance equation for the cubic (111) garnet system, a consistent set of anisotropy parameters was determined for each sampl e. In addition, the damping parameters and contributions were determined from the frequency- a nd angle-dependent linewidth data. The vibrating sample magnetometer (VSM, MicroSense LLC, EZ-9) was used to measure the magnetic moments of the YIG/GGG samples magnetized along the YIG film surface. The external magnetic field H was controlled within an error of ≤0.01 Oe. To est imate the volume magnetization M of the YIG films, the raw VSM signal was corrected from background contributions (due to the sample holder and the GGG substrate) and normalized to the YIG volume. The Curie temperatures TC for the YIG samples were determined by zero-extrap olation of the temperature dependencies M (H=const, T) measured in small in-plane magnetic fields. In or der to verify the Curie temperatures measured by VSM, a differential thermal analysis of a 0.55 mm thick YIG single crystal slice was carried out and then used as a reference sample for the VSM temperature calibration. III. RESULTS AND DISCUSSION A. Microstructural properties of nanometer-thin YIG films The thickness values reported in this study are der ived from the Laue oscillations observed in the θ- 2θ patterns of the high-resolution X-ray diffraction (HR-XRD) measurements and are confirmed by X-ray reflectivity (XRR) measurements (see Fig. 1). The differences between both methods for determining the film thickness are in the range of ±1 nm. The surface roughness of the films, measured by atomic force microscopy (AFM) reveals R MS values ranging between 0.2 and 0.4 nm, independent of the film thickness. Sometimes, howev er, partial remnants of dendritic overgrowth 5 increase the surface roughness to RMS values above 0.4 nm for inspection areas larger than 400 µm2 (see, e.g., the disturbance in the top right corner of the AFM image inset in Fig. 1). -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 10 010 110 210 310 410 510 610 710 810 910 10 t = 10.6 nm Intensity (arb.units) Incidence angle [deg] 20 µm15 10 5 010 15 20 5 0µmnm 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 t = 21.5 nm t = 30.9 nm t = 43.2 nm FIG. 1. XRR plots of LPE-grown YIG films of differe nt thicknesses. Solid lines correspond to the experimental data, while dashed lines represent the fitted curves. The spectra shifted vertically for ease of comparison. The inset shows an AFM image of the surface topography of the 11 nm YIG film with a RMS roughness of 0.4 nm. 1. Epitaxial perfection studied by high-resolution X-ray diffraction Combined high-resolution reciprocal space map (HR-R SM) investigations around asymmetric and symmetric Bragg reflections are useful to evaluate the intergrowth relations of epitaxial films on single-crystalline substrates as well as to disting uish between lattice strain induced by the film lattice distortion or compositional changes due to stoichiometric deviations. 33.10 33.15 33.20 46.0 46.5 47.0 47.5 48.0 48.5 49.0 Qx (nm -1 )Qz (nm -1 ) (088) YIG (GGG) t = 21 nm 5E0 5E1 5E2 5E3 5E4 5E5 5E6 Intensity (a) -0.1 0 0.1 34.5 35.0 35.5 36.0 Qx (nm -1 )Qz (nm -1 ) (444) 6 0 20 40 60 80 100 46810 118.5 119.0 119.5 120.0 10 110 210 310 410 510 6 t = 9 nm t = 11 nm t = 21 nm t = 30 nm t = 42 nm t = 106 nm (888) YIG / GGG Scattering angle θ− 2θ (deg) Intensity (counts) (b) Out-of-plane misfit -δd⊥ film [10 -4] Film thickness (nm) FIG. 2. (a) Combined high-resolution reciprocal spa ce maps around the asymmetric YIG/GGG (088) Bragg reflection of a 21-nm-thin single-cryst alline YIG LPE film. The inset shows the corresponding symmetric YIG/GGG (444) peak: measure ments were carried out using a position- sensitive detector. (b) HR-XRD triple-axis θ–2θ scans around the symmetric YIG/GGG (888) peak for various film thicknesses. The inset shows the ( vertical) out-of-plane misfit vs. film thickness (t he solid line is a guide to the eyes). Figure 2(a) shows the HR-RSM of the 21 nm YIG film grown on GGG (111) substrate, measured at the asymmetric (088) reflection in steep incidence, indicating that both the film and the substrate Bragg peak positions are almost identical. Besides the nearly symmetrical intensity distribution along the [111] out-of-plane direction (i.e. the Qz axis), there is only very weak diffuse scattering close to the Bragg peak visible, pointing towards a nearly perfect crystal lattice without significant compositional strain or geometric mosaicity. In add ition, no shift of in-plane (the Qx axis) film Bragg peak position with respect to the substrate i s observed. This behavior indicates a fully straine d pseudomorphic film growth with a perfect coherent i n-plane lattice match with the GGG substrate. The pattern of the diffuse scattering observed alon g the Qx axis of the symmetric (444) reflection (inset in Fig. 2(a)) is very similar to the one fou nd for a comparable GGG substrate (not shown), indicating that the defect structure of the system is mainly defined by the substrate and/or substrate surface. Within the experimental error of ∆Q/Q ∼ 5×10 −6 nm -1 of the high-resolution diffractometer, the same performance was found for all investigated LPE films with thicknesses below 100 nm, clearly demonstrating coherent YIG film growth with out signs of film relaxation. High-resolution triple-axis coupled θ–2θ scans at the (888) and (444) symmetrical reflectio ns (angular accuracy better than 1.5") were carried ou t to define the strain and film thicknesses of the YIG films. Figure 2(b) shows the results obtained a t the (888) reflection. Under these conditions, the Bragg reflection of the 106 nm thick YIG layer is c learly visible as a shoulder of the (888) GGG substrate reflection at higher diffraction angles a nd this indicates a smaller out-of-plane value for the lattice parameter d888 than for the GGG substrate. This is characteristic for tensely stressed “pure” YIG LPE films [50,63]. For LPE films with a thickne ss significantly less than 100 nm, however, only simulations can provide the structural paramet ers. For this reason, the diffracted signals shown in Fig. 2(b) were simulated and fitted. Using the b est fit of both, the (444) and (888) reflections, t he out-of-plane lattice misfit values ( )⊥ ⊥ ⊥ ⊥ ⊥ ⊥∆ − = − = QQ d d d d / /substrate substrate film filmδ were determined (see Table 1). Assuming a fully pseudomorphic [111]-orie nted system, the in-plane stress of the YIG film can be calculated by σ′|| = -2c 44 δd⊥ film (see the Supplemental Material [69 ] for a detailed derivation and references therein [61,70,71]). The in-plane biaxial ε|| and out-of-plane uniaxial ε⊥ 7 strains can be calculated as well using the stiffne ss tensor components c11 , c12 , and c44 for which we use averaged values taken from [72,73] (see also Su pplemental Material [69 ]). The resulting parameters are listed in Table I. The inset in Fig. 2(b) shows the out-of-plane misfi t as a function of the film thickness. A weak monotonous increase of ⊥ filmdδ with decreasing film thickness is observed between 106 nm and 21 nm. The same behavior was reported by Ortiz et al . [61] for compressively strained EuIG and TbIG PLD-grown films with film thicknesses down to 4 nm and 5 nm, respectively. However, for our thinnest LPE films with t ∼ 10 nm, the out-of-plane misfit rapidly drops. Such a significant change of the misfit with respect to the film thick ness was only mentioned for considerably compressively strained YIG PLD films by O. d’Allivy Kelly et al. [17]. They assume, that this effect indicated a critical film thickness (below 1 5 nm) for strain relaxation, but did not explain it in their letter. For semiconductor LPE films, however, it is known, that interdiffusion processes at the film/substrate interfaces can generate continuous c omposition profiles in the diffusion zone without abrupt changes in the lattice parameters, which lea d to modified stress profiles depending on the thickness of the epilayers (see e.g. [74]). A possi ble explanation for the observed behavior could therefore be the presence of a smoothly changing la ttice parameter value in the interface region. Such composition profiles have recently been discus sed for YIG films grown on GGG substrates by high-temperature and long-time laser MBE deposition experiments [75] , and transition layer thicknesses have been modeled based on polarized ne utron and X-ray reflectometry techniques. The probability of the existence of such a thin continu ous transition layer and its influence on the magneto-static film properties will be discussed be low. TABLE I. Structural parameters of the YIG LPE films grown on GGG (111) substrates: film thickness t measured by HR-XRD, RMS roughness obtained by AFM, vertical lattice misfit ⊥ filmdδ obtained by HR-XRD, in-plane strain ε|| and out-of-plane strain ε⊥ and the resulting in-plane stress σ′||. t (nm) roughness (nm) δd⊥ film ×10 -4 ε|| ×10 -4 ε⊥ ×10 -4 σ′|| ×10 8 Pa 9 - -4.3 2.3 -2.0 0.7 11 0.4 -6.1 3.3 -2.8 0.9 21 0.2 -10.4 5.6 -4.8 1.6 30 0.2 -9.4 5.1 -4.3 1.4 42 0.3 -9.2 5.0 -4.2 1.4 106 0.4 -8.5 4.6 -3.9 1.3 ±1 ±0.1 ±0.7 ±0.4 ±0.4 ±0.1 2. Chemical composition studied by Rutherford Backs cattering spectrometry Besides the epitaxial perfection, the chemical comp osition of the films is of interest to estimate deviations from the ideal Y 3Fe 5O12 stoichiometry and to detect impurity elements. The refore, RBS measurements were performed for selected LPE films. As an example, Fig. 3 shows the random spectrum of a 30 nm thick YIG film on GGG substrate . The inset presents the main part of the spectrum. Applying the NDF software, the computed c urve (solid line) matches perfectly the experimental one (symbols). This enables us to dete rmine the Fe:Y ratio. As for all investigated LPE films, the Fe:Y ratio was determined to be R = 1.67, which corresponds to the ideal iron garnet 8 stoichiometry with Fe:Y = 5:3. At higher magnificat ions of the backscattering yield in Fig. 3, a very low intensity signal can be observed at ion energie s higher than for backscattering on gadolinium atoms from the GGG substrate. Although the intensit y is rather low, it can be attributed to heavy impurity elements present to a very low amount over all in the YIG film. We assign this signal to lead and platinum. These elements may come from the solvent and the crucible during the deposition of the YIG film. After background correc tion a total quantity of (0.08 ± 0.02) at.% for the sum of both elements could be determined. This corr esponds to 0.01 < x + y < 0.02 formula units of the nominal film composition (Y 3-x-yPb xPt y)(Fe 5-x-yPb xPt y)O 12 . In a first approximation, for the calculation of the RBS spectra, it was assumed, tha t both elements contribute in equal parts to the high-energy signal. So, the calculated spectrum tak es into account the existence of 0.04 at.% lead and 0.04 at.% platinum within the YIG layer. This y ields a good representation of the separated signal for these two elements. 1550 1600 1650 1700 1750 1800 0100 200 300 400 measurement simulation separated Pb + Pt signal Backscattering yield (counts) Ion energy (keV) YIG/GGG t = 30 nm Gd signal from GGG substrate 1000 1500 02000 4000 6000 8000 Gd Ga YFe FIG. 3. Energy spectrum of 1.8 MeV He ions backscat tered on the YIG/GGG sample with a YIG film thickness of t = 30 nm. The inset shows the main part of the spec trum with the edges of the substrate elements Gd and Ga and the Fe and Y peak from the YIG film. 3. Crystalline perfection studied by high-resolutio n transmission electron microscopy To analyze the film lattice perfection as well as t he heteroepitaxial intergrowth behavior, HR-TEM investigations were performed. A cross-sectional im age of an 11 nm thin YIG film on a GGG substrate makes it possible to visualize both, the entire YIG film volume up to the film surface and the interface in a magnified HR-TEM microscope imag e (see Fig. 4(a)). Besides the perfect film/substrate interface, neither structural lattic e defects nor significant misalignment could be observed in the coherently strained YIG film lattic e up to the film surface. To prove the homogeneity of the bulk composition an d the performance of the film/substrate interface, HAADF-STEM imaging (Fig. 4(b)) together with element mapping, based on EDXS analysis (Figs. 4(c)-(g)), were performed. The corr esponding HAADF-STEM image in Fig. 4(b) allows clearly resolving the film/interface region due to the significant difference of the atomic number contrast. Because of the uniform spatial dis tribution of both, the film (Y, Fe, O) and the substrate elements (Gd, Ga, O), which are independe ntly represented by different colors in Figs. 9 4(c)-(g), a homogeneous composition over the entire YIG film can be confirmed. Small brightness variations within the element maps (on the right ha nd side) result from slight thickness variations of the classically prepared TEM lamella. Neither an in termixing of the substrate nor of the film elements at the YIG/GGG interface is observed in th e element maps within the EDXS detection limit, which is estimated to be slightly below 1 at .-% for the measuring conditions used. For that reason, tiny Pb and Pt contributions in the YIG fil m, as shown by RBS (see Fig. 3), where not detected here. To evaluate the lateral element distributions acros s the film near the film/substrate interface, quantified line scans were performed as presented i n Fig. 4(h). Using the 10%-to-90% edge response criterion, it shows a transition width of (1.9 ± 0.4) nm at the interface . This is lower than the observed 4-6 nm non-magnetic dead layer reporte d for YIG films deposited by RF magnetron sputtering [76], and the about 4 nm or the 5–7 nm d eep Ga diffusion observed for PLD [77] or laser molecular beam epitaxy (MBE) [75], respectively . However, at some positions of the sample’s cross-section we found a reduced YIG film thickness on a wavy GGG surface (not shown), which we attribute to a possible etch-back of the substra te at the beginning of film growth or an already existing wavy substrate surface. For further growth experiments, a careful characterization of the substrate surfaces by AFM should, therefore, be per formed. The TEM investigations show, that the LPE technology is suitable for growing nanometer-th in YIG films without lattice defects and without significant interdiffusion at the film/subs trate interface, which are necessary preconditions for undisturbed spin-wave propagation and low ferro magnetic damping losses. GGG YIG resist (a) 10 0 2 4 6 8 10 12 14 16 18 Composition (at.-%) Distance (nm) (c) Y (d) Fe (e) Gd (f) Ga (g) O (h) Gd 3Ga 5O12 Y3Fe 5O12 surface interface (b) HAADF Line scan Y Fe Gd Ga O FIG. 4. (a) Cross-sectional high-resolution TEM ima ge of the 11-nm-thin YIG/GGG (111) film. The arrows mark the YIG/GGG interface. (b) HAADF-STEM i mage highlighting the well-separated YIG/GGG interface. (c-g) EDXS element maps of the 1 1-nm-thin YIG/GGG (111) film cross- section. (h) Line scan as marked in (b) of the elem ental concentrations across the film thickness. B. Static and dynamic magnetization characterizatio n of nanometer-thin YIG films After gaining insight into the YIG film microstruct ure, we want to link these properties to the FMR performance to find out, which of them plays an ess ential role in the observed magneto-static and dynamic behavior. Therefore, FMR measurements were carried out within a frequency range of 1 to 40 GHz, with the external magnetic field either par allel to the surface plane of the sample along the H || [11-2] film direction or perpendicular to it ( H || [111 ]). In addition, angle-dependent measurements, i.e., varying the angle θH of the external magnetic field (polar angular depe ndence, where θH = 0 is the sample’s normal [111 ] direction) or the azimuth angle φH (in-plane angular dependence, where φH = 0 is the sample’s horizontal [1-10] direction), were performed at f = 10 GHz. These four measurement ‘geometries’ allo w to determine Landé’s g-factor, effective magnetization 4π Meff , and anisotropy fields from the resonance field de pendence and to disentangle the damping contributions from the linewidth depend ence [78,79]. The FMR resonance equations to fit the angle- and f requency-dependencies (see eqs. (S22), (S23) in the Supplemental Material [69] for in-plane and out -of-plane bias field conditions after Baselgia et al. [80 ]) are derived from the free energy density of a cubi c (111) system [81]: ( ) [ ] ( ) ( ) − + +− − − ++ − ⋅ −= ⊥ ϕ θ θ θ θϕϕ θ θ πθ θ ϕϕ θ θ 3sin cos sin32sin41cos31cos sin cos 2cos cos cos sin sin 3 4 4 42 2 || 22 22 KK K MH M F u sH H H s , (1) 11 where K2⊥, K2|| , and K4 are the uniaxial out-of plane, uniaxial in-plane, and cubic anisotropy constants, respectively. φ and θ are the angles of the magnetization. Angle φu allows for a rotation of the uniaxial anisotropy direction with respect to t he cubic anisotropy direction. 1. Frequency-dependent FMR linewidth analysis To investigate the influence of different contribut ions on the overall magnetic damping, we model the field-swept peak-to-peak linewidth Δ Hpp of our YIG (111) films as a sum of four contributi ons [79,82]: TMS 0 mos G pp H H H H H ∆ + ∆ + ∆ + ∆ = ∆ , (2) where Δ HG is the Gilbert damping, Δ Hmos the mosaicity, Δ H0 the inhomogeneous broadening, and ΔHTMS is the two-magnon scattering contribution, respect ively. Note, that all linewidths in this paper are peak-to-peak linewidths, even if not explicitly stated. The intrinsic Gilbert damping is given by f H Ξ= ∆ γπα 34 G , (3) where γ = gµBħ is the gyromagnetic ratio and Ξ is the dragging function. The dragging function is a correction factor to the linewidth needed in field- swept FMR measurements if H and M are not collinear (see e.g . [82]). For H || M follows Ξ = 1. The inhomogeneity term ∆Hmos accounts for a spread (distribution) of the effect ive magnetization 4π Meff [82,83] given by the parameter δ4πMeff : eff effres mos 4432MMHH πδπ∂∂= ∆ . (4) ∆H0, i.e. the zero-frequency linewidth, is a general b roadening term accounting for other inhomogeneities of the sample, such as the microwav e power dependence of the linewidth in YIG (see, e.g., [84]) and systematic fit errors: for ex ample, consistently narrower total full-width at ha lf- maximum linewidths ∆HFWHM of up to 0.5 Oe were determined by additional freq uency-swept measurements at a microwave power of -10 dBm compar ed to the field-swept measurements at 0 dBm discussed here. All kinds of inhomogeneous broadening (including ∆Hmos ) are caused by slightly different resonance fields in parts of the sample. These indi vidual resonance lines might be still resolvable at low frequencies, where Gilbert damping is not large enough yet–especially for YIG. However, at higher frequencies, these lines become broader and eventually coalesce to a single (apparently broadened) line, which even might exhibit small sho ulders or other kinds of asymmetry. Hence, what might be nicely fit with a single line at high frequencies might cause difficulties at low frequencies and sub-mT linewidths. The effect on fi tting the anisotropy constants from the resonance fields is not so sensitive. If the resona nce lines cannot be disentangled or the line is not entirely Lorentzian-shaped anymore, the fit might o verestimate the true linewidth resulting in a systematic broader line accounted for by ∆H0. 12 The last term in Eq. (2), ∆HTMS , covers the two-magnon scattering contribution, wh ich is an extrinsic damping mechanism due to randomly distrib uted defects. For the in-plane frequency- dependence it reads [78,79,82,85-87]: 2 22 2sin 32 02 0 202 0 2 1 TMS f fff ff H + +− + ⋅ Γ Ξ= ∆−, (5) where f0 = γ4π Meff and Γ is the two-magnon scattering strength. Each of the contributions has a characteristic angl e and frequency dependence. Overall, the linewidth vs. frequency dependencies and the linewi dth vs. angle dependencies can be described with one set of parameters. As we will see, the applied model fits very well to the experimental results and allows for disentangling the contributions that are responsibl e for the frequency dependence of the linewidth. At first, we discuss the different damping contribu tions. Then, we go into details for the individual magneto-static parameters, the relevant anisotropy contributions mentioned above, which provided also the base input for the fit parameters for the frequency-dependent FMR linewidth of our YIG films. In Figure 5, the obtained frequency-dependent peak- to-peak linewidths ∆Hpp (symbols) for the four thicknesses 11 nm, 21 nm, 30 nm, and 42 nm are pres ented. The red (solid) curves represent fits using Eq. (2). Figure 5(a) shows data and fits for the out-of-plane bias field configuration ( θH = 0°) and Fig. 5(b) for field-in-plane ( θH = 90°), respectively. As mentioned above, due to a quite complex shape of the resonance lines below ~15 GHz for θH = 0° (with more absorption lines needed to reflect the shape of the spectrum than for higher f requencies) the linewidths could not anymore be evaluated unambiguously with the required precision for films with thicknesses above 11 nm. However, for the thinnest film, the evaluation was possible and the overall fit exhibits a linear behavior down to 1 GHz. This means, in the field-ou t-of-plane geometry, the main contribution to the damping is the Gilbert damping α, which can be determined from the linear slope according to Eq. (3). As it is known from two-magnon scattering (TMS) theory [85,86], there is no TMS contribution if M is perpendicular to the sample plane. The only rem aining contribution is the inhomogeneous broadening given by the zero-frequenc y offset 13 0369 036 036 0 5 10 15 20 25 30 35 40 036θ H= 0 deg 11 nm 21 nm ∆Hpp (Oe) 30 nm f (GHz) 42 nm (a) FIG. 5: Frequency dependence of the linewidth with magnetic field (a) perpendicular-to-plane and (b) in-plane. The red (solid) lines are fits to the data. For the 11-nm sample the individual contributions to the total linewidth are shown in t he top-right panel. Note the different y-axis scaling for the 11 nm sample in the top-left panel. From these out-of-plane measurements, the Gilbert d amping coefficients could be determined, ranging from α = 0.9 × 10 -4 for the 42-nm-thick sample to α = 2.0 × 10 -4 for the 21 nm sample, which is about twice the value obtained from in-pla ne measurements (as discussed below). For the ultrathin 11 nm film, a slightly increased Gilbert damping coefficient of α = 2.7 × 10 -4 and a significantly enlarged zero-frequency linewidth of 2.8 Oe were found. As mentioned above, the reason for the larger offset might be an apparent u nresolvable broadening due to inhomogeneity. For the 21 and 30 nm sample, the zero-frequency interce pt is about ∆H0 = 0.5 Oe, in contrast to ∆H0 = 1.5 Oe for the 42 nm sample. This indicates, that t he 42 nm sample, in contrast to the thinner samples, seems to have additional microstructural d efects, leading to a superposition of lines. This i s very likely, because the inhomogeneous broadening p reviously reported for 100 nm YIG LPE films was also in the range of ∆H0 = 0.5-0.7 Oe [50]. In Fig. 5(b), the results of the corresponding in-p lane field configuration are given. For the 11 nm sample, the four individual fit contributions consi dered in the fit according to Eq. (2) are depicted by solid curves. This sample shows a significant curva ture. The 42 nm sample also shows a small curvature, whereas the other two samples only have a weak curvature at lower frequencies. This curvature usually hints to a contribution from two- magnon scattering, but can also be due to a spread of the effective magnetization. Note, the frequency -dependence of the mosaicity and TMS term look quite similar at higher frequencies, but show a dif ferent curvature at lower frequencies. Hence, the shape of the curve and, thus, the fit reveal, that it is due to a spread of the effective magnetizatio n, δ4πMeff as given by Eq. (4), which lies in the range of 0. 4 to 0.9 G. For the 11 nm sample, this value 036 036 036 0 5 10 15 20 25 30 35 40 03611 nm Exp. ∆H ∆HG ∆Hmos ∆HTMS ∆H0 θ H= 90 deg 21 nm 30 nm f (GHz) 42 nm ∆Hpp (Oe) (b) 036 036 036 0 5 10 15 20 25 30 35 40 03611 nm Exp. ∆H ∆HG ∆Hmos ∆HTMS ∆H0 θ H= 90 deg 21 nm 30 nm f (GHz) 42 nm ∆Hpp (Oe) (b) 14 is larger, i.e., δ4π Meff = 3.2 G, and in addition one needs a small TMS dam ping contribution of Γ = 1.5×10 7 Hz for a proper fit (see Table II). This is again a distinctive sign, that the 11 nm sample has significantly different structural and/or magnetic properties, leading to the additional linewidth contributions. The Gilbert damping coefficients of all four samples in in-plane configuration are α ≤ 1.3 × 10 -4 and correspond to the best values reported earlier for 100 nm YIG LPE films [50]. These are also lower than for a recently reported 1 8 nm YIG LPE film [51]. Thus, at room temperature, no significant increase in Gilbert dam ping could be observed for LPE films down to 10 nm with decreasing thickness. This contrasts wit h various references for PLD and RF-sputtered YIG films grown on (111) GGG substrates [88-92]. TABLE II. Magnetic damping parameters of the LPE (1 11) YIG films: film thickness t, in-plane Gilbert damping parameter α||, inhomogeneous broadening ∆H0|| , spread of effective magnetization δ4πMeff and two-magnon scattering contribution Γ. t (nm) α|| (×10 -4) ∆H0|| (G) δ4π Meff (Oe) Γ (10 7 Hz) 11 1.2 0.4 3.2 1.5 21 1.3 0.6 0.4 0 30 1.2 0.4 0.7 0 42 1.0 0.4 0.9 0 accuracy ±0.2 ±0.2 ±0.3 ±0.3 All field-in-plane linewidth parameters of the inve stigated samples are summarized in Table II. It is obvious, that inhomogeneous contributions, i.e., th ose originating from magnetic mosaicity δ4πMeff , are very small for the samples without two-magnon s cattering. This confirms the high microstructural perfection and homogeneity of the v olume and interfaces of the LPE-grown films with film thicknesses larger than 11 nm. Contributi ons to two-magnon scattering appear to occur only for LPE films with a thickness of less than 21 nm thick. 2. Analysis of magnetic anisotropy contributions In the following, we will discuss the anisotropy co ntributions, which provided the base input for the fit parameters used for the frequency-dependent FMR linewidth curves shown above. All curves were fitted iteratively with the respective resonan ce equation (see Eqs. (S22) and (S23) in the Supplemental Material [69]) to retrieve a coherent set of fit parameters. The fit parameters are liste d in table III. Since the saturation magnetization an d the in-plane stress are known from VSM measurements and HR-XRD investigations, the anisotr opy constants K can be calculated from the anisotropy fields determined by FMR. TABLE III. Magneto-static parameters of the YIG LPE films of t hickness t: Landé’s g-factor, effective magnetization 4 πMeff exp , cubic anisotropy field 2 K4/Ms, and uniaxial in-plane anisotropy field 2 K2||/Ms determined from FMR, saturation magnetization 4 πMs determined from VSM, stress- induced anisotropy field 2 Kσ/Ms calculated from X-ray diffraction data, resulting o ut-of-plane uniaxial anisotropy field 2 K2⊥/Ms and effective magnetization 4 πMeff cal , cubic anisotropy constant K4, stress-induced anisotropy constant Kσ, and out-of-plane uniaxial anisotropy constant K2⊥. 15 t (nm) g 4πMeff exp (G) 2K4/Ms (Oe) 2K2||/Ms (Oe) 4πMs (G) 11 2.015 1566 -93 2.0 1494 21 2.016 1647 -79 0.8 1819 30 2.015 1677 -79 0.6 1830 42 2.014 1699 -86 1.1 1860 accuracy ±0.002 ±13 ±2 ±3 ±41 t (nm) 2Kσ/Ms (G) 2K2⊥/Ms (G) 4πMeff cal (G) K4 (10 3 erg/cm 3) Kσ (10 3 erg/cm 3) K2⊥ (10 3 erg/cm 3) 11 65 127 1368 -5.5 3.9 7.5 21 91 143 1676 -5.7 6.6 10.4 30 82 135 1696 -5.8 6.0 9.8 42 79 136 1724 -6.4 5.8 10.1 accuracy ±7 ±4 ±40 ±0.3 ±0.4 ±0.5 The g-factor of the samples was determined from the freq uency dependencies of the resonance field. There was no significant thickness dependence obser ved yielding a value of g = 2.015(1) for all samples. The cubic anisotropy field 2 K4/Ms was found to be nearly constant, and the average v alue is -84(2) Oe, which is in good agreement to reporte d values of -85 Oe for a 120 micrometer thick LPE film [81] and of about -80 Oe for a 18 nm thin LPE film [51]. Our calculated anisotropy constants K4 are almost always in the range between -5.7×10 3 and -6.4×10 3 erg/cm 3, which corresponds to YIG single crystal bulk values at 29 5 K [93] . Furthermore, a rather weak in-plane uniaxial anisotropy field 2 K2|| /Ms of about 0.6–2 Oe was found, which had already be en determined for 100 nm YIG LPE films [50]. The stress-induced anisotropy constant Kσ and anisotropy field 2 Kσ/Ms are calculated according to Ref. [94 ] (for details, see Eqs. (S14), (S15), (S18) in the Supplemental Material [69]). 2 Kσ/Ms is small and in the same order of magnitude as the cub ic anisotropy field 2 K4/Ms, but with opposite sign. Due to the observed monotonous increase of th e out-of-plane lattice misfit (see inset in Fig. 2(b)), 2 Kσ/Ms grows with decreasing film thickness until it decl ines significantly at a film thickness below 21 nm. However, the observed stress values ar e almost an order of magnitude smaller than, e.g., for as-deposited YIG PLD films on GGG (111) u nder compressive strain (see, e.g., Refs. [17,23,35,36]). Only by a complex procedure, applyi ng mid-temperature deposition, cooling, and post-annealing treatment, authors of Ref. [95] succ eeded in a change from compressively to tensely strained YIG films. These samples then exhibited th e same stress-induced anisotropy constant as it was observed for our YIG LPE films. In the following, we take a closer look to the cont ributions to the out-of-plane uniaxial anisotropy field H2⊥ = 2 K2⊥/Ms. A general description for magnetic garnets has been given for example by Hansen [94]. Applied to thick [43,64] as well as to thin epitaxial iron garnet films (see , e.g., [37,56,59,60,62]) , the out-of-plane uniaxial anisotropy field H2⊥ is mainly determined by the magnetocrystalline and uniaxial anisotropy contributions. While the former refers to the direction of magnetization to preferred crystallographic directi ons in the cubic garnet lattice, the latter origina tes from lattice strain and growth conditions. Due to t he very low supercooling ( ≤5K), growth-induced contributions, usually observed for micrometer YIG films with larger Pb impurity contents, can be neglected in the case of our nanometer-thin YIG LPE films (see e.g. [64]). Thus, H 2⊥ can be 16 determined quantitatively by summing the cubic magn etocrystalline anisotropy (first term, determined by FMR) and the stress-induced anisotrop y (second term, determined by XRD), s sMK MKHσ2 344 2 + − =⊥ , (6) or expressed for the (111) substrate orientation (s ee also SM [69 ] and Ref. [93,96]) by: sMKH39 4111|| 4 2λσ′ − −=⊥ . (7) Using the experimentally determined first-order cub ic anisotropy constant K4 and the in-plane stress component ||σ′from Tables I and III along with the room-temperatu re magnetostriction coefficient λ111 [94 ], the uniaxial anisotropy field H2⊥ can be calculated, if the saturation magnetization Ms is known. Ms can be obtained with appropriate accuracy for exam ple from VSM or SQUID measurements, if the sample volume is exactly known . Magnetic hysteresis loops of YIG LPE films recorded at room-temperature by VSM measurements with in-plane applied magnetic field are shown in F ig. 6. The paramagnetic contribution of the GGG substrate was subtracted as described in Ref. [50]. Extremely small coercivity fields with Hc values of ∼ 0.2 Oe were obtained for all YIG/GGG samples with the exception of the 21 nm film. These values are comparable with the best gas phase epita xial films [17,39,76], but the measured saturation fields with Hs < 2.0 Oe are significantly smaller. All films exhi bit nearly in-plane magnetization due to the dominant contribution of form anisotropy. Ap art from the thinnest sample, the saturation moments determined are not thickness-dependent (see Table III and Fig. 6) and are very close to YIG volume values determined for YIG single crystal s at room temperature (4 πMs ∼ 1800 G) [93,97]. However, the observed decrease of the satu ration magnetization in such films with a thickness of about 10 nm is significant and will be discussed below. -20 -15 -10 -5 0 5 10 15 20 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 10 100 1.4 1.5 1.6 1.7 1.8 1.9 H (Oe) 4 πM (10 4 G) 106 nm 43 nm 30 nm 21 nm 11 nm 4 πM (10 4 G) d (nm) FIG.6: Magnetization loops M(H) of YIG films at room temperature as a function of the in-plane magnetic field. The inset shows the thickness-depen dent saturation magnetization (the solid line is a guide to the eyes). 17 However, for nanometer-thin films, it is a big chal lenge to determine Ms precisely enough, because too large errors can arise from the film’s volume c alculation. While the surface area of the sample can be determined with sufficient precision by opti cal microscopy, thickness measurements with X- ray or ellipsometry methods can lead to thickness e rrors in the range of ±1 nm due to very small macroscopic morphology or roughness fluctuations. T herefore, for films with thicknesses below 20 nm, for example, uncertainties up to a maximum o f 10 percent must be considered. This could significantly affect the effective magnetization 4 πMeff , which can be calculated based on the measured Ms values by ⊥ − =2 eff 4 4 H M Msπ π . (8) This fact can explain the large difference between the calculated 4 πMeff cal and the measured 4πMeff exp values for the 11 nm thin film discussed below, wh ile a much better agreement was achieved for the thicker films (see Table III). As expected from micrometer-thick YIG LPE films gro wn on GGG (111) substrates [81], the out- of-plane uniaxial anisotropy field H2⊥ and the out-of-plane uniaxial anisotropy constants K2⊥ show, that completely pseudomorphically strained, nanomet er-thin LPE films exhibit no pronounced magnetic anisotropy. Small changes of the in-plane stress σ′|| (see Table I) and thus also in the stress-induced anisotropy 2 Kσ/Ms (or Kσ) have no significant influence on the out-of-plane uniaxial anisotropy H2⊥ (see Table III). A comparable H2⊥ value is also expected for films thicker than 42 nm, since the out-of-plane lattice misfit δd⊥ film tends to a constant value (see inset in Fig. 2 (b) ). This is in contrast to Ref. [51], where the uniaxia l anisotropy field of YIG LPE films becomes negative above a film thickness of about 50 nm. 3. Thickness-dependent analysis of the effective ma gnetization field To verify the trend of the calculated 4 πMeff cal values for decreasing film thicknesses, one can compare the effective magnetization with the experi mentally determined one. This was done for 18 YIG films with thicknesses ranging from 10 to 120 n m, including the four samples from above. All films were grown during the same run under nearly i dentical conditions. Only the growth temperature was varied within a range of 5 K. This time, the FMR was measured with a constant external magnetic field applied in-plane and sweepi ng the frequency. 0 20 40 60 80 100 120 1500 1550 1600 1650 1700 1750 1800 frequency field sweep magnetic field sweep Effective Magnetization (G) Film thickness (nm) (a) 18 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 535 540 545 550 555 560 fitted VSM values Outlier values Curie temperature (K) YIG thickness (m) (b) Fig. 7. (a) Thickness dependence of the effective m agnetization 4 πMeff . Blue circles denote measurements taken by field-sweep and red squares d enote frequency-swept measurements, respectively. (b) Thickness dependence of the Curie temperature Tc. The dashed line is a guide to the eyes. In Fig. 7(a), the obtained thickness dependence of the effective magnetization 4 πMeff (squares) is presented and a monotonous decrease of 4 πMeff with film thickness reduction can be observed. Below 40 nm, the slope of the curve increases and, for the thinnest films, there is a significant drop of about 100 G. This behavior has been confirmed by in-plane FMR magnetic field-sweep measurements for selected samples (circles), as dis cussed before. The values are listed in Table III. Similar results have been reported for YIG PLD film s by Kumar et al. [53]. If we compare the experimental values with the calc ulated ones in Table III, then the same trend of a steady reduction of the effective saturation magnet ization with decreasing film thickness can be observed. The deviation between both 4 πMeff values is approximately 1–2 %, except for the 11 n m film. Hence, the saturation magnetization used to c alculate the effective saturation (according to equation (8)) does not appear to be as error-prone as it could be due to an inaccuracy in the film thickness determination. Therefore, we speculate, t hat the significant drop of 4 πMeff for the 11 nm thin film can be explained by the observed reductio n of 4 πMs (see Table III). A similar behavior for 4 πMs was reported for thin PLD or magnetron-sputtered Y IG films, and different explanations were given [76,56,91]. One r eason for a reduced saturation magnetization could be an intermixing of substrate and film eleme nts at the GGG/YIG interface, whereby a gradual change of the film composition is assumed [75,77]. In particular, gallium ion diffusion into the first YIG atomic layers will lead to magnetically diluted ferrimagnetic layers at the interface, due to the fact, that magnetic Fe ions are replaced by diamagn etic Ga ions in the various magnetic sublattices. This assumption is supported by recent reports of Y IG films on GGG substrates. One reports about a 5–7 nm deep Ga penetration found in laser-MBE films [75 ]. Another group found a Ga penetration throughout a 13-nm-thin PLD film [98]. In these cas es, high-temperature film growth above 850°C or prolonged post-annealing at temperatures of 850° C could promote such diffusion processes. In contrast, the deposition time during which the LPE samples were exposed to high temperatures above 860°C was only 5 minutes. Though, the assumed Ga diffusion depth in our YIG films should not exceed more than 2 nm according to the EDXS ele ment maps in Fig. 4(h). In addition, the Gd 19 diffusion in YIG films, as discussed for RF-magnetr on sputtered [76,99] or PLD films [98], could lead to the incorporation of paramagnetic ions into the diamagnetic rare earth sublattice sites, which would also alter the magnetization [98]. However, n o extended interdiffusion layer was observed at the film/substrate interface for our LPE films, so that the presence of a ‘separate, abrupt’ gadoliniu m iron garnet interface layer, as reported by Ref. [9 8], is not expected. Therefore, due to possible interdiffusion effects at temperatures of about 860 °C, a gradual reduction of Ms at a postulated interface layer could be the reason for the observe d low saturation value for the thinnest LPE film, listed in Tab. III. To further rule out a discrete magnetic dead layer, Curie temperature ( Tc) measurements were performed by VSM. It is known from literature, that Tc remains constant up to a film thickness of approximately four YIG unit cells [100], i.e. 2.8 n m, since one YIG unit cell length along the [111 ] direction amounts to d111 ∼ 0.7 nm. Accordingly, the Tc of “pure” YIG films with abrupt interfaces and a film thickness of ∼10 nm should be equal to that of bulk material. In order to check this, temperature-dependent VSM measurements (see Fig. 7( b)) were carried out for our LPE films as well as for a bulk YIG single crystal slice, which was used as a reference. We found almost constant values of Tc = (551±2) K for sample thicknesses between 46 nm ( thin film) and 0.55 mm (bulk). This is in good agreement with the literature, in w hich a Tc of ∼550 K has been reported, e.g. for a 100-nm-thin sputtered YIG film [76], while 559 K has been reported for YIG single crysta ls [97]. However, for our about 10-nm-thin YIG films, Tc decreased significantly to ∼534±1 K (Fig 7(b)), which is consistent with the observed reduction of 4 πMs listed in Table III. Hence, the most likely explanation for the observed reduction of 4 πMs is that the YIG layers at the substrate/film interface exhibit a reduced saturati on magnetization due to a magnetically diluted iron sublattice, resulting from high-temperature diffusi on of gallium ions from the GGG substrate into the YIG film . While nearly zero gallium content at the film surfa ce leads to a bulk-like value of 4πMs ∼ 1800 G [93], an increased content of gallium at th e film/substrate interface should, therefore, result in significantly reduced 4 πMs values. In this case, the average saturation magne tization for the entire film volume should be reduced and that could explain the observed decrease in 4 πMs to about 1500 G for the 11 nm thin LPE film. For thicker fil ms, however, the influence of thin gallium- enriched interface layers on the entire film magnet ization decreases, which explains the fast achievement of a constant Curie temperature, and th us, a constant Ms with increasing thickness of the YIG volume. In order to confirm our assumptions , additional analyses, such as detailed secondary ion mass spectroscopy (SIMS) investigatio ns, are necessary which, however, go beyond the scope of this report. IV. CONCLUSIONS AND OUTLOOK In summary, we have demonstrated that LPE can be us ed to fabricate sub-40 nm YIG films with high microstructural perfection, smooth surfaces an d sharp interfaces as well as excellent microwave properties down to a minimum film thickness of 11 n m. All LPE films with ≥21 nm thickness exhibit extremely narrow FMR linewidths of ∆Hpp <1.5 Oe at 15 GHz and very low magnetic damping coefficients of α ≤1.3 × 10 -4 which are the lowest values reported within an ext ended frequency range of 1 to 40 GHz. We were able to sho w that LPE-grown YIG films down to a thickness of 21 nm have the same magnetization dyna mics influenced by small cubic and stress- induced anisotropy fields. The deviating magnetizat ion dynamics of ultrathin LPE films with thicknesses of ∼10 nm are probably caused by an increased inhomogen eous damping and by small two-magnon scattering contributions, and we specula te that possible inhomogeneities of the composition in the vicinity of the film/substrate i nterface might be the reason for this. Therefore, i n 20 further studies we will address detailed investigat ions of the composition of the film/substrate interface by high-resolution SIMS measurements and advanced STEM analyses to confirm a gradual change of the LPE film composition at the interface . The results presented here encourage us to take the next step towards nano- and microscaled magnonic structures, such as directional couplers, logic gates, transistors etc. for a next-generation of computing circuits. The development of nanoscopi c YIG waveguides and nanostructures is already underway and the first circuits are current ly being fabricated [10,12,29]. With its scalabilit y to large wafer diameters of up to 3 and 4 inches, L PE technology opens up an alternative way for efficient circuit manufacturing for a future YIG pl anar technology on a wafer scale. ACKNOWLEDGMENTS We thank P. Landeros and R. Gallardo for fruitful d iscussions and A. Khudorozhkov for his help during the measurements. C. D. and O. S. thank R. K öcher for AFM measurements, A. Hartmann for the DSC measurements and R. Meyer and B. Wenzel for technical support. J. G. thanks A. Scholz for the support during the XRD measurements. We would like to thank Romy Aniol for the TEM specimen preparation. The use of HZDR’s Ion Bea m Center TEM facilities and the funding of TEM Talos by the German Federal Ministry of Educati on of Research (BMBF), Grant No. 03SF0451 in the framework of HEMCP are gratefully a cknowledged. This research was financially supported by the Deut sche Forschungsgemeinschaft (DFG), via Grant No. DU 1427/2-1. References [1] A. A. Serga, A.V. 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Casanova, Synthetic Antiferromagnetic Coupling Between Ultrat hin Insulating Garnets , Phys. Rev. Appl. 10, 044046 (2018) . [99] J. F. K. Cooper, C. J. Kinane, S. Langridge, M. Ali, B. J. Hickey, T. Niizeki, K. Uchida, E. Saitoh, H. Ambaye, and A. Glavic, Unexpected structural and magnetic depth dependence of YIG thin films, Phys. Rev. B 96 , 104404 (2017). [100] M. Chern, W. Lee, and D. Liou, Curie temperatures of Y 3Fe 5O12 /Gd 3Fe 5O12 superlattices, J. Magn. Magn. Mater. 170 , L243 (1997). [101 ] Strictly speaking this is valid only for [001] ori ented systems for all others, if the cubic lattice constant values of afilm and asubstrate are almost identical. 27 Supplemental Material: Low damping and microstructural perfection of sub-4 0nm-thin yttrium iron garnet films grown by liquid phase epitaxy Carsten Dubs, 1 Oleksii Surzhenko, 1 Ronny Thomas, 2 Julia Osten, 2 Tobias Schneider, 2 Kilian Lenz, 2 Jörg Grenzer, 2 René Hübner,2 Elke Wendler 3 1 INNOVENT e.V. Technologieentwicklung, Prüssingstr. 27B, 07745 Jena, Germany 2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany 3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, Germany I. STRAIN CALCULATIONS In the following we will derive the in-plane (horiz ontal) stress σ|| as a function of the out-of-plane (vertical) lattice misfit ⊥ filmdδ for a [111] oriented cubic system. These calculati ons are based on the elasticity theory following mainly Hinckley [70 ], Sander [71 ] and Ortiz et al . [61 ]. Assuming a fully pseudomorphic system there is only one parameter to be determined: The out-of-plane lattice misfit ⊥ filmdδ . This value can be directly obtained from the data of the HR-XRD measurements a nd/or from the corresponding simulations of the symmetrical (4 44) and (888) reflections. The vertical and parallel lattice misfits can be ca lculated by , 0|| substrate|| substrate|| film || film substratesubstrate film film =−=−=⊥⊥ ⊥ ⊥ dd dddd dd δ δ (S1) where dk i with i = [substrate, (pseudomorph) film or (cubic) relaxe d film] is the (measured) net plane distances for the k = ⊥ (vertical) or || (parallel) direction with respect to the substrate surface. Ignoring any dynamical diffraction effects, the out -of-plane lattice misfit can be directly determined from the measurement as follows: B qq qq qq dθθδ δtansubstrate substratesubstrate film film film∆− =∆− =−− = − =⊥⊥ ⊥⊥ ⊥ ⊥ ⊥, (S2) where q⊥film and q⊥substrate are the derived peak positions in the Q-space of th e thin film and the substrate, respectively. ∆θB is the (kinematical) Bragg angular difference “thi n film – substrate” and θB is the Bragg Peak position of the substrate, respe ctively. These formulas follow directly from the derivative of Bragg’s law. The in- and out-of-plane strains are given as follo ws: .|| film relaxed|| film relaxed|| film || film relaxedfilm relaxed film dd d dd d −=−=⊥⊥ ⊥ ⊥ε ε (S3) The general relationship between stress and strain is defined as follows: ,kl ijkl ij ε σ C= (S4) where σij are the stress, ε kl the strain and Cijkl the second order stiffness tensors and the summati on is done over the repeated indices. The subscripts ij and kl refer to the axes of the coordinates system of 28 the unit cell (1,2,3 = x,y,z ). The samples under investigations have a [111] ou t-of-plane orientation; therefore, the corresponding rotation matrices have to be applied: ijkll k j i CUUUU Cδ γ β α αβγδ=′ . (S5) For the [111] oriented surfaces U 111 yields to − = 310 3231 21 6131 21 61 111U . (S6) Figure S1 : Representation of the cubic unprimed and the rota ted, primed coordinate system for an <111> oriented thin film; where x´, y´ correspond to the in-plane (||) directions and z´ to the out-of- plane ( ⊥) direction; after [71 ]. The in-plane stress ||σ′ can be expressed in terms of the out-of-plane latt ice misfit ⊥ filmdδ obtained by XRD measurements. For a cubic pseudomorphic system we can write: 1313 1212 1111 11 ε ε ε σ c c c + + = (S7) ( )⊥′+′+′=′ ε ε σ12 || 12 11 || c c c (S8) with ⊥′− =ε ν ε|| (S9) resulting in: || 44 12 1112 11 44 ||4 226 ε σc c cc cc+ ++=′ . (S10) 29 Taking the corresponding rotation matrices and rela tionships into account [101 ]: ⊥ ⊥ ⊥ +− =+=film 111111 || film 1111and11d d δννε δνε , (S11) where .4 4 24 2 44 12 1144 12 11 111 c c cc c c − ++ +=ν (S12) The in-plane stress can be now expressed in terms o f the out-of-plane lattice misfit by: ⊥− =′film 44 || 2 dcδ σ . (S13) Here, c44 is the component from the stiffness tensor and ⊥ filmdδ is the out-of-plane lattice misfit as defined above. II. ANISOTROPY CALCULATIONS FOR (111) ORIENTED EPIT AXIAL GARNET FILMS The stress-induced anisotropy parameter for the cub ic (111) orientation can be calculated according to [94] by 111||23λ =Kσ σ′ − , (S14) where σ´|| is the above calculated in-plane stress for {111} oriented thin films, and λ 111 is the corresponding magnetostriction constant. The stress-induced anisotropy parameter is therefor e given by: 111 film 443 λdc=Kσ⊥δ . (S15) The perpendicular magnetic anisotropy field can be calculated according to [43]: growth cub 2 H+ H+ H= Hstress ⊥ . (S16) Assuming negligible growth-induced contributions Hgrowth and applying the cubic anisotropy field for (111) film orientation obtained by FMR measurem ents scubMK= H4 34− , (S17) and taking into account the stress-induced anisotro py field s sσ stressMλ=MK= H111||3 2 σ′ − , (S18) the effective perpendicular anisotropy field result s in sMλ +KH39 4111|| 4 2σ′ − =⊥ . (S19) 30 From the resonance conditions for the perpendicular (M || [111 ]) magnetized epitaxial thin film, the effective saturation magnetization can be obtained by [64 ] + − − =⊥ s ssMK MKM Hf2 4 eff2 344πω, (S20) from which the effective saturation magnetization c an be calculated by ⊥ −2 eff eff 4 4 H πM= πM=Hs . (S21) III. FERROMAGNETIC RESONANCE From the free energy density given by Eq. (1) of th e main text, the resonance equations have been calculated applying the approach of Baselgia et al. [80 ]. The resonance conditions for the frequency- dependences with field out-of-plane ( f⊥) and in-plane ( f|| ) read: + − − − − =⊥MK MKM HMKM H fe e|| 2 4 ff4 ff 23443442π ππγ, (S22) ( ) ( ) ( ) − − − − + × − − = ϕ ϕϕ π ϕϕπγ3 cos 2 cos24 2cos2 222 4 2 || 2 4 ff|| 2 ||MK MK MKM HMKH fu e u (S23). Examples of angle- and frequency-dependent FMR meas urements with out- and in-plane configuration of the magnetic bias field are shown in Figure S2. 31 -30° 0° 30° 60° 90° 120° 345f = 10 GHz 11 nm 21 nm 30 nm 42 nm Fits Hres (kOe) θH 0 5 10 15 010 20 30 40 f (GHz) H (kOe) 0 5 10 15 010 20 30 40 θH=0° f(GHz) H (kOe) 11 nm 21 nm 30 nm 42 nm Fit (11 nm) θH=90° 11 nm 21 nm 30 nm 42 nm (a) (b) (c) 2.76 2.77 2.78 2 .7 9 2 .8 0 FMR- S igna l (arb . units) H (kOe) 42 nm f = 10 GHz 4.30 4.32 4.34 4.36 4.38 4 .4 0 FMR-Signa l (arb . units) H (kOe) 11 nm f = 8 GHz θH φHH→ [110] _ [112] _M→ Y IG(111) φθ Figure S2. (a) Polar angular dependencies of the FM R measured at f = 10 GHz. The inset shows the FMR coordinate system. Solid lines are fits accordi ng to the resonance equation. (b) Frequency dependencies of the resonance field measured with f ield in-plane and (c) out-of-plane. The solid black line is a fit to the 11 nm dataset. Other fit curves have been omitted for visual clarity. Inset s show FMR spectra and the indicated positions includ ing Lorentzian fits.

1210.6879v1.Decay_rates_for_the_damped_wave_equation_on_the_torus.pdf

arXiv:1210.6879v1 [math.AP] 25 Oct 2012Decay rates for the damped wave equation on the torus With an appendix by St´ ephane Nonnenmacher∗ Nalini Anantharaman†and Matthieu L´ eautaud‡, Universit´ e Paris-Sud 11, Math´ ematiques, Bˆ atiment 425, 91405 Orsay Cedex, France October 26, 2012 Abstract We address the decay rates of the energy for the damped wave eq uation when the damping coefficient bdoes not satisfy the Geometric Control Condition (GCC). Fir st, we give a link with the controllability of the associated Schr¨ odinger equati on. We prove in an abstract setting that the observability of the Schr¨ odinger group implies that th e semigroup associated to the damped wave equation decays at rate 1 /√ t(which is a stronger rate than the general logarithmic one predicted by the Lebeau Theorem). Second, we focus on the 2-dimensional torus. We prove that th e best decay one can expect is 1/t, as soon as the damping region does not satisfy GCC. Converse ly, for smooth damping coefficients b, we show that the semigroup decays at rate 1 /t1−ε, for allε >0. The proof relies on a second microlocalization around trapped directions, a nd resolvent estimates. In the case where the damping coefficient is a characteristic f unction of a strip (hence dis- continuous), St´ ephane Nonnenmacher computes in an append ix part of the spectrum of the associated damped wave operator, proving that the semigrou p cannot decay faster than 1 /t2/3. In particular, our study shows that the decay rate highly dep ends on the way bvanishes. Keywords Dampedwaveequation, polynomialdecay, observability, Sc hr¨ odingergroup, torus, two-microlocal semiclassical measures, spectrum of the damped wave operat or. Contents I The damped wave equation 2 1 Decay of energy: a survey of existing results 2 2 Main results of the paper 5 2.1 The damped wave equation in an abstract setting . . . . . . . . . . . . . . . . . . . . . 5 2.2 Decay rates for the damped wave equation on the torus . . . . . . . . . . . . . . . . . 7 2.3 Some related open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 II Resolvent estimates and stabilization in the abstract se tting 10 3 Proof of Theorem 2.3 assuming Proposition 2.4 10 ∗snonnenmacher@cea.fr †Nalini.Anantharaman@math.u-psud.fr ‡Matthieu.Leautaud@math.u-psud.fr 14 Proof of Proposition 2.4 10 III Proof of Theorem 2.6: smooth damping coefficients on the to rus 15 5 Semiclassical measures 16 6 Zero-th and first order informations on µ 17 7 Geometry on the torus and decomposition of invariant measu res 19 7.1 Resonant and non-resonant vectors on the torus . . . . . . . . . . . . . . . . . . . . . . 19 7.2 Decomposition of invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7.3 Geometry of the subtori TΛandTΛ⊥. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8 Change of quasimode and construction of an invariant cutoff function 22 9 Second microlocalization on a resonant affine subspace 23 10 Propagation laws for the two-microlocal measures νΛandρΛ 28 10.1 Propagation of νΛ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 10.2 Propagation of ρΛ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 11 The measures νΛandρΛvanish identically. End of the proof of Theorem 2.6 32 12 Proof of Proposition 8.2 33 13 Proof of Proposition 8.3: existence of the cutoff function 35 IV An a priori lower bound for decay rates on the torus: proof of Theorem 2.5 41 A Pseudodifferential calculus 42 B Spectrum of P(z)for a piecewise constant damping (by St´ ephane Nonnenmacher) 42 Part I The damped wave equation 1 Decay of energy: a survey of existing results Let (M,g) be a smooth compact connected Riemannian d-dimensional manifold with or without boundary∂M. We denote by ∆ the (non-positive) Laplace-Beltrami operator on Mfor the metric g. Given a bounded nonnegative function, b∈L∞(M),b(x)≥0 onM, we want to understand the asymptotic behaviour as t→+∞of the solution uof the problem ∂2 tu−∆u+b(x)∂tu= 0 inR+×M, u= 0 on R+×∂M(if∂M/\e}atio\slash=∅), (u,∂tu)|t=0= (u0,u1) inM.(1.1) The energy of a solution is defined by E(u,t) =1 2(/ba∇dbl∇u(t)/ba∇dbl2 L2(M)+/ba∇dbl∂tu(t)/ba∇dbl2 L2(M)). (1.2) 2Multiplying (1.1) by ∂tuand integrating on Myields the following dissipation identity d dtE(u,t) =−/integraldisplay Mb|∂tu|2dx, which, asbis nonnegative, implies a decay of the energy. As soon as b≥C >0 on a nonempty open subset ofM, the decay is strict and E(u,t)→0 ast→+∞. The question is then to know at which rate the energy goes to zero. The first interesting issue concerns uniform stabilization: under wh ich condition does there exist a functionF(t),F(t)→0, such that E(u,t)≤F(t)E(u,0) ? (1.3) The answer was given by Rauch and Taylor [RT74] in the case ∂M=∅and by Bardos, Lebeau and Rauch [BLR92] in the general case (see also [BG97] for the necessit y of this condition): assuming thatb∈C0(M), uniform stabilisation occurs if and only if the set {b >0}satisfies the Geometric Control Condition (GCC). Recall that a set ω⊂Mis said to satisfy GCC if there exists L0>0 such that every geodesic γ(resp. generalised geodesic in the case ∂M/\e}atio\slash=∅) ofMwith length larger thanL0satisfiesγ∩ω/\e}atio\slash=∅. Under this condition, one can take F(t) =Ce−κt(for some constants C,κ>0) in (1.3), and the energy decays exponentially. Finally, Lebeau give s in [Leb96] the explicit (and optimal) value of the best decay rate κin terms of the spectral abscissa of the generator of the semigroup and the mean value of the function balong the rays of geometrical optics. In the case where {b >0}does not satisfy GCC, i.e. in the presence of “trapped rays” that d o not meet {b>0}, what can be said about the decay rate of the energy? As soon as b≥C >0 on a nonempty open subset of M, Lebeau shows in [Leb96] that the energy (of smoother initial data ) goes at least logarithmically to zero (see also [Bur98]): E(u,t)≤C/parenleftbig f(t)/parenrightbig2/parenleftBig /ba∇dblu0/ba∇dbl2 H2(M)∩H1 0(M)+/ba∇dblu1/ba∇dbl2 H1 0(M)/parenrightBig ,for allt>0, (1.4) withf(t) =1 log(2+t)(whereH2(M)∩H1 0(M) andH1 0(M) have to be replaced by H2(M) andH1(M) respectively if ∂M=∅). Note that here,/parenleftbig f(t)/parenrightbig2characterizes the decay of the energy, whereas f(t) is that of the associated semigroup. Moreover, the author const ructed a series of explicit examples of geometries for which this rate is optimal, including for instance the case where M=S2is the two-dimensional sphere and {b >0} ∩Nε=∅, whereNεis a neighbourhood of an equator of S2. This result is generalised in [LR97] for a wave equation damped on a (sm all) part of the boundary. In this paper, the authors also make the following comment about th e result they obtain: “Notons toutefois qu’une ´ etude plus approfondie de la localisation s pectrale et des taux de d´ ecroissance de l’´ energie pour des donn´ ees r´ eguli` eres doit faire intervenir la dynamique globale du flot g´ eod´ esique g´ en´ eralis´ esur M. Les th´ eor` emes [LR97, Th´ eor` eme 1] et [LR97, Th´ eor` eme 2] ne four- nissent donc que les bornes a prioriqu’on peut obtenir sans aucune hypoth` ese sur la dynamique, en n’utilisant que les in´ egalit´ es de Carleman qui traduisent “l’effet tunn el”.” In all examples where the optimal decay rate is logarithmic, the trap ped ray is a stable trajectory from the point of view of the dynamics of the geodesic flow. This mean s basically that an important amount of the energy can stay concentrated, for a long time, in a n eighbourhood of the trapped ray, i.e. away from the damping region. If the trapped trajectories are less stable, or unstable, one can expect to obtain an intermediate decay rate, between exponential and logarithmic. We shall say tha t the energy decays at rate f(t) if (1.4) is satisfied (more generally, see Definition 2.2 below in the abstr act setting). This problem has already been adressed and, in some particular geometries, sev eral different behaviours have been exhibited. Two main directions have been investigated. On the one hand, Liu and Rao considered in [LR05] the case where Mis a square and the set {b >0}contains a vertical strip. In this situation, the trapped trajecto ries consist in a family 3of parallel vertical geodesics; these are unstable, in the sense th at nearby geodesics diverge at a linear rate. They proved that the energy decays at rate/parenleftBig log(t) t/parenrightBig1 2(i.e., that (1.4) is satisfied with f(t) =/parenleftBig log(t) t/parenrightBig1 2). This was extended by Burq and Hitrik [BH07] (see also [Nis09]) to the case of partially rectangular two-dimensional domains, if the set {b >0}contains a neighbourhood of the non-rectangular part. In [Phu07], Phung proved a decay at rate t−δfor some (unprecised) δ >0 in a three-dimensional domain having two parallel faces. In all these s ituations, the only obstruction to GCC is due to a “cylinder of periodic orbits”. The geometry is flat an d the unstabilities of the geodesic flow around the trapped rays are relatively weak (geodes ics diverge at a linear rate). In [BH07], the authors argue that the optimal decay in their geomet ry should be of the form1 t1−ε, for allε>0. They provide conditions on the damping coefficient b(x) under which one can obtain such decay rates, and wonder whether this is true in general. Our m ain theorem (see Theorem 2.6 below) extends these results to more general damping functions bon the two-dimensional torus. On the other hand, Christianson [Chr10] proved that the energy d ecays at rate e−C√ tfor some C >0, in the case where the trapped set is a hyperbolic closed geodesic. Schenck [Sch11] proved an energy decay at rate e−Cton manifolds with negative sectional curvature, if the trapped set is “small enough” in terms of topological pressure (for instance, a small ne ighbourhood of a closed geodesic), and if the damping is “large enough” (that is, starting from a damping functionb,βbwill work for anyβ >0 sufficiently large). In these two papers, the geodesic flow near th e trapped set enjoys strong instability properties: the flow on the trapped set is uniform ly hyperbolic, in particular all trajectories are exponentially unstable. These cases confirm the idea that the decay rates of the energy s trongly depends on the stability of trapped trajectories. One may now want to compare these geometric situations to situatio ns where the Schr¨ odinger group is observable (or, equivalently, controllable), i.e. for which th ere existC >0 andT >0 such that, for all u0∈L2(M), we have /ba∇dblu0/ba∇dbl2 L2(M)≤C/integraldisplayT 0/ba∇dbl√ b e−it∆u0/ba∇dbl2 L2(M)dt. (1.5) The conditions under which this property holds are also known to be r elated to stability of the geodesic flow. In particular, the works [BLR92], [LR05], [BH07, Nis09] and [Chr10, Sch11] can be seen as counterparts for damped wave equations of the articles [L eb92], [Har89a, Jaf90], [BZ04] and [AR10], respectively, in the context of observation of the Schr¨ od inger group. Our main results are twofold. First, we clarify (in an abstract settin g) the link between the ob- servability (or the controllability) of the Schr¨ odinger equation and polynomial decay for the damped wave equation. This follows the spirit of [Har89b], [Mil05], exploring the lin ks between the different equations and their control properties (e.g. observability, contr ollability, stabilization...). More pre- cisely, we prove that the controllability of the Schr¨ odinger equatio n implies a polynomial decay at rate1√ tfor the damped wave equation (Theorem 2.3). Second, we study precisely the damped wave equation on the flat to rusT2in case GCC fails. We give the following a priori lower bound on the decay rate, revisiting the argument of [BH07]: (1.1) is not stable at a better rate than1 t, provided that GCC is not satisfied. In this situation, the Schr¨ odingergroupisknowntobecontrollable(see[Jaf90], [Kom92] andthemorerecentworks[AM11] and [BZ11]). Thus, one cannot hope to have a decay better than po lynomial in our previous result, i.e. under the mere assumption that the Schr¨ odinger flow is observ able. The remainder of the paper is devoted to studying the gap between thea priorilower and upper bounds given respectively by1 tand1√ ton flat tori. For smoothnonvanishing damping coefficient b(x), we prove that the energy decays at rate1 t1−εfor allε >0. This result holds without making any dynamical assumption on the damping coefficient, but only on the order of vanishing of b. It generalises a result of [BH07], which holds in the case where bis invariant in one direction. Our 4analysis is, again, inspired by the recent microlocal approach propo sed in [AM11] and [BZ11] for the observability of the Schr¨ odinger group. More precisely, we fo llow here several ideas and tools introduced in [Mac10] and [AM11]. In the situation where bis a characteristic function of a vertical strip of the torus (hence discon- tinuous), St´ ephane Nonnenmacher proves in Appendix B that the decay rate cannot be faster than 1 t2/3. This is done by explicitly computing the high frequency eigenvalues of the damped wave oper- ator which are closest to the imaginary axis (see for instance the fig ures in [AL03, AL12]). The fact that the decay rate 1 /tis not achieved in this situation was observed in the numerical comput ations presented in [AL12]. In contrast to the control problem for the Sch¨ odinger equation , this result shows that the stabi- lization of the wave equation is not only sensitive to the global proper ties of the geodesic flow, but also to the rate at which the damping function vanishes. 2 Main results of the paper Our first result can be stated in a general abstract setting that w e now introduce. We come back to the case of the torus afterwards. 2.1 The damped wave equation in an abstract setting LetHandYbe two Hilbert spaces (resp. the state space and the observation /control space) with norms/ba∇dbl·/ba∇dblHand/ba∇dbl·/ba∇dblY, and associated inner products ( ·,·)Hand (·,·)Y. We denote by A:D(A)⊂H→Hanonnegative selfadjoint operatorwith compact resolvent, and B∈ L(Y;H) a control operator. We recall that B∗∈ L(H;Y) is defined by ( B∗h,y)Y= (h,By)H for allh∈Handy∈Y. Definition 2.1. We say that the system ∂tu+iAu= 0, y=B∗u, (2.1) is observable in time Tif there exists a constant KT>0 such that, for all solution of (2.1), we have /ba∇dblu(0)/ba∇dbl2 H≤KT/integraldisplayT 0/ba∇dbly(t)/ba∇dbl2 Ydt. We recall that the observability of (2.1) in time Tis equivalent to the exact controllability in timeTof the adjoint problem ∂tu+iAu=Bf, u(0) =u0, (2.2) (see for instance [Leb92] or [RTTT05]). More precisely, given T >0, the exact controllability in time Tis the ability of finding for any u0,u1∈Ha control function f∈L2(0,T;Y) so that the solution of (2.2) satisfies u(T) =u1. We equip H=D(A1 2)×Hwith the graph norm /ba∇dbl(u0,u1)/ba∇dbl2 H=/ba∇dbl(A+Id)1 2u0/ba∇dbl2 H+/ba∇dblu1/ba∇dbl2 H, and define the seminorm |(u0,u1)|2 H=/ba∇dblA1 2u0/ba∇dbl2 H+/ba∇dblu1/ba∇dbl2 H. Of course, if Ais coercive on H,|·|His a norm on Hequivalent to /ba∇dbl·/ba∇dblH. We also introduce in this abstract setting the damped wave equation on the space H, /braceleftBigg ∂2 tu+Au+BB∗∂tu= 0, (u,∂tu)|t=0= (u0,u1)∈ H,(2.3) 5which can be recast on Has a first order system /braceleftbigg ∂tU=AU, U|t=0=t(u0,u1),U=/parenleftbigg u ∂tu/parenrightbigg ,A=/parenleftbigg 0 Id −A−BB∗/parenrightbigg , D(A) =D(A)×D(A1 2).(2.4) The compact injections D(A)֒→D(A1 2)֒→Himply that D(A)֒→ Hcompactly, and that the operator Ahas a compact resolvent. We define the energy of solutions of (2.3) by E(u,t) =1 2/parenleftbig /ba∇dblA1 2u/ba∇dbl2 H+/ba∇dbl∂tu/ba∇dbl2 H/parenrightbig =1 2|(u,∂tu)|2 H2. Definition 2.2. Letfbe a function such that f(t)→0 whent→+∞. We say that System (2.3) is stable at rate f(t) if there exists a constant C >0 such that for all ( u0,u1)∈D(A), we have E(u,t)1 2≤Cf(t)|A(u0,u1)|H,for allt>0. If it is the case, for all k >0, there exists a constant Ck>0 such that for all ( u0,u1)∈D(Ak), we have (see for instance [BD08, page 767]) E(u,t)1 2≤Ck/parenleftbig f(t)/parenrightbigk/ba∇dblAk(u0,u1)/ba∇dblH,for allt>0. Theorem 2.3. Suppose that there exists T >0such that System (2.1)is observable in time T. Then System(2.3)is stable at rate1√ t. Note that the gain of the log( t)1 2with respect to [LR05, BH07] is not essential in our work. It is due to the optimal characterization of polynomially decaying semigro ups obtained by Borichev and Tomilov [BT10]. This Theorem may be compared with the works (both presented in a s imilar abstract setting) [Har89b]byHaraux,provingthatthe controllabilityofwave-typee quationsinsometimeisequivalent to uniform stabilization of (2.3), and [Mil05] by Miller, showing that the c ontrollability of wave-type equations in some time implies the controllability of Schr¨ odinger-type equations in any time. Note that the link between this abstract setting and that of Proble m (1.1) isH=Y=L2(M), A=−∆ withD(A) =H2(M) if∂M=∅andH2(M)∩H1 0(M) otherwise, Bis the multiplication in L2(M) by the bounded function√ b. As a first application of Theorem 2.3 we obtain a different proof of the polynomial decay results for wave equations of [LR05] and [BH07] as consequences of the as sociated control results for the Schr¨ odinger equation of [Har89a] and [BZ04] respectively. Moreover, Theorem 2.3 provides also several new stability results f or System (1.1) in particu- lar geometric situations; namely, in all following situations, the Schr¨ odinger group is proved to be observable, and Theorem 2.3 gives the polynomial stability at rate1√ tfor (1.1): •For any nonvanishing b(x)≥0 in the 2-dimensional square (resp. torus), as a consequence of [Jaf90] (resp. [Mac10, BZ11]); for any nonvanishing b(x)≥0 in thed-dimensional rectangle (resp.d-dimensional torus) as a consequence of [Kom92] (resp. [AM11]); •IfMis the Bunimovich stadium and b(x)>0 on the neighbourhood of one half disc and on one point of the opposite side, as a consequence of [BZ04]; •IfMis ad-dimensional manifold of constant negative curvature and the set of trapped tra- jectories (as a subset of S∗M, see [AR10, Theorem 2.5] for a precise definition) has Hausdorff dimension lower than d, as a consequence of [AR10]; 6Moreover, Lebeau gives in [Leb96, Th´ eor` eme 1 (ii)] several 2-dim ensional examples for which the decay rate1 log(2+t)is optimal. For all these geometrical situations, Theorem 2.3 implies th at the Schr¨ odinger group is not observable. The proof of Theorem 2.3 relies on the following characterization of p olynomial decay for Sys- tem (2.3). For z∈C, we define on Hthe operator P(z) =A+z2Id+zBB∗, with domain D(P(z)) =D(A). We prove in Lemma 4.2 below that P(is) is invertible for all s∈R,s/\e}atio\slash= 0. Proposition 2.4. Suppose that for any eigenvector ϕofA, we haveB∗ϕ/\e}atio\slash= 0. (2.5) Then, for all α>0, the five following assertions are equivalent: The system (2.3)is stable at rate1 tα, (2.6) There exist C >0ands0≥0such that for all s∈R,|s| ≥s0,/ba∇dbl(isId−A)−1/ba∇dblL(H)≤C|s|1 α,(2.7) There exist C >0ands0≥0such that for all z∈C,satisfying |z| ≥s0, and|Re(z)| ≤1 C|Im(z)|1 α,we have /ba∇dbl(zId−A)−1/ba∇dblL(H)≤C|Im(z)|1 α,(2.8) There exist C >0ands0≥0such that for all s∈R,|s| ≥s0,/ba∇dblP(is)−1/ba∇dblL(H)≤C|s|1 α−1,(2.9) There exists C >0ands0≥0such that for all s∈R,|s| ≥s0andu∈D(A), /ba∇dblu/ba∇dbl2 H≤C/parenleftbig |s|2 α−2/ba∇dblP(is)u/ba∇dbl2 H+|s|1 α/ba∇dblB∗u/ba∇dbl2 Y/parenrightbig .(2.10) Thispropositionisprovedasaconsequenceofthecharacterizatio nofpolynomialdecayforgeneral semigroups in terms of resolvent estimates given in [BT10], providing t he equivalence between (2.6) and (2.7). See also [BD08] for generaldecay rates in Banach space s. Note in particular that the proof of a decay rate is reduced to the proof of a resolvent estimate on t he imaginary axes. By the way, this estimate implies the existence of a “spectral gap” between the spectrum of Aand the imaginary axis, given by (2.8). Note finally that the estimates (2.7), (2.9) and (2.10) can be equivale ntly restricted to s >0, sinceP(−is)u=P(is)u. 2.2 Decay rates for the damped wave equation on the torus The main results of this article deal with the decay rate for Problem ( 1.1) on the torus T2:= (R/2πZ)2. In this setting, as well as in the abstract setting, we shall write P(z) =−∆+z2+zb(x). First, we give an a priori lower bound for the decay rate of the damped wave equation, on T2, when GCC is “strongly violated”, i.e. assuming that supp( b) does not satisfy GCC (instead of {b>0}). This theorem is proved by constructing explicit quasimodes for the operator P(is). Theorem 2.5. Suppose that there exists (x0,ξ0)∈T∗T2,ξ0/\e}atio\slash= 0, such that {b>0}∩{x0+τξ0,τ∈R}=∅. Then there exist two constants C >0andκ0>0such that for all n∈N, /ba∇dblP(inκ0)−1/ba∇dblL(L2(T2))≥C. (2.11) As a consequence of Proposition 2.4, polynomial stabilization at rate1 t1+εforε >0 is not possible if there is a strongly trapped ray (i.e. that does not interse ct supp(b)). More precisely, in such geometry, Theorem 2.5 combined with Lemma 4.6 and [BD08, Pro position 1.3] shows that m1(t)≥C 1+t, for someC >0 (with the notation of [BD08] where m1(t) denotes the best decay rate). Then, the main goal of this paper is to explore the gap between the a prioriupper bound1√ tfor the decay rate, given by Theorem 2.3, and the a priorilower bound1 tof Theorem 2.5. Our results are twofold (somehow in two opposite directions) and concern eithe r the case of smooth damping functionsb, or the case b=1U, withU⊂T2. 72.2.1 The case of smooth damping coefficients Our main result deals with the case of smooth damping coefficients. Wit hout any geometric assump- tion, but with an additional hypothesis on the order of vanishing of t he damping function b, we prove a weak converse of Theorem 2.5. Theorem 2.6. LetM=T2with the standard flat metric. There exists ε0>0satisfying the following property. Suppose that bis a nonnegative nonvanishing function on T2satisfying√ b∈C∞(T2)and that there exist ε∈(0,ε0)andCε>0such that |∇b(x)| ≤Cεb1−ε(x),forx∈T2. (2.12) Then, there exist C >0ands0≥0such that for all s∈R,|s| ≥s0, /ba∇dblP(is)−1/ba∇dblL(L2(T2))≤C|s|δ,withδ= 8ε (2.13) As a consequence of Proposition 2.4, in this situation, the d amped wave equation (1.1)is stable at rate1 t1 1+δ. Following carefully the steps of the proof, one sees that ε0=1 76works, but the proof is not optimized with respect to this parameter, and it is likely that it could be much improved. One of the main difficulties in understanding the decay rates is that th ere exists no general monotonicity property of the type “ b1(x)≤b2(x) for allx=⇒the decay rate associated to the dampingb2is larger (or smaller) than the decay rate associated to the damping b1”. This makes a significant difference with observability or controllability problems of t he type (1.5). Assumption (2.12) is only a local assumption in a neighbourhood of ∂{b>0}(even if it is stated here globally on T2). Far from this set, i.e. on each compact set {b≥b0}forb0>0, the constant Cεcan be choosen uniformly, depending only on b0, and not on ε. Hence,εsomehow quantifies the vanishing rate of the damping function b. An interesting situation is when the smooth function bvanishes like e−1 xαin smooth local coordi- nates, for some α>0. In this case, Assumption (2.12) is satisfied for any ε>0, and the associated damped wave equation (1.1) is stable at rate1 t1−δfor anyδ >0. This shows that the lower bound given by Theorem 2.5, as well as the decay rate1 t, are sharp in general. This phenomenon had already been remarked by Burq and Hitrik in [BH07] in the case where bis invariant in one direction. Typical smooth functions not satisfying Assumption (2.12) are for instance functions vanishing like sin(1 x)2e−1 x. We do not have any idea of the decay rate achieved in this case (exc ept for the a prioribounds1√ tand1 t). Theorem 2.6 generalises the result of [BH07], which only holds if bis assumed to be invariant in one direction. Our proof is based on ideas and tools developped in [Ma c10, AM11] and especially on two-microlocal semiclassical measures. One of the key technica l points appears in Section 13: we have to construct, for each trapped direction, a cutoff funct ion invariant in that direction and adapted to the damping coefficient b. We do not know how to adapt this technical construction to tori of higher dimension, d >2; hence we do not know whether Theorem 2.6 holds in higher dimension (although we have no reason to suspect it should not hold) . Only in the particular case wherebis invariant in d−1 directions can our methods (or those of [BH07]) be applied to prove the analogue of Theorem 2.6. Note that if GCC is satisfied, one has (on a general compact manifold M) for someC >1 and all|s| ≥s0the estimate /ba∇dblP(is)−1/ba∇dblL(L2(M))≤C|s|−1. (2.14) instead of (2.13). Estimate (2.14) is in turn equivalent to uniform sta bilization (see [Hua85] together with Lemma 4.6 below). 8Remark 2.7. As a consequence of Theorem 2.6 on the torus, we can deduce that the decay rate t−1 1+δalso holds for Equation (1.1) if M= (0,π)2is the square, one takes with Dirichlet or Neumann boundary conditions, and the damping function bis smooth, vanishes near ∂Mand satisfies As- sumption (2.12). First, we extend the function bas an even (with respect to both variables) smooth function on the larger square ( −π,π)2, and using the injection ı: (−π,π)2→T2, as a smooth func- tion onT2, still satisfying (2.12). Moreover, D(∆D) (resp.D(∆N)) on (0,π)2can be identified as the closed subspace of odd (resp. even) functions of D(∆D) (resp.D(∆N)) on (−π,π)2. Using again the injection ı, it can also be identified with a closed subspace of H2(T2). The estimate /ba∇dblu/ba∇dblL2(T2)≤C|s|δ/ba∇dblP(is)u/ba∇dblL2(T2)for allu∈H2(T2), is thus also true on the square (0 ,π)2for Dirichlet or Neumann boundary conditions. In particular, this strongly improves the results of [LR05]. The lower bound of Theorem 2.5 can be similarly extended to the case o f a square with Dirichlet or Neumann boundary conditions, implying that the rate1 tis optimal if GCC is strongly violated. 2.2.2 The case of discontinuous damping functions Appendix B (by St´ ephane Nonnenmacher) deals with the case wher ebis the characteristic function of a vertical strip, i.e. b=/tildewideB1U, for some/tildewideB >0 andU= (a,b)×T⊂T2. Due to the invariance of b in one direction, the spectrum of the damped waveoperator Asplits into countably many “branches” of eigenvalues. This structure of the spectrum is illustrated in the n umerics of [AL03, AL12]. The branch closest to the imaginary axis is explicitly computed, it cont ains a sequence of eigen- values (zi)i∈Nsuch that Im zi→ ∞and|Rezi| ≤C0 (Imzi)3/2. This result is in agreement with the numerical tests given in [AL12]. As a consequence, for any ε >0 andC >0, the strip/braceleftbig |Rez| ≤C|Im(z)|−3/2+ε/bracerightbig contains in- finitely many poles of the resolvent ( zId−A)−1, so item (2.8) in Proposition 2.4 implies the following obstruction to the stability of this damped system : Corollary 2.8. For anyε >0, the damped wave equation (1.1)onT2with the damping function (B.1)cannot be stable at the rate1 t2/3+ε. The same result holds on the square with Dirichlet or Neumann boundary conditions. More precisely, in this situation, Lemma 4.6 and [BD08, Proposition 1.3] yield thatm1(t)≥ C (1+t)2/3, for someC >0 (with the notation of [BD08] where m1(t) denotes the best decay rate). This corollary shows in particular that the regularity conditions in The orem 2.6 cannot be com- pletely disposed of if one wants a stability at the rate 1 /t1−εfor smallε. 2.3 Some related open questions The various results obtained in this article lead to several open ques tions. 1. In the case where bis the characteristic function of a vertical strip, our analysis show s that the best decay rate lies somewhere between1 t1 2and1 t2 3, but the “true” decay rate is not yet clear. 2. It would also be interesting to investigate the spectrum and the d ecay rates for damping functionsbinvariant in one direction, but having a less singular behaviour than a c haracteristic function. In particular, is it possible to give a precise link between the vanishing rate of band the decay rate? 3. In the general setting of Section 2.1 (as well as in the case of the damped wave equation on the torus), is the a prioriupper bound1 t1 2for the decay rate optimal? 4. For smooth damping functions vanishing like e−1 xα, Theorem 2.6 yields stability at rate1 t1−δ for allδ>0. Is the decay rate1 treached in this situation? Can one find a damping function b such that the decay rate is exactly1 t? 95. The lower bound of of Theorem 2.5 is still valid in higher dimensional to ri. Is there an analogue of Theorem 2.6 (i.e. for general “smooth” damping functions) for Td, withd≥3? Part II Resolvent estimates and stabilization in the abstract setting 3 Proof of Theorem 2.3 assuming Proposition 2.4 To prove Theorem 2.3, we express the observability condition as a re solvent estimate (also known as the Hautus test), as introduced by Burq and Zworski [BZ04], and f urther developed by Miller [Mil05] and Ramdani, Takahashi, Tenenbaum and Tucsnak [RTTT05]. For a su rvey of this notion, we refer to the book [TW09, Section 6.6]. In particular [Mil05, Theorem 5.1] (or [TW09, Theorem 6.6.1]) yields that System (2.1) is ob- servable in some time T >0 if and only if there exists a constant C >0 such that we have /ba∇dblu/ba∇dbl2 H≤C/parenleftbig /ba∇dbl(A−λId)u/ba∇dbl2 H+/ba∇dblB∗u/ba∇dbl2 Y/parenrightbig ,for allλ∈Randu∈D(A). As a first consequence, Assumption (2.5) is satisfied and Propositio n 2.4 applies in this context. Moreover, we have, for all s∈Randu∈D(A), /ba∇dblu/ba∇dbl2 H≤C/parenleftbig /ba∇dbl(A−s2Id+isBB∗−isBB∗)u/ba∇dbl2 H+/ba∇dblB∗u/ba∇dbl2 Y/parenrightbig ≤C/parenleftbig /ba∇dblP(is)u/ba∇dbl2 H+s2/ba∇dblBB∗u/ba∇dbl2 H+/ba∇dblB∗u/ba∇dbl2 Y/parenrightbig (3.1) SinceB∈ L(Y;H), we obtain for s≥1 and for some C >0, /ba∇dblu/ba∇dbl2 H≤C/parenleftbig /ba∇dblP(is)u/ba∇dbl2 H+s2/ba∇dblB∗u/ba∇dbl2 Y/parenrightbig ≤C/parenleftbig s2/ba∇dblP(is)u/ba∇dbl2 H+s2/ba∇dblB∗u/ba∇dbl2 Y/parenrightbig . Proposition 2.4 then yields the polynomial stability at rate1√ tfor (2.3). This concludes the proof of Theorem 2.3. 4 Proof of Proposition 2.4 Our proof strongly relies on the characterization of polynomially sta ble semigroups, given in [BT10, Theorem 2.4], which can be reformulated as follows. Theorem 4.1 ([BT10], Theorem 2.4) .Let(et˙A)t≥0be a bounded C0-semigroup on a Hilbert space ˙H, generated by ˙A. Suppose that iR∩Sp(˙A) =∅. Then, the following conditions are equivalent: /ba∇dblet˙A˙A−1/ba∇dblL(˙H)=O(t−α),ast→+∞, (4.1) /ba∇dbl(isId−˙A)−1/ba∇dblL(˙H)=O(|s|1 α),ass→ ∞. (4.2) Let us first describe some spectral properties of the operator Adefined in (2.4). Lemma 4.2. The spectrum of Acontains only isolated eigenvalues and we have Sp(A)⊂/parenleftbigg/parenleftbig −1 2/ba∇dblB∗/ba∇dbl2 L(H;Y),0/parenrightbig +iR/parenrightbigg ∪/parenleftBig [−/ba∇dblB∗/ba∇dbl2 L(H;Y),0]+0i/parenrightBig , withker(A) = ker(A)×{0}. 10Moreover, the operator P(z)is an isomorphism from D(A)ontoHif and only if z /∈Sp(A). If this is satisfied, we have (zId−A)−1=/parenleftbigg P(z)−1(BB∗+zId)P(z)−1 P(z)−1(zBB∗+z2Id)−IdzP(z)−1/parenrightbigg . (4.3) The localization properties for the spectrum of A, stated in the first part of this lemma are illustrated for instance in [AL03] or [AL12]. This Lemma leads us to introduce the spectral projector of Aon ker(A), given by Π0=1 2iπ/integraldisplay γ(zId−A)−1dz∈ L(H), whereγdenotes a positively oriented circle centered on 0 with a radius so sma ll that 0 is the single eigenvalue of Ain the interior of γ. We set ˙H= (Id−Π0)Hand equip this space with the norm /ba∇dbl(u0,u1)/ba∇dbl2 ˙H:=|(u0,u1)|2 H=/ba∇dblA1 2u0/ba∇dbl2 H+/ba∇dblu1/ba∇dbl2 H, and associated inner product. This is indeed a norm on ˙Hsince/ba∇dbl(u0,u1)/ba∇dbl˙H= 0 is equivalent to (u0,u1)∈ker(A)×{0}= Π0H. Besides, we set ˙A=A|˙Hwith domain D(˙A) =D(A)∩˙H. A first remark is that Sp( ˙A) = Sp(A)\{0}, so that Sp( ˙A)∩iR=∅. The remainder of the proof consists in applying Theorem 4.1 to the op erator˙Ain˙H. We first check the assumptions of Theorem 4.1 and describe the solutions of the evolution problem (2.4) (or equivalently (2.3)). Lemma 4.3. The operator ˙Agenerates a contraction C0-semigroup on ˙H, denoted (et˙A)t≥0. More- over, for all initial data U0∈ H, Problem (2.4)(or equivalently (2.3)) has a unique solution U∈C0(R+;H), issued from U0, that can be decomposed as U(t) =et˙A(Id−Π0)U0+Π0U0,for allt≥0. (4.4) As a consequence, we can apply Theorem 4.1 to the semigroup gener ated by ˙A. The proof of Proposition 2.4 will be achieved when the following lemmata are proved. Lemma 4.4. Conditions (2.6)and(4.1)are equivalent. Lemma 4.5. Conditions (2.9)and(2.10)are equivalent. Conditions (2.7)and(2.8)are equivalent. Lemma 4.6. There exist C >1ands0>0such that for s∈R,|s| ≥s0, /ba∇dbl(isId−˙A)−1/ba∇dblL(˙H)−C |s|≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)≤ /ba∇dbl(isId−˙A)−1/ba∇dblL(˙H)+C |s|,(4.5) and C−1|s|/ba∇dblP(is)−1/ba∇dblL(H)≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)≤C/parenleftbig 1+|s|/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig . (4.6) In particular this implies that (4.2),(2.7)and(2.9)are equivalent. The proof of Lemma 4.6 is more or less classical and we follow [Leb96, BH 07]. Proof of Lemma 4.2. AsAhas compact resolvent, its spectrum contains only isolated eigenva lues. Suppose that z∈Sp(A), then we have, for some ( u0,u1)∈D(A)\{0}, /braceleftbiggu1=zu0, −Au0−BB∗u1=zu1, and in particular Au0+z2u0+zBB∗u0= 0, (4.7) 11withu0∈D(A)\{0}. Suppose that z∈iR, then, this yields Au0−Im(z)2u0+iIm(z)BB∗u0= 0. Following [Leb96], takingtheinnerproductofthisequationwith u0yieldsiIm(z)/ba∇dblB∗u0/ba∇dbl2 Y= 0. Hence, eitherIm( z) = 0, orB∗u0= 0. In the first case, Au0= 0, i.e.u0∈ker(A), andu1= 0. This yields ker( A)⊂ ker(A)×{0}(and the otherinclusion is clear). In the second case, u0is an eigenvectorof Aassociated to the eigenvalue Im( z)2and satisfies B∗u0= 0, which is absurd, according to Assumption (2.5). Thus, Sp( A)∩iR⊂ {0}. Now, for a general eigenvalue z∈C, taking the inner product of (4.7) with u0yields /braceleftbigg(Au0,u0)H+(Re(z)2−Im(z)2)/ba∇dblu0/ba∇dbl2 H+Re(z)/ba∇dblB∗u0/ba∇dbl2 Y= 0, 2Re(z)Im(z)/ba∇dblu0/ba∇dbl2 H+Im(z)/ba∇dblB∗u0/ba∇dbl2 Y= 0.(4.8) If Im(z)/\e}atio\slash= 0, then, the second equation of (4.8) together with Sp( ˙A)∩iR⊂ {0}gives 0>Re(z) =−1 2/ba∇dblB∗u0/ba∇dbl2 Y /ba∇dblu0/ba∇dbl2 H≥ −1 2/ba∇dblB∗/ba∇dbl2 L(H;Y). If Im(z) = 0, then, the first equation of (4.8) together with ( ˙Au0,u0)H≥0 gives−Re(z)/ba∇dblB∗u0/ba∇dbl2 Y≥ Re(z)2/ba∇dblu0/ba∇dbl2 H, which yields 0≥Re(z)≥ −/ba∇dblB∗/ba∇dbl2 L(H;Y). Following [Leb96], we now give the link between P(z)−1and (zId−A)−1forz /∈Sp(A). Taking F= (f0,f1)∈ H, andU= (u0,u1), we have F= (zId−A)U⇐⇒/braceleftbiggu1=zu0−f0, P(z)u0=f1+(BB∗+zId)f0.(4.9) As a consequence, we obtain that P(z) :D(A)→His invertible if and only if ( zId−A) :D(A)→ H is invertible, i.e. if and only if z /∈Sp(A). Moreover, for such values of z, System (4.9) is equivalent to /braceleftbiggu0=P(z)−1f1+P(z)−1(BB∗+zId)f0, u1=zP(z)−1f1+zP(z)−1(BB∗+zId)f0−f0, which can be rewritten as (4.3). This concludes the proof of Lemma 4 .2. Proof of Lemma 4.3. Let us check that ˙Ais a maximal dissipative operator on ˙H[Paz83]. First, it is dissipative since, for U= (u0,u1)∈D(˙A), (˙AU,U)˙H= (A1 2u1,A1 2u0)H−(Au0,u1)H−(BB∗u1,u1)H=−/ba∇dblB∗u1/ba∇dbl2 Y≤0. Next, the fact that A −Id is onto is a consequence of Lemma 4.2. Hence, for all F∈˙H ⊂ H, there exists U∈D(A) such that ( A −Id)U=F. Applying (Id −Π0) to this identity yields ( ˙A − Id)(Id−Π0)U=F, sothat ˙A−Id :D(˙A)→˙Hisonto. AccordingtotheLumer-PhillipsTheorem(see for instance [Paz83, Chapter 1, Theorem 4.3]) ˙Agenerates a contraction C0-semigroup on ˙H. Then, Formula (4.4) directly comes from the linearity of Equation (2.4) (or e quivalently (2.3)) together with the decomposition of the initial condition U0= (I−Π0)U0+Π0U0. Proof of Lemma 4.4. Condition (4.1) is equivalent to the existence of C >0 such that for all t>0, and˙U0∈˙H, we have /ba∇dblet˙A˙A−1˙U0/ba∇dbl˙H≤C tα/ba∇dbl˙U0/ba∇dbl˙H. This can be rephrased as /ba∇dblet˙A˙U0/ba∇dbl˙H≤C tα/ba∇dbl˙A˙U0/ba∇dbl˙H, (4.10) for allt >0, and˙U0∈D(˙A). Now, take any U0= (u0,u1)∈D(A), and associated projection ˙U0= (Id−Π0)U0∈D(˙A). According to (4.4), we have E(u,t) =1 2/parenleftbig /ba∇dblA1 2u(t)/ba∇dbl2 H+/ba∇dbl∂tu(t)/ba∇dbl2 H/parenrightbig =1 2|et˙A˙U0+Π0U0|2 H=1 2/ba∇dblet˙A˙U0/ba∇dbl2 ˙H, 12and |AU0|H=|˙A˙U0+AΠ0U0|H=/ba∇dbl˙A˙U0/ba∇dbl˙H. This shows that (4.10) is equivalent to (2.6), and concludes the proo f of Lemma 4.4. Proof of Lemma 4.5. First, (2.9) clearly implies (2.10). To prove the converse, for u∈D(A), we have (P(is)u,u)H=/parenleftbig (A−s2Id)u,u/parenrightbig H+is/ba∇dblB∗u/ba∇dbl2 Y. Taking the imaginarypartof this identity gives s/ba∇dblB∗u/ba∇dbl2 Y= Im(P(is)u,u)H, so that, usingthe Young inequality, we obtain for all ε>0, |s|1 α/ba∇dblB∗u/ba∇dbl2 Y=|s|1 α−1|Im(P(is)u,u)H| ≤|s|2 α−2 4ε/ba∇dblP(is)u/ba∇dbl2 H+ε/ba∇dblu/ba∇dbl2 H. Plugging this into (2.10) and taking εsufficiently small, we obtain that for some C >0 ands0≥0, for anys∈R,|s| ≥s0, /ba∇dblu/ba∇dbl2 H≤C|s|2 α−2/ba∇dblP(is)u/ba∇dbl2 H, which yields (2.9). Hence (2.9) and (2.10) are equivalent. Second, Condition (2.8) clearlyimplies (2.7) and it only remains to prove the converse. For z∈C, we writer= Re(z) ands= Im(z). We have the identity ((r+is)Id−A)−1= (isId−A)−1/parenleftbig Id+r(isId−A)−1/parenrightbig−1. (4.11) Hence, assuming /ba∇dblr(isId−A)−1/ba∇dblL(H)≤1 2, (4.12) this gives /vextenddouble/vextenddouble/vextenddouble/parenleftbig Id+r(isId−A)−1/parenrightbig−1/vextenddouble/vextenddouble/vextenddouble L(H)=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay k=0(−1)k/parenleftbig r(isId−A)−1/parenrightbigk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L(H)≤2. As a consequence of (4.11) and (2.7), we then obtain /vextenddouble/vextenddouble((r+is)Id−A)−1/vextenddouble/vextenddouble L(H)≤2/ba∇dbl(isId−A)−1/ba∇dblL(H)≤2C|s|1 α, for alls≥s0, under Condition (4.12). Finally, (2.7) also yields /ba∇dblr(isId−A)−1/ba∇dblL(H)≤ |r|C|s|1 α, so that Condition (4.12) is realised as soon as |r| ≤1 2C|s|1 α. This proves (2.8) and concludes the proof of Lemma 4.5. Proof of Lemma 4.6. To prove (4.5), we first remark that the norms /ba∇dbl · /ba∇dbl˙Hand/ba∇dbl · /ba∇dblHare equiv- alent on ˙H, so that the norms /ba∇dbl · /ba∇dblL(˙H)and/ba∇dbl · /ba∇dblL(H)are equivalent on L(˙H). Next, we have (isId−˙A)−1(Id−Π0) = (isId−A)−1(Id−Π0) and /ba∇dbl(isId−˙A)−1/ba∇dblL(H)=/ba∇dbl(isId−˙A)−1(Id−Π0)/ba∇dblL(H)=/ba∇dbl(isId−A)−1(Id−Π0)/ba∇dblL(H) ≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)+/ba∇dbl(isId−A)−1Π0/ba∇dblL(H), together with /ba∇dbl(isId−A)−1/ba∇dblL(H)=/ba∇dbl(isId−˙A)−1(Id−Π0)+(isId−A)−1Π0/ba∇dblL(H) ≤ /ba∇dbl(isId−˙A)−1/ba∇dblL(H)+/ba∇dbl(isId−A)−1Π0/ba∇dblL(H). 13Moreover, for |s| ≥1, we have /ba∇dbl(isId−A)−1Π0/ba∇dblL(H)=/ba∇dbl(is)−1Π0/ba∇dblL(H)=1 |s|/ba∇dblΠ0/ba∇dblL(H)=C |s|, which concludes the proof of (4.5). Let us now prove (4.6). For concision, we set H1=D(A1 2) endowed with the graph norm /ba∇dblu/ba∇dblH1=/ba∇dbl(A+ Id)1 2u/ba∇dblHand denote by H−1=D(A1 2)′its dual space. The operator Acan be uniquely extended as an operator L(H1;H−1), still denoted Afo simplicity. With this notation, the spaceH−1can be equipped with the natural norm /ba∇dblu/ba∇dblH−1=/ba∇dbl(A+Id)−1 2u/ba∇dblH. As a consequenceof Formula(4.3), and using the fact that Sp( A)∩iR⊂ {0}, there exist constants C >1 ands0>0 such that for all s∈R,|s| ≥s0, C−1M(s)≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)≤CM(s) (4.13) with M(s) =/parenleftBig /ba∇dblP(is)−1(BB∗+isId)/ba∇dblL(H1)+/ba∇dblP(is)−1/ba∇dblL(H;H1) +/ba∇dblP(is)−1(isBB∗−s2Id)−Id/ba∇dblL(H1;H)+/ba∇dblsP(is)−1/ba∇dblL(H)/parenrightBig (4.14) On the one hand, this direcly yields for s∈R,|s| ≥s0, |s|/ba∇dblP(is)−1/ba∇dblL(H)≤C/ba∇dbl(isId−A)−1/ba∇dblL(H). This proves that (4.2) implies (2.9). On the other hand, we have to estimate each term of (4.14). First, usingAu=P(is)u+s2u− isBB∗u, we have /ba∇dblu/ba∇dbl2 H1=/ba∇dblA1 2u/ba∇dbl2 H+/ba∇dblu/ba∇dbl2 H=/parenleftbig P(is)u+s2u−isBB∗u,u/parenrightbig H+/ba∇dblu/ba∇dbl2 H = Re/parenleftbig P(is)u,u/parenrightbig H+(s2+1)/ba∇dblu/ba∇dbl2 H≤C/parenleftbig /ba∇dblP(is)u/ba∇dbl2 H+(s2+1)/ba∇dblu/ba∇dbl2 H/parenrightbig ≤C/parenleftBig 1+(s2+1)/ba∇dblP(is)−1/ba∇dbl2 L(H)/parenrightBig /ba∇dblP(is)u/ba∇dbl2 H, so that /ba∇dblP(is)−1/ba∇dblL(H;H1)≤C/parenleftbig 1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig . (4.15) Second, the same computation for ( P(is)−1)∗= (A−s2Id−isBB∗)−1(the adjoint of P(is)−1in the spaceH) in place of P(is)−1leads to (P(is)−1)∗∈ L(H;H1), together with the estimate /ba∇dbl(P(is)−1)∗/ba∇dblL(H;H1)≤C/parenleftbig 1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig . By transposition, we havet(P(is)−1)∗∈ L(H−1;H), together with the estimate /ba∇dblt(P(is)−1)∗/ba∇dblL(H−1;H)≤ /ba∇dbl(P(is)−1)∗/ba∇dblL(H;H1)≤C/parenleftbig 1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig .(4.16) Moreover,t(P(is)−1)∗is defined, for every u∈H,v∈H−1, by /parenleftbigt(P(is)−1)∗v,u/parenrightbig H=/angbracketleftbig v,(P(is)−1)∗u/angbracketrightbig H−1,H1=/parenleftBig (A+Id)−1 2v,(A+Id)1 2(P(is)−1)∗u/parenrightBig H. In particular, taking v∈Hgives /parenleftbigt(P(is)−1)∗v,u/parenrightbig H=/parenleftbig P(is)−1v,u/parenrightbig H, which implies that the restriction of the operatort(P(is)−1)∗toHcoincides with P(is)−1. For simplicity, we will denote P(is)−1fort(P(is)−1)∗. 14Equation (4.16) can thus be rewritten /ba∇dblP(is)−1/ba∇dblL(H−1;H)≤C/parenleftbig 1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig . (4.17) Then, we have P(is)−1(isBB∗−s2Id)−Id =P(is)−1A, so that /ba∇dblP(is)−1(isBB∗−s2Id)−Id/ba∇dblL(H1;H)=/ba∇dblP(is)−1A/ba∇dblL(H1;H)≤ /ba∇dblP(is)−1/ba∇dblL(H−1;H)/ba∇dblA/ba∇dblL(H1;H−1) ≤/parenleftbig 1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig (4.18) Third, for |s| ≥1 we write P(is)−1(BB∗+isId) =i s/parenleftbig P(is)−1A−Id/parenrightbig , (4.19) and it remains to estimate the term /ba∇dblP(is)−1A/ba∇dblL(H1)in (4.14). For f∈H1, we setu=P(is)−1Af. We haveu∈H1, together with (A−s2Id+isBB∗)u=Af. Taking the real part of the inner product of this identity with u, we find /ba∇dblA1 2u/ba∇dbl2 H−s2/ba∇dblu/ba∇dbl2 H= Re(Af,u)H≤ /ba∇dblAf/ba∇dblH−1/ba∇dblu/ba∇dblH1≤C/ba∇dblf/ba∇dblH1/ba∇dblu/ba∇dblH1, asA∈ L(H1,H−1). Hence /ba∇dblu/ba∇dbl2 H1≤C(1+s2)/ba∇dblu/ba∇dbl2 H+C/ba∇dblf/ba∇dbl2 H1 Using (4.17), this gives /ba∇dblu/ba∇dbl2 H1≤C(1+s2)/ba∇dblP(is)−1A/ba∇dbl2 L(H1;H)/ba∇dblf/ba∇dbl2 H1+C/ba∇dblf/ba∇dbl2 H1 ≤C(1+s2)/ba∇dblP(is)−1/ba∇dbl2 L(H−1;H)/ba∇dblf/ba∇dbl2 H1+C/ba∇dblf/ba∇dbl2 H1 ≤C(1+s2)/parenleftbig 1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig2/ba∇dblf/ba∇dbl2 H1, and finally /ba∇dblP(is)−1A/ba∇dblL(H1)≤C(1 +|s|)/parenleftbig 1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig . Coming back to (4.19), we have, for |s| ≥1, /ba∇dblP(is)−1(BB∗+isId)/ba∇dblL(H1)≤C/parenleftbig 1+|s|/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig . (4.20) Finally, combining (4.15), (4.18) and (4.20), together with (4.13)-(4 .14), we obtain for |s| ≥1, /ba∇dbl(isId−A)−1/ba∇dblL(H)≤C/parenleftbig 1+|s|/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig . This concludes the proof of Lemma 4.6. Part III Proof of Theorem 2.6: smooth damping coefficients on the torus To prove Theorem 2.6, we shall instead prove Estimate (2.9) with α=1 1+δ(which, according to Proposition2.4, is equivalent to the statement ofTheorem 2.6). Let us first recast (2.9) with α=1 1+δ in the semiclassical setting : taking h=s−1, we are left to prove that there exist C >1 andh0>0 such that for all h≤h0, for allu∈H2(T2), we have /ba∇dblu/ba∇dblL2(T2)≤Ch−δ/ba∇dblP(i/h)u/ba∇dblL2(T2) (4.21) 15We prove this inequality by contradiction, using the notion of semiclas sical measures. The idea of developing such a strategy for proving energy estimates, toge ther with the associate technology, originates from Lebeau [Leb96]. We assume that (4.21) is not satisfied, and will obtain a contradiction at the end of Section 11. Hence, for all n∈N, there exists 0 <hn≤1 nandun∈H2(T2) such that /ba∇dblun/ba∇dblL2(T2)>n hδn/ba∇dblP(i/hn)un/ba∇dblL2(T2). Settingvn=un//ba∇dblun/ba∇dblL2(T2), and Phn b=−h2 n∆−1+ihnb(x) =h2 nP(i/hn), we then have, as n→ ∞, hn→0+, /ba∇dblvn/ba∇dblL2(T2)= 1, h−2−δ n/ba∇dblPhn bvn/ba∇dblL2(T2)→0. Our goal is now to associate to the sequence ( un,hn) a semiclassicalmeasure on the cotangent bundle µonT∗T2=T2×(R2)∗(where (R2)∗is the dual space of R2). To obtain a contradiction, we shall prove both that µ(T∗T2) = 1, and that µ= 0 onT∗T2. From now on, we drop the subscript nof the sequences above, and write hin place ofhnandvh in place ofvn. We study sequences ( h,vh) such that h→0+and /braceleftBigg /ba∇dblvh/ba∇dblL2(T2)= 1 /ba∇dblPh bvh/ba∇dblL2(T2)=o(h2+δ),ash→0+.(4.22) In particular, this last equation also yields the key information (bvh,vh)L2(T2)=h−1Im(Ph bvh,vh)L2(T2)=o(h1+δ),ash→0+. In the following, it will be convenient to identify ( R2)∗andR2through the usual inner product. In particular, the cotangent bundle T∗T2=T2×(R2)∗will be identified with T2×R2. 5 Semiclassical measures We denote by T∗T2the compactification of T∗T2obtained by adding a point at infinity to each fiber (i.e., the set T2×(R2∪ {∞})). A neighbourhood of ( x,∞)∈T∗T2is a setU×/parenleftbig {∞}∪R2\K/parenrightbig , whereUis a neighbourhood of xinT2andKa compact set in R2. Endowed with this topology, the setT∗T2is compact. We denote by S0(T∗T2),S0for short, the space of functions a(x,ξ) that satisfy the following properties: 1.a∈C∞(T∗T2). 2. There exists a compact set K⊂R2and a constant k0∈Csuch thata(x,ξ) =k0for all ξ∈R2\K. Note that we have in particular C∞ c(T∗T2)⊂S0(T∗T2). To a symbol a∈S0(T∗T2), we associate its semiclassical Weyl quantization Oph(a) by For- mula (A.1), which, according to the Calder´ on-Vaillancourt Theorem (see Appendix A) defines a uniformly bounded operator on L2(T2). From the sequence ( vh,h) (see for instance [GL93]), we can define (using again the Calder´ on - Vaillancourt Theorem) the associated Wigner distribution Vh∈(S0)′by /angbracketleftbig Vh,a/angbracketrightbig (S0)′,S0= (Oph(a)vh,vh)L2(T2),for alla∈S0(T∗T2). (5.1) 16Decomposing vhandain Fourier series, ˆvh(k) =1 2π/integraldisplay T2e−ik·xvh(x)dx,ˆa(h,k,ξ) =1 2π/integraldisplay T2e−ik·xa(h,x,ξ)dx, the expression (5.1) can be more explicitly rewritten as /angbracketleftbig Vh,a/angbracketrightbig (S0)′,S0=1 2π/summationdisplay k,j∈Z2ˆa/parenleftbigg h,j−k,h 2(k+j)/parenrightbigg ˆvh(k)ˆvh(j). Proposition 5.1. The family (Vh)is bounded in (S0)′. Hence, there exists a subsequence of the sequence (h,vh)and an element µ∈(S0)′, such that Vh⇀µweakly in (S0)′, i.e. (Oph(a)vh,vh)L2(T2)→ /a\}b∇acketle{tµ,a/a\}b∇acket∇i}ht(S0)′,S0for alla∈S0(T∗T2). (5.2) In addition, /a\}b∇acketle{tµ,a/a\}b∇acket∇i}ht(S0)′,S0is nonnegative if ais; in other words, µmay be identified with a nonnegative Radon measure on T∗T2. Notation: in what follows we shall denote by M+(T∗T2) the set of nonnegativeRadon measures onT∗T2. Proof.The proof is an adaptation from the original proof of G´ erard [G´ er 91] (see also [GL93] in the semiclassical setting). The fact that the Wigner distributions Vhare uniformly bounded in ( S0)′follows from the Calder´ on-Vaillancourt theorem (see Appendix A), and from the bo undedness of ( vh) inL2(T2). The sharp G˚ arding inequality gives the existence of C >0 such that, for all a≥0 andh>0, (Oph(a)vh,vh)L2(T2)≥ −Ch/ba∇dblvh/ba∇dbl2 L2(T2), so that the distribution µis nonnegative (and is hence a measure). 6 Zero-th and first order informations on µ To simplify the notation, we set Ph b=Ph 0+ihb(x),withPh 0=−h2∆−1 = Oph(|ξ|2−1). The geodesic flow on the torus φτ:T∗T2→T∗T2forτ∈Ris the flow generated by the Hamiltonian vector field associated to the symbol1 2(|ξ|2−1), i.e. by the vector field ξ·∂xonT∗T2. Explicitely, we have φτ(x,ξ) = (x+τξ,ξ), τ∈R,(x,ξ)∈T∗T2. Notethatφτpreservesthe ξ-component,and,inparticulareveryenergylayer {|ξ|2=C >0} ⊂T∗T2. Now, we describe the first properties of the measure µimplied by (4.22). We recall that for ν∈D′(T∗T2), (φτ)∗ν∈D′(T∗T2) is defined by /a\}b∇acketle{t(φτ)∗ν,a/a\}b∇acket∇i}ht=/a\}b∇acketle{tν,a◦φτ/a\}b∇acket∇i}htfor alla∈C∞ c(T∗T2). In particular, ( φτ)∗νis a measure if νis. We shall say that νis aninvariant measure if it is invariant by the geodesic flow, i.e. ( φτ)∗ν=νfor allτ∈R. Proposition 6.1. Letµbe as in Proposition 5.1. We have 1.supp(µ)⊂ {|ξ|2= 1}(hence is compact in T∗T2), 2.µ(T∗T2) = 1, 3.µis invariant by the geodesic flow, i.e. (φτ)∗µ=µ, 4./a\}b∇acketle{tµ,b/a\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0, whereMc(T∗T2)denotes the space of compactly supported mea- sures onT∗T2. 17In other words, µis an invariant probability measure on T∗T2vanishing on {b>0}. These are standard arguments, that we reproduce here for the reader’s comfort. In particular, we recover all informations required to prove the Bardos-Lebeau -Rauch-Tayloruniform stabilization theorem under GCC. But we do not use here the second order infor mations of (4.22); this will be the key point to prove Theorem 2.6. Proof.First, we take χ∈C∞(T∗T2) depending only on the ξvariable, such that χ≥0,χ(ξ) = 0 for|ξ| ≤2, andχ(ξ) = 1 for |ξ| ≥3. Hence,χ(ξ) |ξ|2−1∈C∞(T∗T2) and we have the exact composition formula Oph(χ) = Oph/parenleftbiggχ(ξ) |ξ|2−1/parenrightbigg Ph 0, since both operators are Fourier multipliers. Moreover, Oph/parenleftBig χ(ξ) |ξ|2−1/parenrightBig is a bounded operator on L2(T2). As a consequence, we have /angbracketleftbig Vh,χ/angbracketrightbig (S0)′,S0→ /a\}b∇acketle{tµ,χ/a\}b∇acket∇i}htM(T∗T2),C0(T∗T2), together with /angbracketleftbig Vh,χ/angbracketrightbig (S0)′,S0=/parenleftbigg Oph/parenleftbiggχ(ξ) |ξ|2−1/parenrightbigg Ph 0vh,vh/parenrightbigg L2(T2) =/parenleftbigg Oph/parenleftbiggχ(ξ) |ξ|2−1/parenrightbigg Ph bvh,vh/parenrightbigg L2(T2)−ih/parenleftbigg Oph/parenleftbiggχ(ξ) |ξ|2−1/parenrightbigg bvh,vh/parenrightbigg L2(T2). Since/ba∇dblPh bvh/ba∇dblL2(T2)=o(1) and/ba∇dblvh/ba∇dblL2(T2)= 1, both terms in this expression vanish in the limit h→0+. This implies that /a\}b∇acketle{tµ,χ/a\}b∇acket∇i}htM(T∗T2),C0(T∗T2)= 0. Since this holds for all χas above, we have supp(µ)⊂ {|ξ|2= 1}, which proves Item 1. In particular, this implies that µ/parenleftBig T∗T2\T∗T2/parenrightBig = 0. Now, Item 2 is a direct consequence of 1 = /ba∇dblvh/ba∇dbl2 L2(T2)→ /a\}b∇acketle{tµ,1/a\}b∇acket∇i}htM(T∗T2),C0(T∗T2)and Item 1. Item 4 is a direct consequence of ( bvh,vh)L2(T2)= o(1). Finally, for a∈C∞ c(T∗T2), we recall that /bracketleftbig Ph 0,Oph(a)/bracketrightbig =h iOph({|ξ|2−1,a}) =2h iOph(ξ·∂xa), is a consequence of the Weyl quantization (any other quantization would have left an error term of orderO(h2)). Hence, (5.1) yields /angbracketleftbig Vh,ξ·∂xa/angbracketrightbig D′(T∗T2),C∞c(T∗T2)→ /a\}b∇acketle{tµ,ξ·∂xa/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2), (6.1) together with /angbracketleftbig Vh,ξ·∂xa/angbracketrightbig D′(T∗T2),C∞c(T∗T2)=i 2h/parenleftbig/bracketleftbig Ph 0,Oph(a)/bracketrightbig vh,vh/parenrightbig L2(T2) =i 2h/parenleftbig Oph(a)vh,Ph 0vh/parenrightbig L2(T2)−i 2h/parenleftbig Oph(a)Ph 0vh,vh/parenrightbig L2(T2) =i 2h/parenleftbig Oph(a)vh,Ph bvh/parenrightbig L2(T2)−i 2h/parenleftbig Oph(a)Ph bvh,vh/parenrightbig L2(T2) −1 2(Oph(a)vh,bvh)L2(T2)−1 2(Oph(a)bvh,vh)L2(T2).(6.2) In this expression, we have1 h/parenleftbig Oph(a)vh,Ph bvh/parenrightbig L2(T2)→0 and1 h/parenleftbig Oph(a)Ph bvh,vh/parenrightbig L2(T2)→0 since /ba∇dblPh bvh/ba∇dblL2(T2)=o(h). Moreover, the last two terms can be estimated by |(Oph(a)bvh,vh)L2(T2)| ≤ /ba∇dbl√ bvh/ba∇dblL2(T2)/ba∇dbl√ bOph(a)vh/ba∇dblL2(T2)=o(1), (6.3) 18since (bvh,vh)L2(T2)=o(1). This yields/angbracketleftbig Vh,ξ·∂xa/angbracketrightbig D′(T∗T2),C∞c(T∗T2)→0, so that, using (6.1), /a\}b∇acketle{tµ,ξ·∂xa/a\}b∇acket∇i}htM(T∗T2),C0 c(T∗T2)= 0 for all a∈C∞ c(T∗T2). Replacing abya◦φτand integrating with respect to the parameter τgives (φτ)∗µ=µ, which concludes the proof of Item 3. 7 Geometry on thetorus and decomposition of invariant mea- sures 7.1 Resonant and non-resonant vectors on the torus In this section, we collect several facts concerning the geometry ofT∗T2and its resonant subspaces. Most of the setting and the notation comes from [AM11, Section 2]. We shall say that a submodule Λ ⊂Z2is primitive if /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht∩Z2= Λ, where /a\}b∇acketle{tΛ/a\}b∇acket∇i}htdenotes the linear subspace of R2spanned by Λ. The family of all primitive submodules will be denoted by P. Let us denote by Ω j⊂R2, forj= 0,1,2, the set of resonant vectors of order exactly j, i.e., Ωj:={ξ∈R2such that rk(Λ ξ) = 2−j},with Λ ξ:=/braceleftbig k∈Z2such thatξ·k= 0/bracerightbig =ξ⊥∩Z2. Note that the sets Ω jform a partition of R2, and that we have •Ω0={0}; •ξ∈Ω1if and only if the geodesic issued from any x∈T2in the direction ξis periodic; •ξ∈Ω2if and only if the geodesic issued from any x∈T2in the direction ξis dense in T2. For each Λ ∈ Psuch that rk(Λ) = 1, we define Λ⊥:=/braceleftbig ξ∈R2such thatξ·k= 0 for allk∈Λ/bracerightbig , TΛ:=/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht/2πΛ, TΛ⊥:= Λ⊥/(2πZ2∩Λ⊥). Note that TΛandTΛ⊥are two submanifolds of T2diffeomorphic to one-dimensional tori. Their cotangent bundles admit the global trivialisations T∗TΛ=TΛ×/a\}b∇acketle{tΛ/a\}b∇acket∇i}htandT∗TΛ⊥=TΛ⊥×Λ⊥. For a function fonT2with Fourier coefficients ( ˆf(k))k∈Z2, and Λ∈ P, we shall say that fhas only Fourier modes in Λ if ˆf(k) = 0 fork /∈Λ. This means that fis constant in the direction Λ⊥, or, equivalently, that σ·∂xf= 0 for all σ∈Λ⊥. We denote by Lp Λ(T2) the subspace of Lp(T2) consisting of functions having only Fourier modes in Λ. For a function f∈L2(T2) (resp. a symbol a∈S0(T∗T2)), we denote by /a\}b∇acketle{tf/a\}b∇acket∇i}htΛits orthogonal projection on L2 Λ(T2), i.e. the average of falong Λ⊥: /a\}b∇acketle{tf/a\}b∇acket∇i}htΛ(x) :=/summationdisplay k∈Λeik·x 2πˆf(k)/parenleftBigg resp./a\}b∇acketle{ta/a\}b∇acket∇i}htΛ(x,ξ) :=/summationdisplay k∈Λeik·x 2πˆa(k,ξ)/parenrightBigg . If rk(Λ) = 1 and vis a vector in Λ⊥\{0}, we also have /a\}b∇acketle{tf/a\}b∇acket∇i}htΛ(x) = lim T→∞1 T/integraldisplayT 0f(x+tv)dt. (7.1) In particular, note that /a\}b∇acketle{tf/a\}b∇acket∇i}htΛ(resp./a\}b∇acketle{ta/a\}b∇acket∇i}htΛ) is nonnegative if f(resp.a) is, and that /a\}b∇acketle{tf/a\}b∇acket∇i}htΛ∈C∞(T2) (resp./a\}b∇acketle{ta/a\}b∇acket∇i}htΛ∈S0(T∗T2)) iff∈C∞(T2) (resp.a∈S0(T∗T2)). Finally, given f∈L∞ Λ(T2), we denote by mfthe bounded operator on L2 Λ(T2), consisting in the multiplication by f. 197.2 Decomposition of invariant measures We denote by M+(T∗T2) the set of finite, nonnegative measures on T∗T2. With the definitions above, we have the following decomposition Lemmata, proved in [Mac1 0] or [AM11, Section 2]. These properties are given for general measures µ∈ M+(T∗T2). Of course, they apply in particular to the measure µdefined by Proposition 5.1. Lemma 7.1. Letµ∈ M+(T∗T2). Thenµdecomposes as a sum of nonnegative measures µ=µ|T2×{0}+µ|T2×Ω2+/summationdisplay Λ∈P,rk(Λ)=1µ|T2×(Λ⊥\{0}) (7.2) Givenµ∈ M+(T∗T2), we define its Fourier coefficients by the complex measures on R2: ˆµ(k,·) :=/integraldisplay T2e−ik·x 2πµ(dx,·), k∈Z. One has, in the sense of distributions, the following Fourier inversion formula: µ(x,ξ) =/summationdisplay k∈Z2eik·x 2πˆµ(k,ξ). Lemma 7.2. Letµ∈ M+(T∗T2)andΛ∈ P. Then, the distribution /a\}b∇acketle{tµ/a\}b∇acket∇i}htΛ(x,ξ) :=/summationdisplay k∈Λeik·x 2πˆµ(k,ξ), is inM+(T∗T2)and satisfies, for all a∈C∞ c(T∗T2), /a\}b∇acketle{t/a\}b∇acketle{tµ/a\}b∇acket∇i}htΛ,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)=/a\}b∇acketle{tµ,/a\}b∇acketle{ta/a\}b∇acket∇i}htΛ/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2). Lemma 7.3. Letµ∈ M+(T∗T2)be aninvariant measure. Then, for all Λ∈ P,µ|T2×(Λ⊥\{0})is also a nonnegative invariant measure and µ|T2×(Λ⊥\{0})=/a\}b∇acketle{tµ/a\}b∇acket∇i}htΛ|T2×(Λ⊥\{0}). Let us now come back to the measure µgiven by Proposition 5.1, which satisfies all properties listed in Proposition 6.1. In particular, this measure vanishes on the n on-empty open subset of T2 given by {b >0}(see Item 4 in Proposition 6.1). As a consequence of Proposition 6.1, and of the three lemmata above, this yields the following lemma. Lemma 7.4. We haveµ=/summationtext Λ∈P,rk(Λ)=1µ|T2×(Λ⊥\{0}). As a consequence of Proposition 6.1, we have indeed that the measu reµis supported in {|ξ|= 1}, which implies µ|T2×{0}= 0. In addition, Lemma 7.3 applied with Λ = {0}implies that µ|T2×Ω2is constant in x– and thus vanishes everywhere since it vanishes on {b>0}. Remark 7.5. Since the measure µis supported in {|ξ|= 1}(Proposition 6.1, Item 1), we have µ|T2×Λ⊥=µ|T2×(Λ⊥\{0}) (which simplifies the notation). As a consequence of these lemmata and the last remark, the study of the measure µis now reduced to that of all nonnegative invariant measures µ|T2×Λ⊥with rk(Λ) = 1. The aim of the next sections is to prove that the measure µ|T2×Λ⊥vanishes identically, for each periodic direction Λ⊥. 207.3 Geometry of the subtori TΛandTΛ⊥ To study the measure µ|T2×(Λ⊥\{0}), we need to describe briefly the geometry of the subtori TΛand TΛ⊥ofT2, and introduce adapted coordinates. We defineχΛthe linear isomorphism χΛ: Λ⊥×/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht →R2: (s,y)/ma√sto→s+y, and denote by ˜ χΛ:T∗Λ⊥×T∗/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht →T∗R2its extension to the cotangent bundle. This application can be defined as follows: for ( s,σ)∈T∗Λ⊥= Λ⊥×(Λ⊥)∗and (y,η)∈T∗/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht=/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht×/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht∗, we can extendσto a covector of R2vanishing on /a\}b∇acketle{tΛ/a\}b∇acket∇i}htandηto a covector of R2vanishing on Λ⊥. Remember that we identify ( R2)∗withR2through the usual inner product; thus we can also see σas an element of Λ⊥andηas an element of /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht. Then, we have ˜χΛ(s,σ,y,η) = (s+y,σ+η)∈T∗R2=R2×(R2)∗. Conversely, any ξ∈(R2)∗can be decomposed into ξ=σ+ηwhereσ∈Λ⊥andη∈ /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht. We denote byPΛthe orthogonal projection of R2onto/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht, i.e.PΛξ=η. Next, the map χΛgoes to the quotient, giving a smooth Riemannian covering of T2by πΛ:TΛ⊥×TΛ→T2: (s,y)/ma√sto→s+y. We shall denote by ˜ πΛits extension to cotangent bundles: ˜πΛ:T∗TΛ⊥×T∗TΛ→T∗T2. As the map πΛis not an injection (because the torus TΛ⊥×TΛcontains several copies of T2), we introduce its degree pΛ, which is also equal toVol(TΛ⊥×TΛ) Vol(T2). Then, the application TΛu:=1√pΛu◦χΛ, defines a linear isomorphism L2 loc(R2)→L2 loc(Λ⊥× /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht). Note that because of the factor1√pΛ, TΛmapsL2(T2) isometrically into a subspace of L2(TΛ⊥×TΛ). Moreover, TΛmapsL2 Λ(T2) into L2(TΛ)⊂L2(TΛ⊥×TΛ), since the nonvanishing Fourier modes of u∈L2 Λ(T2) correspond only to frequencies k∈Λ. This reads TΛu(s,y) =1√pΛu(y) for (s,y)∈TΛ⊥×TΛ. (7.3) Since ˜χΛis linear, we have, for any a∈C∞(T∗R2) TΛOph(a) = Oph(a◦˜χΛ)TΛ, (7.4) where on the left Ophis the Weyl quantization on R2(A.1), and on the right Ophis the Weyl quantization on Λ⊥× /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht. Next, we denote by OpΛ⊥ hand OpΛ hthe Weyl quantization operators defined on smooth test functions on T∗Λ⊥×T∗/a\}b∇acketle{tΛ/a\}b∇acket∇i}htand acting only on the variables in T∗Λ⊥and T∗/a\}b∇acketle{tΛ/a\}b∇acket∇i}htrespectively, leaving the other frozen. For any a∈C∞ c(T∗Λ⊥×T∗/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht), we have : Oph(a) = OpΛ⊥ h◦OpΛ h(a) = OpΛ h◦OpΛ⊥ h(a). (7.5) Now, if the symbol a∈C∞ c(T∗T2) has only Fourier modes in Λ, we remark, in view of (7.3), that a◦˜πΛdoes not depend on s∈TΛ⊥. Therefore, we sometimes write a◦˜πΛ(σ,y,η) fora◦˜πΛ(s,σ,y,η) and (7.4)-(7.5) give TΛOph(a) = OpΛ h◦OpΛ⊥ h(a◦˜πΛ)TΛ= OpΛ h(a◦˜πΛ(hDs,·,·))TΛ. (7.6) Note that for every σ∈Λ⊥, the operator OpΛ h(a◦˜πΛ(σ,·,·)) mapsL2(TΛ) into itself. More precisely, it maps the subspace TΛ(L2 Λ(T2)) into itself. 218 Change of quasimode and construction of an invariant cut- off function In this section, we first construct from the quasimode vhanother quasimode wh, that will be easier to handle when studying the measure µ|T2×Λ⊥. Indeedwhis basically a microlocalization of vhin the direction Λ⊥at a precise concentration rate. Moreover, we introduce a cutoff function χΛ h(x) =χΛ h(y,s), well-adapted to the damping coeffi- cientband to the invariance of the measure µ|T2×Λ⊥in the direction Λ⊥(this cutoff function plays the role of the function χ(b/h) used in [BH07] in the case where bis itself invariant in the direction Λ⊥). Its construction is a key point in the proof of Theorem 2.6. Letχ∈C∞ c(R) be a nonnegative function such that χ= 1 in a neighbourhood of the origin. We first define wh:= Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg vh, which implicitely depends on α∈(0,1). The following lemma implies that, for δandαsufficiently small,whis as well a o(h2+δ)-quasimode for Ph b. Lemma 8.1. For anyα>0such that δ+ε 2+α≤1 2,3α+2δ<1, (8.1) we have /ba∇dblPh bwh/ba∇dblL2(T2)=o(h2+δ). As a consequence of this lemma, the semiclassical measures associa ted towhsatisfy in particular the conclusions of Proposition 6.1. Moreover, the following proposit ion implies that the sequence wh contains all the information in the direction Λ⊥. Proposition 8.2. For anya∈C∞ c(T∗T2)and anyα∈(0,3/4)satisfying the assumptions of Lemma 8.1, we have /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim h→0(Oph(a)wh,wh)L2(T2). Next, we state the desired properties of the cutoff function χΛ h. The proof of its existence is a crucial point in the proof of Theorem 2.6. Proposition 8.3. Forδ= 8ε, andε<1 76, there exists αsatisfying (8.1), such that for any constant c0>0, there exists a cutoff function χΛ h∈C∞(T2)valued in [0,1], such that 1.χΛ h=χΛ h(y)does not depend on the variable s(i.e.χΛ hisΛ⊥-invariant), 2./ba∇dbl(1−χΛ h)wh/ba∇dblL2(T2)=o(1), 3.b≤c0honsupp(χΛ h), 4./ba∇dbl∂yχΛ hwh/ba∇dblL2(T2)=o(1), 5./ba∇dbl∂2 yχΛ hwh/ba∇dblL2(T2)=o(1). Note that the function χΛ himplicitely depends on the constant c0, that will be taken arbitrarily small in Section 10. In the particular case where the damping function bis invariant in one direction, this proposition is not needed. In this case, one can take as in [BH07] χΛ h=χ(b c0h). In thed-dimensional torus, this cutoff functions works as well if bis invariant in d−1 directions, and an analogue of Theorem 2.6 can be stated in this setting. Unfortunately, our construction of the function χΛ h(see the proof of Proposition 8.3 in Section 13) strongly relies on the fact that all trap ped directions are periodic, and fails in higher dimensions. We give here a proof of Lemma 8.1. Because of their technicality, we p ostpone the proofs of Propositions 8.2 and 8.3 to Sections 12 and 13 respectively. 22Proof of Lemma 8.1. First, we develop Ph bwh=Ph bOph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg vh= Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg Ph bvh+ih/bracketleftbigg b,Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg/bracketrightbigg vh, (8.2) sincePh 0and Oph/parenleftBig χ/parenleftBig |PΛξ| hα/parenrightBig/parenrightBig are both Fourier mutipliers. We know that /vextenddouble/vextenddouble/vextenddouble/vextenddoubleOph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg Ph bvh/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(T2)≤ /ba∇dblPh bvh/ba∇dblL2(T2)=o(h2+δ). It only remains to study the operator /bracketleftbigg b,Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg/bracketrightbigg =ih1−αOph/parenleftbigg ∂yb χ′/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg +OL(L2)(h2(1−α)) (8.3) according to the symbolic calculus. Moreover, using Assumption (2.1 2), we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yb χ′/parenleftbigg|PΛξ| hα/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cb1−ε. The sharp G˚ arding inequality applied to the nonnegative symbol C2b2(1−ε)−/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yb χ′/parenleftbigg|PΛξ| hα/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 , then yields/parenleftBigg Oph/parenleftBigg C2b2(1−ε)−/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yb χ′/parenleftbigg|PΛξ| hα/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg vh,vh/parenrightBigg L2(T2)≥ −Ch1−α, and hence /vextenddouble/vextenddouble/vextenddouble/vextenddoubleOph/parenleftbigg ∂yb χ′/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg vh/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(T2)≤C2(b2(1−ε)vh,vh)L2(T2)+O(h1−α). (8.4) When using the inequality/integraltext f1−εdν≤/parenleftbig/integraltext fdν/parenrightbig1−εfor nonnegative functions (with dν=|vh(x)|2dx), we obtain (b2(1−ε)vh,vh)L2(T2)≤(b2vh,vh)(1−ε) L2(T2)≤C/ba∇dblbvh/ba∇dbl2(1−ε) L2(T2)=o(h1−ε). Combining this estimate together with (8.3) and (8.4) gives /vextenddouble/vextenddouble/vextenddouble/vextenddoubleih/bracketleftbigg b,Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg/bracketrightbigg vh/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(T2)=o(h5 2−α−ε 2)+O(h5−3α 2). Coming back to the expression of Ph bwhgiven in (8.2), this concludes the proof of Lemma 8.1. 9 Second microlocalization on a resonant affine subspace We want to analyse precisely the structure of the restriction µ|T2×(Λ⊥\{0}), using the full information contained in o(h2+δ)-quasimodes like vhandwh. From now on, we want to take advantage of the family whofo(h2+δ)-quasimodes constructed in Section 8, which are microlocalised in the direction Λ⊥. Hence, we define the Wigner distribution Wh∈D′(T∗T2) associated to the functions whand the scale h, by /angbracketleftbig Wh,a/angbracketrightbig (S0)′,S0= (Oph(a)wh,wh)L2(T2)for alla∈S0(T∗T2). 23According to Proposition 8.2, we recover in the limit h→0, /angbracketleftbig Wh,a/angbracketrightbig (S0)′,S0→ /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0 c(T∗T2), for anya∈C∞ c(T∗T2) (andαsatisfying (8.1)). To provide a precise study of µ|T2×Λ⊥, we shall introduce as in [Mac10, AM11] two-microlocal semiclassicalmeasures,describingat afinerlevel the concentrat ionofthe sequence vhonthe resonant subspace Λ⊥={ξ∈R2such thatPΛξ= 0}. These objects have been introduced in the local Euclidean case by N ier [Nie96] and Fermanian- Kammerer [FK00b, FK00a]. A specific concentration scale may also be chosen in the in the two- microlocal variable, giving rise to the two-scales semiclassical measu res studied by Miller [Mil96, Mil97] and Fermanian-Kammerer and G´ erard [FKG02]. We first have to describe the adapted symbol class (inspired by [FK0 0a] and used in [AM11]). According to Lemma 7.3 (see also Remark 7.5), it suffices to test the m easureµ|T2×Λ⊥with functions constant in the direction Λ⊥(or equivalently, having only x-Fourier modes in Λ, in the sense of the following definition). Definition 9.1. Given Λ ∈ P, we shall say that a∈S1 Λifa=a(x,ξ,η)∈C∞(T∗T2×/a\}b∇acketle{tΛ/a\}b∇acket∇i}ht) and 1. there exists a compact set Ka⊂T∗T2such that, for all η∈ /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht, the function ( x,ξ)/ma√sto→a(x,ξ,η) is compactly supported in Ka; 2.ais homogeneous of order zero at infinity in the variable η∈ /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht; i.e., if we denote by SΛ:= S1∩/a\}b∇acketle{tΛ/a\}b∇acket∇i}httheunit spherein /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht, thereexists R0>0(depending on a)andahom∈C∞ c(T∗T2×SΛ) such that a(x,ξ,η) =ahom/parenleftbigg x,ξ,η |η|/parenrightbigg ,for|η| ≥R0and (x,ξ)∈T∗T2; forη/\e}atio\slash= 0, we will also use the notation a(x,ξ,∞η) :=ahom/parenleftBig x,ξ,η |η|/parenrightBig . 3.ahas onlyx-Fourier modes in Λ, i.e. a(x,ξ,η) =/summationdisplay k∈Λeik·x 2πˆa(k,ξ,η). Note that this last assumption is equivalent to saying that σ·∂xa= 0 for any σ∈Λ⊥. We denote byS1 Λ′the topological dual space of S1 Λ. Letχ∈C∞ c(R) be a nonnegative function such that χ= 1 in a neighbourhood of the origin. Let R>0. The previous remark allows us to define, for a∈S1 Λthe two following elements of S1 Λ′: /angbracketleftBig Wh,Λ R,a/angbracketrightBig S1 Λ′,S1 Λ:=/angbracketleftbigg Wh,/parenleftbigg 1−χ/parenleftbigg|PΛξ| Rh/parenrightbigg/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/angbracketrightbigg D′(T∗T2),C∞ c(T∗T2),(9.1) /angbracketleftbig Wh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ:=/angbracketleftbigg Wh,χ/parenleftbigg|PΛξ| Rh/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/angbracketrightbigg D′(T∗T2),C∞c(T∗T2). (9.2) In particular, for any R>0 anda∈S1 Λ, we have /angbracketleftbigg Wh,a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/angbracketrightbigg D′(T∗T2),C∞c(T∗T2)=/angbracketleftBig Wh,Λ R,a/angbracketrightBig S1 Λ′,S1 Λ+/angbracketleftbig Wh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ.(9.3) The following two propositions are the analogues of [FK00a] in our con text. They state the existence of the two-microlocal semiclassical measures, as the limit objects ofWh,Λ RandWh R,Λ. 24Proposition 9.2. There exists a subsequence (h,wh)and a nonnegative measure νΛ∈ M+(T∗T2× SΛ)such that, for all a∈S1 Λ, we have lim R→∞lim h→0/angbracketleftBig Wh,Λ R,a/angbracketrightBig S1 Λ′,S1 Λ=/angbracketleftbigg νΛ,ahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg M(T∗T2×SΛ),C0c(T∗T2×SΛ). To define the limit of the distributions Wh R,Λ, we need first to introduce operator spaces and operator-valued measures, following [G´ er91]. Given a Hilbert space H(in the following, we shall useH=L2(TΛ)), we denote respectively by K(H),L1(H) the spaces of compact and trace class operators on H. We recall that they are both two-sided ideals of the ring L(H) of bounded operators onH. We refer for instance to [RS80, Chapter VI.6] for a description of the space L1(H) and its basic properties. Given a Polish space T(in the following, we shall use T=T∗TΛ⊥), we denote by M+(T;L1(H)) the space of nonnegative measures on T, taking values in L1(H). More precisely, we haveρ∈ M+(T;L1(H)) ifρis a bounded linear form on C0 c(T) such that, for every nonnegative functiona∈C0 c(T),/a\}b∇acketle{tρ,a/a\}b∇acket∇i}htM(T),C0 c(T)∈ L1(H) isanonnegativehermitian operator. As aconsequence of [RS80, Theorem VI.26], these measures can be identified in a natur al way to nonnegative linear functionals on C0 c(T;K(H)). Proposition 9.3. There exists a subsequence (h,wh)and a nonnegative measure ρΛ∈ M+(T∗TΛ⊥;L1(L2(TΛ))), such that, for all K∈C∞ c(T∗TΛ⊥;K(L2(TΛ))), lim h→0(K(s,hDs)TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ))= tr/braceleftBigg/integraldisplay T∗TΛ⊥K(s,σ)ρΛ(ds,dσ)/bracerightBigg .(9.4) Moreover (for the same subsequence), for all a∈S1 Λ, we have lim R→∞lim h→0/angbracketleftbig Wh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ= tr/braceleftBigg/integraldisplay T∗TΛ⊥OpΛ 1/parenleftbig a(˜πΛ(σ,y,0),η)/parenrightbig ρΛ(ds,dσ)/bracerightBigg .(9.5) In the left hand-side of (9.4), the inner product actually means (K(s,hDs)TΛwh,TΛwh)L2(TΛ⊥L2(TΛ)) =/integraldisplay s∈TΛ⊥,s′∈Λ⊥,σ∈Λ⊥ei h(s−s′)·σ/parenleftbigg K/parenleftbigs+s′ 2,σ/parenrightbig TΛwh(s′,y),TΛwh(s,y)/parenrightbigg L2y(TΛ)ds ds′dσ. In the expression (9.5), remark that for each σ∈Λ⊥, the operator OpΛ 1/parenleftbig a(˜πΛ(σ,y,0),η) is in L(L2(TΛ)). Hence, its product with the operator ρΛ(ds,dσ) defines a trace-class operator. Before proving Propositions 9.2 and 9.3, we explain how to reconstru ct the measure µ|T2×Λ⊥ from the two-microlocal measures νΛandρΛ. This reduces the study of the measure µto that of all two-microlocal measures νΛandρΛ, for Λ∈ P. We denote by M+ c(T) the set of compactly supported measures on T, and by /a\}b∇acketle{t·,·/a\}b∇acket∇i}htMc(T),C0(T) the associated duality bracket. Proposition 9.4. For alla∈C∞ c(T∗T2)having only x-Fourier modes in Λ(i.e. for all a∈S1 Λ independent of the third variable η∈ /a\}b∇acketle{tΛ/a\}b∇acket∇i}ht), we have /a\}b∇acketle{tµ,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)=/angbracketleftbig νΛ,a/angbracketrightbig M(T∗T2×SΛ),C0c(T∗T2×SΛ)+tr/braceleftBigg/integraldisplay T∗TΛ⊥ma◦˜πΛ(σ)ρΛ(ds,dσ)/bracerightBigg ,(9.6) 25and /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0 c(T∗T2)=/angbracketleftbig νΛ|T2×Λ⊥×SΛ,a/angbracketrightbig M(T∗T2×SΛ),C0c(T∗T2×SΛ) +tr/braceleftBigg/integraldisplay T∗TΛ⊥ma◦˜πΛ(σ)ρΛ(ds,dσ)/bracerightBigg , (9.7) where forσ∈Λ⊥,ma◦˜πΛ(σ)denotes the multiplication in L2(TΛ)by the function y/ma√sto→a◦˜πΛ(σ,y). Moreover, we have νΛ∈ M+ c(T∗T2×SΛ)andρΛ∈ M+ c(T∗TΛ⊥;L2(TΛ))(i.e. both measures are compactly supported). Formula (9.7) follows immediately from (9.6) by restriction. By the defi nition of the measure ρΛ, we see that it is already supported on T2×Λ⊥(see expression (9.2)). The end of this section is devoted to the proofs of the three propo sitions, inspired by [FK00a, AM11]. Proof of Proposition 9.2. The Calder´ on-Vaillancourt theorem implies that the operators Oph/parenleftbigg/parenleftbigg 1−χ/parenleftbigg|PΛξ| Rh/parenrightbigg/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/parenrightbigg = Op1/parenleftbigg/parenleftbigg 1−χ/parenleftbigg|PΛξ| R/parenrightbigg/parenrightbigg a(x,hξ,P Λξ)/parenrightbigg are uniformly bounded as h→0 andR→+∞. It follows that the family Wh,Λ Ris bounded in S1 Λ′, and thus there exists a subsequence ( h,wh) and a distribution ˜ µΛsuch that lim R→∞lim h→0/angbracketleftBig Wh,Λ R,a/angbracketrightBig S1 Λ′,S1 Λ=/angbracketleftbig ˜µΛ,a(x,ξ,η)/angbracketrightbig S1 Λ′,S1 Λ. Because of the support properties of the function χ, we notice that/angbracketleftbig ˜µΛ,a/angbracketrightbig S1 Λ′,S1 Λ= 0 as soon as the support of ais compact in the variable η. Hence, there exists a distribution νΛ∈D′(T∗T2×SΛ) such that/angbracketleftbig ˜µΛ,a(x,ξ,η)/angbracketrightbig S1 Λ′,S1 Λ=/angbracketleftbigg νΛ,ahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg D′(T∗T2×SΛ),C∞c(T∗T2×SΛ). Next, suppose that a >0 (and that√1−χis smooth). Then, using [AM11, Corollary 35], and setting bR(x,ξ) =/parenleftbigg/parenleftbigg 1−χ/parenleftbigg|PΛξ| Rh/parenrightbigg/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/parenrightbigg1 2 , there exists C >0 such that for all h≤h0andR≥1, we have /vextenddouble/vextenddouble/vextenddouble/vextenddoubleOph/parenleftbigg/parenleftbigg 1−χ/parenleftbigg|PΛξ| Rh/parenrightbigg/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/parenrightbigg −Oph(bR)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble L(L2(T2))≤C R. As a consequence, we have, /angbracketleftBig Wh,Λ R,a/angbracketrightBig S1 Λ′,S1 Λ≥ /ba∇dblOph(bR)wh/ba∇dbl2 L2(T2)−C R/ba∇dblwh/ba∇dbl2 L2(T2), so that the limit/angbracketleftBig νΛ,ahom/parenleftBig x,ξ,η |η|/parenrightBig/angbracketrightBig D′(T∗T2×SΛ),C∞c(T∗T2×SΛ)is nonnegative. The distribution νΛ is nonnegative, and is hence a measure. This concludes the proof of Proposition 9.2. Proof of Proposition 9.3. First, the proof of the existence of a subsequence ( h,wh) and the measure ρΛsatisfying (9.4) is the analogue of Proposition 5.1 in the context of op erator valued measures, viewing the sequence whas a bounded sequence of L2(TΛ⊥;L2(TΛ)). It follows the lines of this result, after the adaptation of the symbolic calculus to operator v alued symbols (or more precisely, of [G´ er91] in the semiclassical setting). 26Second, using the definition (9.2) together with (7.6), we have /angbracketleftbig Wh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ=/parenleftbigg Oph/parenleftbigg χ/parenleftbigg|PΛξ| Rh/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/parenrightbigg wh,wh/parenrightbigg L2(T2) =/parenleftbigg OpΛ⊥ h◦OpΛ h/parenleftbigg χ/parenleftbigg|η| Rh/parenrightbigg a/parenleftBig ˜πΛ(σ,y,η),η h/parenrightBig/parenrightbigg TΛwh,TΛwh/parenrightbigg L2(TΛ⊥×TΛ). Hence, setting ah R,Λ(σ,y,η) =χ/parenleftbigg|η| R/parenrightbigg a(˜πΛ(σ,y,hη),η), we obtain /angbracketleftbig Wh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ=/parenleftBig OpΛ⊥ h◦OpΛ 1/parenleftbig ah R,Λ(σ,y,η)/parenrightbig TΛwh,TΛwh/parenrightBig L2(TΛ⊥×TΛ). We also notice that OpΛ 1/parenleftbig ah R,Λ/parenrightbig ∈ K(L2(TΛ)), for anyσ∈Λ⊥sinceah R,Λhas compact support with respect toη. Moreover, for any R>0 fixed and a∈S1 Λ, the Calder´ on-Vaillancourt theorem yields OpΛ 1/parenleftbig ah R,Λ/parenrightbig = OpΛ 1/parenleftbig a0 R,Λ/parenrightbig +hB for someB∈ L(L2(TΛ)), uniformly bounded with respect to h. Using (9.4), this implies that for anyR>0 fixed and a∈S1 Λ, we have lim h→0/angbracketleftbig Wh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ= tr/braceleftBigg/integraldisplay T∗TΛ⊥OpΛ 1/parenleftbig a0 R,Λ/parenrightbig ρΛ(ds,dσ)/bracerightBigg . Moreover, we have lim R→+∞OpΛ 1/parenleftbig a0 R,Λ/parenrightbig = OpΛ 1/parenleftbig a0 ∞,Λ/parenrightbig = OpΛ 1/parenleftbig a(˜πΛ(σ,y,0),η)/parenrightbig , in the strong topology of C∞ c(T∗TΛ⊥;L(L2(TΛ))). This proves (9.5) and concludes the proof of Proposition 9.3. Proof of Proposition 9.4. Takinga∈S1 Λ, independent of the third variable η∈ /a\}b∇acketle{tΛ/a\}b∇acket∇i}htgives /angbracketleftbig Wh,a(x,ξ)/angbracketrightbig D′(T∗T2),C∞c(T∗T2)→ /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2), together with/angbracketleftBig Wh,Λ R,a/angbracketrightBig S1 Λ′,S1 Λ→/angbracketleftbig νΛ,a/angbracketrightbig M(T∗T2×SΛ),C0c(T∗T2×SΛ), (according to Proposition 9.2) and /angbracketleftbig Wh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ→tr/braceleftBigg/integraldisplay T∗TΛ⊥OpΛ 1/parenleftbig a(˜πΛ(σ,y,0))/parenrightbig ρΛ(ds,dσ)/bracerightBigg = tr/braceleftBigg/integraldisplay T∗TΛ⊥ma◦˜πΛ(σ)ρΛ(ds,dσ)/bracerightBigg , (according to Proposition 9.3). Now, using the last three equations together with Equation (9.3) directly gives (9.7). As both terms in the right hand-side of (9.7) are nonnegative measu res and the left-hand side is a compactly supported nonnegative measure, this implies that νΛandρΛare both compactly supported. 2710 Propagation laws for the two-microlocal measures νΛand ρΛ In this section, we study the propagation properties of νΛandρΛ. The key point here is the use of the cutoff function introduced in Proposition 8.3. We will use repeatedly the following fact, which follows from Item 2 in Pr oposition 8.3: if Ais a bounded operator on L2(T2), we have (Awh,wh)L2(T2)=/parenleftbig AχΛ hwh,χΛ hwh/parenrightbig L2(T2)+/ba∇dblA/ba∇dblL(L2)o(1). (10.1) To simplify the notation, we shall write Ac0,hforχΛ hAχΛ h. 10.1 Propagation of νΛ We define for ( x,ξ,η)∈T∗T2×/a\}b∇acketle{tΛ/a\}b∇acket∇i}htandτ∈Rthe flows φ0 τ(x,ξ,η) := (x+τξ,ξ,η), generated by the vector field ξ·∂xand, forη/\e}atio\slash= 0, φ1 τ(x,ξ,η) :=/parenleftbigg x+τη |η|,ξ,η/parenrightbigg , generated by the vector fieldη |η|·∂x. With these definitions, we have the following propagation laws for the two-microlocal measure νΛ. Proposition 10.1. The measure νΛisφ0 τ- andφ1 τ-invariant, i.e. (φ0 τ)∗νΛ=νΛand(φ1 τ)∗νΛ=νΛ,for everyτ∈R. The key result here is the additional “transverse propagation law” given by the flow φ1 τ. The measureνΛnot only propagates along the geodesic flow φ0 τ, but also along directions transverse to Λ⊥. Proof.Fixa∈S1 Λ. Thecomputationdonein(6.2)isstillvalidreplacing aby/parenleftBig 1−χ/parenleftBig |PΛξ| Rh/parenrightBig/parenrightBig a/parenleftBig x,ξ,PΛξ h/parenrightBig , sinceitonlyusesthefactthatOph/parenleftBig/parenleftBig 1−χ/parenleftBig |PΛξ| Rh/parenrightBig/parenrightBig a/parenleftBig x,ξ,PΛξ h/parenrightBig/parenrightBig isboundedandthat /ba∇dblPh bwh/ba∇dblL2(T2)= o(h) and (bwh,wh)L2(T2)=o(1). This yields lim h→0/angbracketleftBig Wh,Λ R,ξ·∂xa/angbracketrightBig S1 Λ′,S1 Λ = lim h→0/angbracketleftbigg Wh,ξ·∂x/braceleftbigg/parenleftbigg 1−χ/parenleftbigg|PΛξ| Rh/parenrightbigg/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/bracerightbigg/angbracketrightbigg D′(T∗T2),C∞ c(T∗T2)= 0, and hence, in the limit R→+∞, we obtain /angbracketleftbigg νΛ,ξ·∂xahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg M(T∗T2×SΛ),C0c(T∗T2×SΛ)= 0. Replacingahombyahom◦φ0 τand integrating with respect to the parameter τgives (φ0 τ)∗νΛ=νΛ, which concludes the first part of the proof. Second, to prove the φ1 τ-invariance of νΛwe compute /angbracketleftbigg νΛ,η |η|·∂xahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg M(T∗T2×SΛ),C0c(T∗T2×SΛ)= lim R→∞lim h→0/angbracketleftbigg Wh,Λ R,η |η|·∂xa/angbracketrightbigg S1 Λ′,S1 Λ.(10.2) 28Setting aR(x,ξ,η) =1 |η|/parenleftbigg 1−χ/parenleftbigg|η| R/parenrightbigg/parenrightbigg a(x,ξ,η), and AR:= Oph/parenleftbigg aR/parenleftBig x,ξ,PΛξ h/parenrightBig/parenrightbigg (10.3) we have the relation /angbracketleftbigg Wh,Λ R,η |η|·∂xa/angbracketrightbigg S1 Λ′,S1 Λ=−i 2([∆Λ,AR]wh,wh)L2(T2) where ∆ Λ=∂2 yis the laplacian in the direction Λ. Lemma 10.2. For any given c0>0andR>0, we have ([∆Λ,AR]wh,wh)L2(T2)= ([∆Λ,AR c0,h]wh,wh)L2(T2)+o(1). We postpone the proof of Lemma 10.2 and first indicate how it allows to prove Proposition 10.1. We now know that /angbracketleftbigg νΛ,η |η|·∂xahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg M(T∗T2×SΛ),C0 c(T∗T2×SΛ)= lim R→∞lim h→0−i 2([∆Λ,AR c0,h]wh,wh)L2(T2). Recall that a∈S1 Λimplies that ahas onlyx-Fourier modes in Λ, i.e. PΛξ·∂xa=ξ·∂xa. We have also assumed in this section that bhas onlyx-Fourier modes in Λ. As a consequence, we have −i 2([∆Λ,AR c0,h]wh,wh)L2(T2)=−i 2([∆,AR c0,h]wh,wh)L2(T2) =i 2h2/parenleftbig/bracketleftbig Ph 0,AR c0,h/bracketrightbig wh,wh/parenrightbig L2(T2). (10.4) Developing the last expression of (10.4), we obtain i 2h2/parenleftbig/bracketleftbig Ph 0,AR c0,h/bracketrightbig wh,wh/parenrightbig L2(T2)=i 2h2/parenleftbig AR c0,hwh,Ph bwh/parenrightbig L2(T2)−i 2h2/parenleftbig AR c0,hPh bwh,wh/parenrightbig L2(T2) −1 2h/parenleftbig AR c0,hwh,bwh/parenrightbig L2(T2)−1 2h/parenleftbig AR c0,hbwh,wh/parenrightbig L2(T2).(10.5) SinceAR c0,his bounded in L(L2(T2)), its adjoint AR c0,his also bounded so that the first two terms in the last expression vanish in the limit h→0, using/ba∇dblPh bwh/ba∇dblL2(T2)=o(h2). To estimate the last two terms, we use again the boundedness of ARand (AR)∗and write |/parenleftbig AR c0,hwh,bwh/parenrightbig L2(T2)| ≤ /ba∇dblAR/ba∇dbl/ba∇dblχΛ hbwh/ba∇dblL2(T2)≤2c0h/ba∇dblAR/ba∇dbl, according to Item 3 in Proposition 8.3. It follows that limsup h→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2h/parenleftbig AR c0,hwh,bwh/parenrightbig L2(T2)+1 2h/parenleftbig AR c0,hbwh,wh/parenrightbig L2(T2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2c0sup/ba∇dblAR/ba∇dbl. Coming back to the expression (10.2), we obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg νΛ,η |η|·∂xahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg M(T∗T2×SΛ),C0c(T∗T2×SΛ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2c0sup/ba∇dblAR/ba∇dbl and sincec0was arbitrary, /angbracketleftbigg νΛ,η |η|·∂xahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg M(T∗T2×SΛ),C0c(T∗T2×SΛ)= 0 Replacingahombyahom◦φ1 τand integrating with respect to the parameter τgives (φ1 τ)∗νΛ=νΛ, which concludes the proof of Proposition 10.1. 29Proof of Lemma 10.2. We are going to show that ([∆Λ,AR c0,h]wh,wh)L2(T2)= ([∆Λ,AR]c0,hwh,wh)L2(T2)+o(1). (10.6) Then, usingthefactthat[∆ Λ,AR]isaboundedoperator(itssymbolis/parenleftBig 1−χ/parenleftBig |η| R/parenrightBig/parenrightBig η |η|·∂xa(x,ξ,η)), together with (10.1), this is also ([∆ Λ,AR]wh,wh)L2(T2)+o(1). To prove (10.6), we develop the difference [∆ Λ,AR c0,h]−[∆Λ,AR]c0,has [∆Λ,AR c0,h]−[∆Λ,AR]c0,h=/bracketleftbig ∂2 y,χΛ h/bracketrightbig ARχΛ h+χΛ hAR/bracketleftbig ∂2 y,χΛ h/bracketrightbig . (10.7) Then, writing/bracketleftbig ∂2 y,χΛ h/bracketrightbig =∂2 yχΛ h+2∂yχΛ h∂y, we have /parenleftbig/bracketleftbig ∂2 y,χΛ h/bracketrightbig ARχΛ hwh,wh/parenrightbig L2(T2)=/parenleftbig ARχΛ hwh,∂2 yχΛ hwh/parenrightbig L2(T2)+/parenleftbig ∂y◦ARχΛ hwh,2∂yχΛ hwh/parenrightbig L2(T2). Recalling that the operator ∂y◦ARis bounded, and using Items 4 and 5 in Proposition 8.3, we obtain /vextendsingle/vextendsingle/vextendsingle/parenleftbig/bracketleftbig ∂2 y,χΛ h/bracketrightbig ARχΛ hwh,wh/parenrightbig L2(T2)/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dbl∂2 yχΛ hwh/ba∇dblL2(T2)++C/ba∇dbl∂yχΛ hwh/ba∇dblL2(T2)=o(1). The last term in (10.7) is handled similarly. This finally implies (10.6) and con cludes the proof of Lemma 10.2. 10.2 Propagation of ρΛ We denote by ( ωj Λ,ej Λ)j∈Nthe eigenvalues and associated eigenfunctions of the operator −∆Λ=−∂2 y forming a Hilbert basis of L2(TΛ). We shall use the projector onto low frequencies of −∆Λ, i.e., for anyω∈R+, the operator Πω Λ:=/summationdisplay ωj Λ≤ω(·,ej Λ)L2(TΛ)ej Λ, which has finite rank. We have the following propagation laws for the two-microlocal measu reρΛ. Proposition 10.3. 1. For any K∈C∞ c/parenleftbig T∗TΛ⊥;K(L2(TΛ))/parenrightbig , independent of s(i.e.K(s,σ) = K(σ)), and any ω>0, we have tr/braceleftBigg/integraldisplay T∗TΛ⊥[∆Λ,Πω ΛK(σ)Πω Λ]ρΛ(ds,dσ)/bracerightBigg = 0. 2. Moreover, defining MΛ:=/integraldisplay TΛ⊥×Λ⊥ρΛ(ds,dσ)∈ L1(L2(TΛ)), we have [∆Λ,MΛ] = 0. Remark that for any σ∈Λ⊥, the operator [∆Λ,Πω ΛK(σ)Πω Λ] = Πω Λ[∆Λ,K(σ)]Πω Λ, has finite rank, so the right hand-side of Item 1 is well-defined. Note that the definition of MΛhas a signification since ρΛhas a compact support, according to Proposition 9.4. The commutation relations of Items 1 and 2 in this proposition corres pond to propagation laws at the operator level. They are formulated here in a “derivated for m”, which, for Item 2 for instance, is equivalent to eiτ∆ΛMΛe−iτ∆Λ=MΛ,for allτ∈R, in the “integrated form”. 30Proof.ForK∈C∞ c(Λ⊥;K(L2(TΛ))) (in other words K∈C∞ c(T∗TΛ⊥;K(L2(TΛ))) independent of s∈TΛ⊥), we denote Kω(σ) := Πω ΛK(σ)Πω Λ and we note that Kωis also in C∞ c(Λ⊥;K(L2(TΛ))). Hence, we have tr/braceleftBigg/integraldisplay T∗TΛ⊥[∆Λ,Πω ΛK(σ)Πω Λ]ρΛ(ds,dσ)/bracerightBigg =−lim h→0([−∆Λ,Kω(hDs)]TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ)) To show that this limit vanishes, we proceed as in lines (10.4), (10.5) an d in the subsequent calcula- tion, replacing the operator ARbyKω(hDs). With the notation ∆ Λ=∂2 yand ∆ Λ⊥=∂2 s, we first note that ([−∆Λ,Kω(hDs)]TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ))= ([−∆,Kω(hDs)]TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ)), since ∆ = ∆ Λ+∆Λ⊥and since [∆ Λ⊥,Kω(hDs)] = 0.As a matter of fact, Kω(hDs) = OpΛ h(Kω(σ)) and ∆ Λ⊥=−h−2OpΛ h(|σ|2) are both Fourier multipliers. The following lemma is proved the same way as Lemma 10.2 Lemma 10.4. For any given c0>0, we have ([∆Λ,Kω(σ)]TΛwh,TΛwh)L2(T2)= ([∆Λ,Kω c0,h(hDs)]TΛwh,TΛwh)L2(T2)+o(1). HereKω c0,h(hDs) meansχΛ hKω(hDs)χΛ h. Writing −h2∆ =TΛPh bT∗ Λ−ihb◦πΛ, we have /parenleftbig [−∆,Kω c0,h(hDs)]TΛwh,TΛwh/parenrightbig L2(TΛ⊥;L2(TΛ)) =1 h2/parenleftbig Kω c0,h(hDs)TΛwh,TΛPh bwh/parenrightbig L2(TΛ⊥;L2(TΛ))−1 h2/parenleftbig Kω c0,h(hDs)TΛPh bwh,TΛwh/parenrightbig L2(TΛ⊥;L2(TΛ)) +i h/parenleftbig Kω c0,h(hDs)TΛwh,TΛ(bwh)/parenrightbig L2(TΛ⊥;L2(TΛ))+i h/parenleftbig Kω c0,h(hDs)TΛ(bwh),TΛwh/parenrightbig L2(TΛ⊥;L2(TΛ)). It follows, as in (10.5), that limsup h→0|/parenleftbig [−∆,Kω c0,h(hDs)]TΛwh,TΛwh/parenrightbig L2(TΛ⊥;L2(TΛ))| ≤2c0/ba∇dblK/ba∇dbl and sincec0was arbitrary, we can conclude that lim h→0([∆Λ,Kω(σ)]TΛwh,TΛwh)L2(T2)= 0, which concludes the proof of Item 1. Item 1 gives, for all K∈ K(L2(TΛ)) constant (which is possible since ρΛ(ds,dσ) has compact support), 0 = tr/braceleftBigg/integraldisplay T∗TΛ⊥[∆Λ,Kω]ρΛ(ds,dσ)/bracerightBigg = tr/braceleftBigg [∆Λ,Kω]/integraldisplay T∗TΛ⊥ρΛ(ds,dσ)/bracerightBigg = tr{[∆Λ,Kω]MΛ}. Using that tr( AB) = tr(BA) for allA∈ L1andB∈ Ltogether with the linearity of the trace (see [RS80, Theorem VI.25]), we now obtain, for all K∈ K(L2(TΛ)), and allω>0, 0 = tr{[∆Λ,Πω ΛKΠω Λ]MΛ}= tr{KΠω Λ[∆Λ,MΛ]Πω Λ}. Consequently, we have for all ω>0, Πω Λ[∆Λ,MΛ]Πω Λ= 0 (see [RS80, Theorem VI.26]). Letting ωgo to +∞, this yields [∆ Λ,MΛ] = 0 and concludes the proof of Item 2. 3111 The measures νΛandρΛvanish identically. End of the proof of Theorem 2.6 In this section, we prove that both measures νΛandρΛvanish when paired with the function /a\}b∇acketle{tb/a\}b∇acket∇i}htΛ. Then, we deduce that these two measures vanish identically. In tur n, this implies that µ|T2×Λ⊥= 0, and finally that µ= 0, which will conclude the proof of Theorem 2.6. Proposition 11.1. We have /angbracketleftbig νΛ|T2×Λ⊥×SΛ,/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/angbracketrightbig Mc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0,andtr{m/a\}bracketle{tb/a\}bracketri}htΛMΛ}= 0. As a consequence, we prove that ρΛandνΛ|T2×Λ⊥×SΛvanish. Proposition 11.2. We haveρΛ= 0andνΛ|T2×Λ⊥×SΛ= 0. Henceµ|T2×Λ⊥= 0. This allows to conclude the proof of Theorem 2.6. Indeed, as a conse quence of the decomposition formula of Proposition 9.4, we obtain, for all Λ ∈ P, such that rk(Λ) = 1, µ|T2×Λ⊥= 0. Using the decomposition of the measure µgiven in Lemma 7.1 together with Lemma 7.4, this yields µ= 0 onT2. This is in contradiction with µ(T∗T2) = 1 (Proposition 6.1), and this contradiction proves Theorem 2.6. We now prove Propositions 11.1 and 11.2 Proof of Proposition 11.1. First, (4.22) implies that ( bvh,vh)L2(T2)→0, and hence /a\}b∇acketle{tµ,b/a\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0. Then the decomposition given in Lemma 7.1 into a sum of nonnegative me asures yields that, for all Λ∈ P, /a\}b∇acketle{tµ|T2×Λ⊥,b/a\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0, (11.1) sincebis also nonnegative. Lemmata 7.2, 7.3 and 7.4 (see also Remark 7.5), th en give /a\}b∇acketle{tµ|T2×Λ⊥,/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/a\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)=/angbracketleftbig µ|T2×(Λ⊥\{0}),/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/angbracketrightbig Mc(T∗T2),C0(T∗T2) =/a\}b∇acketle{tµ|T2×Λ⊥,b/a\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0, (11.2) where the function /a\}b∇acketle{tb/a\}b∇acket∇i}htΛis also nonnegative. The decomposition formula of Proposition 9.4 into the two-microlocal semiclassical measures then yields /a\}b∇acketle{tµ|T2×Λ⊥,/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/a\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)=/angbracketleftbig νΛ|T2×Λ⊥×SΛ,/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/angbracketrightbig Mc(T∗T2×SΛ),C0(T∗T2×SΛ) +tr/braceleftBigg/integraldisplay T∗TΛ⊥m/a\}bracketle{tb/a\}bracketri}htΛρΛ(ds,dσ)/bracerightBigg . Besides,themeasure νΛ|T2×Λ⊥×SΛisnonnegative,hence/angbracketleftbig νΛ|T2×Λ⊥×SΛ,/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/angbracketrightbig Mc(T∗T2×SΛ),C0(T∗T2×SΛ)≥ 0. Similarly, ρΛ∈ M+ c(T∗TΛ⊥;L1(TΛ)) and the operator m/a\}bracketle{tb/a\}bracketri}htΛ∈ L(L2(TΛ)) is selfadjoint and non- negative, which gives tr/braceleftBig/integraltext T∗TΛ⊥m/a\}bracketle{tb/a\}bracketri}htΛρΛ(ds,dσ)/bracerightBig ≥0. Using (11.1) and (11.2), this yields /angbracketleftbig νΛ|T2×Λ⊥×SΛ,/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/angbracketrightbig Mc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0, and tr/braceleftBigg/integraldisplay T∗TΛ⊥m/a\}bracketle{tb/a\}bracketri}htΛρΛ(ds,dσ)/bracerightBigg = 0. In this expression, the operator m/a\}bracketle{tb/a\}bracketri}htΛdoes not depend on ( s,σ), so that 0 = tr/braceleftBigg m/a\}bracketle{tb/a\}bracketri}htΛ/integraldisplay T∗TΛ⊥ρΛ(ds,dσ)/bracerightBigg = tr{m/a\}bracketle{tb/a\}bracketri}htΛMΛ}, which concludes the proof of Proposition 11.1. 32Proof of Proposition 11.2. Let us first prove that ρΛ= 0. We recall that the operator MΛis a selfadjoint nonnegative trace-class operator. Moreover, Prop osition 10.3 implies that the operators MΛand ∆ Λcommute. As a consequence, there exists a Hilbert basis (˜ ej Λ)j∈NofL2(TΛ) in which MΛand ∆ Λare simultaneously diagonal, i.e. such that −∆Λ˜ej Λ=ωj Λ˜ej Λ,andMΛ˜ej Λ=γj Λ˜ej Λ, where (γj Λ)j∈Nare the associated eigenvalues of MΛ. In particular, we have γj Λ≥0 for allj∈N(and γj Λ∈ℓ1). Note that the basis (˜ ej Λ)j∈Nis not necessarily the same as the basis ( ej Λ)j∈Nintroduced in Section 10.2. Using Proposition 11.1, together with the definition of the trace (se e for instance [RS80, Theorem VI.18]) we have 0 = tr{m/a\}bracketle{tb/a\}bracketri}htΛMΛ}=/summationdisplay j∈N/parenleftBig m/a\}bracketle{tb/a\}bracketri}htΛMΛ˜ej Λ,˜ej Λ/parenrightBig L2(TΛ)=/summationdisplay j∈Nγj Λ/parenleftBig /a\}b∇acketle{tb/a\}b∇acket∇i}htΛ˜ej Λ,˜ej Λ/parenrightBig L2(TΛ). Since all terms in this sum are nonnegative (because both γj Λand/a\}b∇acketle{tb/a\}b∇acket∇i}htΛare), we deduce that for all j∈N, γj Λ/parenleftBig /a\}b∇acketle{tb/a\}b∇acket∇i}htΛ˜ej Λ,˜ej Λ/parenrightBig L2(TΛ)= 0. Suppose that γj Λ/\e}atio\slash= 0 for some j∈N. Then,/parenleftBig /a\}b∇acketle{tb/a\}b∇acket∇i}htΛ˜ej Λ,˜ej Λ/parenrightBig L2(TΛ)= 0 where /a\}b∇acketle{tb/a\}b∇acket∇i}htΛis nonnegative and not identically zero on TΛ. This yields ˜ ej Λ= 0 on the nonempty open set {/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ>0}. Using a unique continuation property for eigenfunctions of the Laplace operato r onTΛ, we finally obtain that the eigenfunction ˜ ej Λvanishes identically on TΛ. This is absurd, and thus we must have γj Λ= 0 for all j∈N, so thatMΛ= 0. SinceρΛ∈ M+(T∗TΛ⊥;L1(TΛ)), this directly gives ρΛ= 0. Next, we prove that νΛ= 0. This is a consequence of the additional propagation law of νΛwith respect to the flow φ1 τ(see Section 10.1). Indeed the torus TΛhas dimension one, ( φ1 τ)∗νΛ=νΛ (according to Proposition 10.1) and, using Proposition 11.1, νΛvanishes on the (nonempty) set {/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ>0}×R2×SΛ(with{/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ>0}clearly satisfying GCC on TΛ). Hence,νΛ= 0. To conclude the proof of Proposition 11.2, it only remains to use the d ecomposition formula (9.7) which directly yields µ|T2×Λ⊥= 0. 12 Proof of Proposition 8.2 In this section, we prove Proposition 8.2. For this, we consider two- microlocal semiclassical measures at the scale hα. The setting is close to that of [FK05]. We shall see that the concentration rate of the sequence vhtowards the direction Λ⊥is of the formhαfor allα≤3+δ 4. First, Lemma 7.3 yields µ|T2×Λ⊥=/a\}b∇acketle{tµ/a\}b∇acket∇i}htΛ|T2×Λ⊥(see also Remark 7.5), i.e. /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)=/a\}b∇acketle{tµ|T2×Λ⊥,/a\}b∇acketle{ta/a\}b∇acket∇i}htΛ/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2), and it suffices to characterize the action of µ|T2×Λ⊥on Λ⊥-invariant symbols. Recall that, for all a∈C∞ c(T∗T2), /a\}b∇acketle{tµ,a/a\}b∇acket∇i}htM(T∗T2),C0 c(T∗T2)= lim h→0(Oph(a)vh,vh)L2(T2). In this section, the assumption√ b∈C∞(T2) is used in an essential way for the propagation result of Lemma 12.2 below. Like in (9.1) and (9.2), let us define : /angbracketleftBig Vh,Λ R,a/angbracketrightBig S1 Λ′,S1 Λ:=/angbracketleftbigg Vh,/parenleftbigg 1−χ/parenleftbigg|PΛξ| Rh/parenrightbigg/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/angbracketrightbigg D′(T∗T2),C∞c(T∗T2),(12.1) /angbracketleftbig Vh R,Λ,a/angbracketrightbig S1 Λ′,S1 Λ:=/angbracketleftbigg Vh,χ/parenleftbigg|PΛξ| Rh/parenrightbigg a/parenleftbigg x,ξ,PΛξ h/parenrightbigg/angbracketrightbigg D′(T∗T2),C∞ c(T∗T2), (12.2) 33fora∈S1 Λ. We takeR=R(h) =h−(1−α)for someα∈(0,1), so thatRh=hα. The proof of Proposition 9.2 applies verbatim and shows the existence of a subsequence ( h,vh) and a nonnegative measure νΛ α∈ M+(T∗T2×SΛ) such that, for all a∈S1 Λ, we have lim h→0/angbracketleftBig Vh,Λ R(h),a/angbracketrightBig S1 Λ′,S1 Λ=/angbracketleftbigg νΛ α,ahom/parenleftbigg x,ξ,η |η|/parenrightbigg/angbracketrightbigg M(T∗T2×SΛ),C0 c(T∗T2×SΛ). Proposition 12.1. LetR(h) =h−(1−α)withα≤3+δ 4. Then νΛ α|T2×(Λ⊥\{0})×SΛ= 0 The proof of Proposition 12.1 relies on the following propagation resu lt. Lemma 12.2. Forα≤3+δ 4the measure νΛ αisφ0 τ- andφ1 τ-invariant, i.e. (φ0 τ)∗νΛ α=νΛ αand(φ1 τ)∗νΛ α=νΛ α,for everyτ∈R. The proof is very similar to that of Proposition 10.1 but does not use A ssumption (2.12). Proof.The proof of φ0 τ-invariance is strictly identical to what has been done for Propositio n 10.1 and thus we focus on the φ1 τ-invariance. Equation (10.5) still holds with R(h) =h−(1−α), now reading /angbracketleftbigg Vh,Λ R(h),η |η|·∂xa/angbracketrightbigg S1 Λ′,S1 Λ=i 2h2/parenleftBig AR(h)vh,Ph bvh/parenrightBig L2(T2)−i 2h2/parenleftBig AR(h)Ph bvh,vh/parenrightBig L2(T2) −1 2h/parenleftBig AR(h)vh,bvh/parenrightBig L2(T2)−1 2h/parenleftBig AR(h)bvh,vh/parenrightBig L2(T2) whereARwas defined in (10.3). Using /ba∇dblPh bvh/ba∇dblL2(T2)=o(h1+δ) together with the boundedness of AR(h), it follows that lim h→0/angbracketleftbigg Vh,Λ R(h),η |η|·∂xa/angbracketrightbigg S1 Λ′,S1 Λ= lim h→0/parenleftBig −1 2h/parenleftbig AR(h)vh,bvh/parenrightbig L2(T2)−1 2h/parenleftbig AR(h)bvh,vh/parenrightbig L2(T2)/parenrightBig . Recall from (4.22) that /ba∇dbl√ bvh/ba∇dblL2(T2)=o(h1+δ 2).In addition, it follows from standard microlocal calculus that [AR(h),√ b] =OL(L2)(R(h)−2). We can thus write /angbracketleftbigg Vh,Λ R(h),η |η|·∂xa/angbracketrightbigg S1 Λ′,S1 Λ=o(1)−1 h/parenleftbig AR(h)√ bvh,√ bvh/parenrightbig L2(T2)+1 2h/parenleftbig√ b[AR(h),√ b]vh,vh/parenrightbig L2(T2) +1 2h/parenleftbig [√ b,AR(h)]√ bvh,vh/parenrightbig L2(T2) =o(1)+o(R(h)−2h−1+δ 2) =o(1)+o(h3 2+δ 2−2α), which vanishes if we take α≤3+δ 4. Proof of Proposition 12.1. To prove Proposition 12.1, we first note that /angbracketleftbig νΛ α|T2×(Λ⊥\{0})×SΛ,/a\}b∇acketle{tb/a\}b∇acket∇i}htΛ/angbracketrightbig Mc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0, sinceνΛ αis (φ0 τ)-invariant and/angbracketleftbig νΛ α,b/angbracketrightbig Mc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0. Then, the φ1 τ-invariance of νΛ α implies that νΛ α|T2×(Λ⊥\{0})×SΛvanishes. 34Proof of Proposition 8.2. Proposition 12.1 implies that /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim h→0/parenleftbigg Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg a(x,ξ)/parenrightbigg vh,vh/parenrightbigg L2(T2) for allα≤3+δ 4anda∈C∞ c(T∗T2). The same holds if we replace χbyχ2: /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim h→0/parenleftbigg Oph/parenleftbigg χ2/parenleftbigg|PΛξ| hα/parenrightbigg a(x,ξ)/parenrightbigg vh,vh/parenrightbigg L2(T2). Since Oph/parenleftbigg χ2/parenleftbigg|PΛξ| hα/parenrightbigg a(x,ξ)/parenrightbigg = Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg Oph(a)Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg +O(h1−α),(12.3) we obtain /a\}b∇acketle{tµ|T2×Λ⊥,a/a\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim h→0/parenleftbigg Oph(a)Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg vh,Oph/parenleftbigg χ/parenleftbigg|PΛξ| hα/parenrightbigg/parenrightbigg vh/parenrightbigg L2(T2), for allα≤3+δ 4anda∈C∞ c(T∗T2). 13 Proof of Proposition 8.3: existence of the cutoff function Given a constant c0>0, we define the following subsets of T2: Eh=/a\}b∇acketle{t{b>c0h}/a\}b∇acket∇i}htΛ,Fh=/angbracketleftBigg/uniondisplay x∈{b>c0h}B(x,(c0h)2ε)/angbracketrightBigg Λ=/uniondisplay x∈EhB(x,(c0h)2ε),Gh=Fh\Eh, where forU⊂T2, we denote /a\}b∇acketle{tU/a\}b∇acket∇i}htΛ:=/uniontext τ∈R{U+τσ}for someσ∈Λ⊥\{0}. Remark that Eh⊂Fh and that T2=Eh∪Gh∪(T2\Fh). Note also that the sets Eh,Fhare non-empty for hsmall enough, and that Ghis non empty (for hsmall enough) as soon as bvanishes somewhere on T2(this condition is assumed here since otherwise, GCC is satisfied). In this section, we construct the cutoff function χΛ hneeded to prove the propagation results of Section 10. In particular, this function will be Λ⊥-invariant and will satisfy χΛ h= 0 on EhandχΛ h= 1 onT2\Fh. The proof of Proposition 8.3 relies on three key lemmata. The first ke y lemma is a precised version of Proposition 6.1 concerning the localization in T∗T2of the semiclassical measure µ. It is an intermediate step towards the propagation result stated in Lem ma 13.2. Lemma 13.1. For anyχ∈C∞ c(R), such that χ= 1in a neighbourhood of the origin, for all a∈C∞ c(T∗T2), andγ≤3+δ 2, we have (Oph(a)wh,wh)L2(T2)=/parenleftbigg Oph(a)Oph/parenleftbigg χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg/parenrightbigg wh,wh/parenrightbigg L2(T2)+o(h3+δ 2−γ)/ba∇dblOph(a)/ba∇dblL(L2), (13.1) For alla∈C∞ c(T∗T2)and allτ∈R, (Oph(a◦φτ)wh,wh)L2(T2)= (Oph(a)wh,wh)L2(T2)+o(τh1+δ 2)/ba∇dblOph(a◦φt)/ba∇dblL∞(0,τ;L(L2(T2))) In this statement, we used the notation /ba∇dblOph(a◦φt)/ba∇dblL∞(0,τ;L(L2(T2))):= sup t∈(0,τ)/ba∇dblOph(a◦φt)/ba∇dblL(L2(T2)). In turn, this lemma implies the following transport property. 35Lemma 13.2. Suppose that the coefficients α,εsatisfy 0<10ε≤α,andα+2ε≤1. (13.2) Then, for any time τ∈Runiformly bounded with respect to h, and any h-family of functions ψ=ψh∈C∞ c(T2)satisfying /ba∇dbl∂k xψ/ba∇dblL∞(T2)≤Ckh−2ε|k|,for allk∈N2, (13.3) we have, (ψ(s,y)wh,wh)L2(T2)= (ψ(s+τ,y)wh,wh)L2(T2)+(ψ(s−τ,y)wh,wh)L2(T2) +O(hα−10ε)+O(h1−α−2ε)+o(h1+δ 2), (13.4) where the coordinates (s,y)are the ones introduced in Section 7.3. In view of Proposition 8.3, this lemma will allow us to propagate the smalln ess of the sequence whabove the set {b>c0h}to all Eh. The third key lemma states a property of the damping function b, as a consequence of Assump- tion 2.12. Lemma 13.3. There exists b0=b0(ε)>0such that for all x∈T2satisfying 0<b(x)<b0and for allz∈B(x,b(x)2ε), we haveb(z)≥b(x) 2. With these three lemmata, we are now able to prove Proposition 8.3. Proof of Proposition 8.3. In the coordinates ( s,y) of Section 7.3, we can write Eh=TΛ⊥×Eh,Fh=TΛ⊥×Fh,withEh⊂Fh⊂TΛ. Here,Fhis a union of intervals and has uniformly bounded total length. We can hence cover Fhwith C1h−2εsubsets of length of order ( c0h)2ε/2, overlapping on intervals of length of order ( c0h)2ε/10. Associated to this covering, we denote by ( ψj)j∈{1,...,J},J=J(h), a smooth partition of unity on Eh, satisfying moreover •ψj∈C∞ c(Fh); •/summationtextJ j=1ψj(y) = 1 fory∈Eh; • /ba∇dbl∂m yψj/ba∇dblL∞(TΛ)≤Cmh−2εm, for allm∈N; •J=J(h)≤Ch−2ε. Similarly, we cover TΛ⊥withC2h−2εsubsets of length of order ( c0h)2ε/2, overlapping on intervals of length of order ( c0h)2ε/10, and define ( ψk)k∈{1,...,K}an associated partition of unity on TΛ⊥ satisfying •ψk∈C∞ c(TΛ⊥); •/summationtextK k=1ψk(s) = 1 fors∈TΛ⊥; • /ba∇dbl∂m sψk/ba∇dblL∞(TΛ⊥)≤Cmh−2εm, for allm∈N; •K=K(h)≤Ch−2ε, •for anyk,k0∈ {1,...,K}2, there exists τksatisfying |τk| ≤Length(TΛ⊥)≤Candψk(s+τk) = ψk0(s). 36We set ψkj(s,y) :=ψk(s)ψj(y),andχΛ h(s,y) = 1−J/summationdisplay j=1K/summationdisplay k=1ψkj(s,y)∈C∞(T2), which satisfies ∂sχΛ h(s,y) = 0, i.e.χΛ his Λ⊥-invariant, together with •χΛ h= 0 on Ehand henceb≤c0hon supp(χΛ h); •χΛ h= 1 onT2\Fh; •χΛ h∈[0,1] onGh, with|∂yχΛ h| ≤Ch−2εand|∂2 yχΛ h| ≤Ch−4ε. To conclude the proof of Proposition 8.3, it remains to check Item 2 ( /ba∇dbl(1−χΛ h)wh/ba∇dblL2(T2)=o(1)), Item 4 (/ba∇dbl∂yχΛ hwh/ba∇dblL2(T2)=o(1)) and Item 5 ( /ba∇dbl∂2 yχΛ hwh/ba∇dblL2(T2)=o(1)). Now, letus fix j0∈ {1,...,J}. Becauseofthe definition ofthe set Eh, there exists k0∈ {1,...,K} andx0∈ {b > c0h}such that supp( ψk0j0)⊂B(x0,(c0h)2ε). According to Lemma 13.3, we have B(x0,(c0h)2ε)⊂ {b>c0h 2}, so that supp( ψk0j0)⊂ {b>c0h 2}. This yields c0h 2(ψk0j0wh,wh)L2(T2)≤(bψk0j0wh,wh)L2(T2)=o(h1+δ), and hence ( ψk0j0wh,wh)L2(T2)=o(hδ). Moreover, for any k∈ {1,...,K}, there exists τksatisfying |τk| ≤C2with ψkj0(s+τk,y) =ψk0j0(s,y). Hence, using (13.4), we obtain o(hδ) = (ψk0j0(s,y)wh,wh)L2(T2)= (ψkj0(s+τk,y)wh,wh)L2(T2) = (ψkj0(s+2τk,y)wh,wh)L2(T2)+(ψkj0(s,y)wh,wh)L2(T2) +O(hα−10ε)+O(h1−α−2ε)+o(h1+δ 2). (13.5) Since both termsonthe righthand-sidearenonnegative,this implies (ψkj0(s,y)wh,wh)L2(T2)=o(hδ) as soon as α−10ε>δ, 1−α−2ε>δ, 1+δ 2≥δ, (which implies (13.2)). From now on we will take δ= 8ε(this choice is explained in the following lines). The existence of αsatisfying this condition together with (8.1) and α<3/4, is equivalent to havingε<1 76. To conclude the proof of Proposition 8.3, we first compute ((1−χΛ h)wh,wh)L2(T2)=J/summationdisplay j=1K/summationdisplay k=1(ψkjwh,wh)L2(T2)=Ch−4εo(hδ) =o(1), sinceδ≥4ε. This proves Item 2. Next, we have by construction supp( ∂2 yχΛ h)⊂supp(∂yχΛ h)⊂ Ghwith/ba∇dbl∂yχΛ h/ba∇dblL∞(T2)=O(h−2ε),/ba∇dbl∂2 yχΛ h/ba∇dblL∞(T2)=O(h−4ε). Hence, covering supp( ∂yχΛ h)) by balls of radius ( c0h)2εand using a propagation argument similar to (13.5) shows that we hav e /ba∇dblwh/ba∇dblL2(supp(∂yχΛ h))=o(hδ 2). We thus obtain /ba∇dbl∂yχΛ hwh/ba∇dblL2(T2)=o(hδ 2−2ε) =o(1),/ba∇dbl∂2 yχΛ hwh/ba∇dblL2(T2)=o(hδ 2−4ε) =o(1), (sinceδ≥8ε) which concludes the proof of Items 4 and 5, and that of Propositio n 8.3. 37To conclude this section, it remains to prove Lemmata 13.2, 13.1 and 1 3.3. In the following proofs, we shall systematically write ηin place of PΛξandσin place of (1 −PΛ)ξto lighten the notation. Hence, ξ∈R2is decomposed as ξ=η+σwithη∈ /a\}b∇acketle{tΛ/a\}b∇acket∇i}htandσ∈Λ⊥, in accordance to Section 7.3. Proof of Lemma 13.2 from Lemma 13.1. First, given a function ψ∈C∞ c(T2) satisfying (13.3), we have, (ψwh,wh)L2(T2)= (Oph(ψ◦φτ)wh,wh)L2(T2)+o(τh1+δ 2)/ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2)) =/parenleftbigg Oph(ψ◦φτ)Oph/parenleftbigg χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg/parenrightbigg Oph/parenleftBig χ/parenleftBigη 2hα/parenrightBig/parenrightBig wh,wh/parenrightbigg L2(T2) +/parenleftbig o(τh1+δ 2)+o(τh3+δ 2−γ)/parenrightbig /ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2)), when using Lemma 13.1 together with Oph/parenleftbig χ/parenleftbigη 2hα/parenrightbig/parenrightbig wh=wh. Next, the pseudodifferential calculus yields (ψwh,wh)L2(T2)=/parenleftbigg Oph/parenleftbigg ψ◦φτχ/parenleftbigg|ξ|2−1 hγ/parenrightbigg χ/parenleftBigη 2hα/parenrightBig/parenrightbigg wh,wh/parenrightbigg L2(T2)+O(h2−γ−2ε)+O(h1−α−2ε) +/parenleftbig o(τh1+δ 2)+o(τh3+δ 2−γ)/parenrightbig /ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2)). (13.6) A particular feature of the Weyl quantization in the Euclidean settin g is that the Egorov theorem provides an exact formula (see for instance [DS99]): Oph(ψ◦φt) =e−ith∆ 2Oph(ψ)eith∆ 2, so that /ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2))≤C0uniformly with respect to h. Now, remark that the cutoff function χ/parenleftbigη 2hα/parenrightbig χ/parenleftBig |ξ|2−1 hγ/parenrightBig can be decomposed (for hsmall enough) as χ/parenleftBigη 2hα/parenrightBig χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg =χ/parenleftBigη 2hα/parenrightBig/parenleftbig ˜χh η(σ)+ ˜χh η(−σ)/parenrightbig for some nonnegative function ˜ χh ηsuch that (σ,η)/ma√sto→˜χh η(σ)∈C∞ c(R2), such that ˜ χh η(σ) =χ/parenleftBig |ξ|2−1 hγ/parenrightBig forη∈suppχ/parenleftbig· 2hα/parenrightbig andσ>0, and ˜χh η(σ) = 0 forη /∈suppχ/parenleftbig· 2hα/parenrightbig orσ≤0. Choosingγ=α, we have in particular |σ−1| ≤Chαon supp/parenleftBig χ/parenleftBigη 2hα/parenrightBig ˜χh η(σ)/parenrightBig . Next, we recallthat ψ◦φτ(s,y,σ,η) =ψ(s+τσ,y+τη), and we focus on the first term (corresponding toσ>0) in the right-hand side of the identity χ/parenleftbigg|ξ|2−1 hα/parenrightbigg χ/parenleftBigη 2hα/parenrightBig ψ◦φτ=χ/parenleftBigη 2hα/parenrightBig/parenleftbig ˜χh η(σ)+ ˜χh η(−σ)/parenrightbig ψ◦φτ. (13.7) We set ζ(1) τ(s,y,σ,η) =χ/parenleftBigη 2hα/parenrightBig ˜χh η(σ)ψ(s+τσ,y+τη),andζ(2) τ(s,y,σ,η) =χ/parenleftBigη 2hα/parenrightBig ˜χh η(σ)ψ(s+τ,y), andwewanttocompareOph(ζ(1) τ)andOph(ζ(2) τ). Forthis, letusestimate, formultiindices ℓ,m∈N2, /vextendsingle/vextendsingle/vextendsingle∂ℓ (s,y)∂m (σ,η)/parenleftBig ζ(2) τ−ζ(1) τ/parenrightBig (s,y,σ,η)/vextendsingle/vextendsingle/vextendsingle ≤Cm/summationdisplay ν≤m/vextendsingle/vextendsingle/vextendsingle∂m−ν (σ,η)/parenleftBig χ/parenleftBigη 2hα/parenrightBig ˜χh η(σ)/parenrightBig ∂ℓ (s,y)∂ν (σ,η)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle.(13.8) On the one hand, we have /vextendsingle/vextendsingle/vextendsingle∂m−ν (σ,η)/parenleftBig χ/parenleftBigη 2hα/parenrightBig ˜χh η(σ)/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤Cm,νh−α|m−ν|. (13.9) 38On the other hand, for |ν|>0 we can also write /vextendsingle/vextendsingle/vextendsingle∂ℓ (s,y)∂ν (σ,η)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle∂ℓ (s,y)∂ν (σ,η)ψ(s+τσ,y+τη)/vextendsingle/vextendsingle/vextendsingle ≤Cℓ,ν|τ||ν|h−2ε(|ℓ|+|ν|)≤Cℓ,νh−2ε(|ℓ|+|ν|), since|τ| ≤C. Finally, for |ν|= 0, we apply the mean value theorem to the function (σ,η)/ma√sto→∂ℓ (s,y)ψ(s+τσ,y+τη) and write /vextendsingle/vextendsingle/vextendsingle∂ℓ (s,y)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle ≤(|η|+|σ−1|) sup T∗T2/vextendsingle/vextendsingle/vextendsingle∇(σ,η)∂ℓ (s,y)(ψ(s+τσ,y+τη))/vextendsingle/vextendsingle/vextendsingle. With (13.3), this yields /vextendsingle/vextendsingle/vextendsingle∂ℓ (s,y)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle≤(|η|+|σ−1|)Cℓh−2ε|ℓ||τ|h−2ε ≤(|η|+|σ−1|)Cℓh−2ε(|ℓ|+1), (13.10) for|τ| ≤C. Usingnowthat |η| ≤Chαand|σ−1| ≤Chαonsupp/parenleftbig χ/parenleftbigη 2hα/parenrightbig ˜χh η(σ)/parenrightbig ,andcombining(13.8),(13.9) and (13.10), we obtain, for all m∈N2,ℓ∈N2and 0<h≤h0sufficiently small, h|m|/vextendsingle/vextendsingle/vextendsingle∂ℓ (s,y)∂m (σ,η)/parenleftBig ζ(2) τ−ζ(1) τ/parenrightBig (s,y,σ,η)/vextendsingle/vextendsingle/vextendsingle≤Cℓ,mhα−2ε(|ℓ|+1)h|m|h−α|m| +Cℓ,m/summationdisplay 0<ν≤mh|m|h−2ε(|ℓ|+|ν|)h−α|m−ν| ≤Cℓ,m/parenleftBig h(1−α)|m|hα−2ε(|ℓ|+1)+|m|h|m|(1−α)h−2ε|ℓ|hα−2ε/parenrightBig ≤Cℓ,mhα−2ε(|ℓ|+1). Using a precised version of the Calder´ on-Vaillancourt theorem, as presented in Theorem A.1 below (in which only |ℓ|= 4 derivations are needed with respect to xin dimension two), we obtain Oph(ζ(2) τ) = Oph(ζ(1) τ)+OL(L2)(hα−10ε). Similarly, we have Oph/parenleftBig χ/parenleftBigη 2hα/parenrightBig ˜χh η(−σ)ψ(s+τσ,y+τη)/parenrightBig = Oph/parenleftBig χ/parenleftBigη 2hα/parenrightBig ˜χh η(−σ)ψ(s−τ,y)/parenrightBig +OL(L2)(hα−10ε). Coming back to (13.6) and using (13.7), we finally obtain, for all |τ| ≤C, (ψwh,wh)L2(T2)=/parenleftBig Oph/parenleftBig χ/parenleftBigη 2hα/parenrightBig ˜χh η(σ)ψ(s+τ,y)/parenrightBig wh,wh/parenrightBig L2(T2) +/parenleftBig Oph/parenleftBig χ/parenleftBigη 2hα/parenrightBig ˜χh η(−σ)ψ(s−τ,y)/parenrightBig wh,wh/parenrightBig L2(T2) +O(hα−10ε)+O(h1−α−2ε)+o(h1+δ 2)+o(h3+δ 2−α). With the pseudodifferential calculus, this yields (13.4), which conclud es the proofofLemma 13.2. Proof of Lemma 13.1. Here, we only have to make more precise some arguments in the proo f of Lemma 6.1. Recall that according to Lemma 8.1, whsatisfiesPh bwh=o(h2+δ). 39First, we take χ∈C∞ c(R), such that χ= 1 in a neighbourhood of the origin. Hence,1−χ(r) r∈ C∞(R) and we have the exact composition formula Oph/parenleftbigg 1−χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg/parenrightbigg = Oph/parenleftbigg/parenleftbigg 1−χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg/parenrightbigghγ |ξ|2−1/parenrightbiggPh 0 hγ, since both operators are Fourier multipliers. Moreover, Oph/parenleftBig/parenleftBig 1−χ/parenleftBig |ξ|2−1 hγ/parenrightBig/parenrightBig hγ |ξ|2−1/parenrightBig is uniformly bounded as an operator of L(L2(T2)). As a consequence, we have /parenleftbigg Oph(a)Oph/parenleftbigg 1−χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg/parenrightbigg wh,wh/parenrightbigg L2(T2) =/parenleftbigg Oph(a)Oph/parenleftbigg/parenleftbigg 1−χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg/parenrightbigghγ |ξ|2−1/parenrightbiggPh 0 hγwh,wh/parenrightbigg L2(T2) =/parenleftbigg APh b hγwh,wh/parenrightbigg L2(T2)−/parenleftbigg Aihb hγwh,wh/parenrightbigg L2(T2), whereA= Oph(a)Oph/parenleftBig/parenleftBig 1−χ/parenleftBig |ξ|2−1 hγ/parenrightBig/parenrightBig hγ |ξ|2−1/parenrightBig is bounded on L2(T2). UsingPh bwh=o(h2+δ) and (bwh,wh)L2(T2)=o(h1+δ), this gives /parenleftbigg Oph(a)Oph/parenleftbigg 1−χ/parenleftbigg|ξ|2−1 hγ/parenrightbigg/parenrightbigg wh,wh/parenrightbigg L2(T2)=o(h3+δ 2−γ)/ba∇dblOph(a)/ba∇dblL(L2), which in turn implies (13.1). Next, Identity (6.2) yields, for all a∈C∞ c(T2), (Oph(ξ·∂xa)wh,wh)L2(T2)=i 2h/parenleftbig Oph(a)wh,Ph bwh/parenrightbig L2(T2)−i 2h/parenleftbig Oph(a)Ph bwh,wh/parenrightbig L2(T2) −1 2(Oph(a)wh,bwh)L2(T2)−1 2(Oph(a)bwh,wh)L2(T2) =o(h1+δ)/ba∇dblOph(a)/ba∇dblL(L2)+o(h1+δ 2)/ba∇dblOph(a)/ba∇dblL(L2), as a consequence of Ph bwh=o(h2+δ) and (bwh,wh)L2(T2)=o(h1+δ). Applying this identity to a◦φt in place ofa, and integrating on t∈[0,τ] finally gives (Oph(a◦φτ)wh,wh)L2(T2)= (Oph(a)wh,wh)L2(T2)+o(τh1+δ 2)/ba∇dblOph(a◦φt)/ba∇dblL∞(0,τ;L(L2)), which concludes the proof of Lemma 13.1. Proof of Lemma 13.3. Here,B:=B(x,b(x)2ε) denotes the euclidian ball in T2centered at xof radiusb(x)2ε. Setting M:= sup z∈Bb(z), m:= inf z∈Bb(z), we have |∇b(z)| ≤Cεb1−ε(z)≤CεM1−ε,for allz∈B, as a consequence of Assumption 2.12. Moreover, the mean value th eorem yields b(x)−CεM1−εb(x)2ε≤b(z)≤b(x)+CεM1−εb(x)2ε,forz∈B, and, in particular, m≥b(x)−CεM1−εb(x)2ε,andM≤b(x)+CεM1−εb(x)2ε. (13.11) 40Now, defining f(M) :=b(x) +CεM1−εb(x)2ε, we see that fis a strictly concave function with f(0) =b(x)>0. There exists a unique M0∈R+satisfyingf(M0) =M0. Moreover, we have M≤f(M) if and only if M≤M0. Takingb0sufficiently small so that b0+Cεb1+ε2 0≤b1−ε 0, we obtainf(b(x)1−ε)≤b(x)1−ε. In particular, this gives M0≤b(x)1−εand hence M≤b(x)1−ε according to the second estimate of (13.11). Coming back to the fir st estimate of (13.11), this yields m≥b(x)−Cεb(x)(1−ε)2b(x)2ε=b(x)−Cεb(x)1+ε2. Takingb0sufficiently small so that b0−Cεb1+ε2 0≥b0 2, we obtain m≥b(x) 2, which concludes the proof of Lemma 13.3. Part IV Ana priori lower bound for decay rates on the torus: proof of Theorem 2.5 Under the assumption {b>0}∩{x0+τξ0,τ∈R}=∅, (13.12) for some (x0,ξ0)∈T∗T2,ξ0/\e}atio\slash= 0, we construct in this section a constant κ0>0 and a sequence (ϕn)n∈NofO(1)-quasimodes in the limit n→+∞for the family of operators P(inκ0). WeusethenotationintroducedinSections7.1and9. First, notetha t, asaconsequenceof (13.12), ξ0is necessarily a rational direction, and the set {x0+τξ0,τ∈R}is a one-dimensional subtorus of T2, given by {x0+τξ0,τ∈R}={x0+τξ0,τ∈R}=x0+TΛ⊥ ξ0,with Λ ξ0∈ P. Letχ∈C∞ c(T2) such that χhas onlyx-Fourier modes in Λ ξ0,χ= 0 on a neighbourhood of {b>0}andχ= 1 onx0+TΛ⊥ ξ0. From Assumption (13.12), we have rk(Λ ξ0) = 1, so that one can find k∈Λ⊥ ξ0∩Z2\{0}. Besides, for alln∈Nwe havenk∈Λ⊥ ξ0∩Z2\{0}. We then define the sequence of quasimodes (ϕn)n∈Nby ϕn(x) =χ(x)eink·x, n∈N, x∈T2. We haveϕn∈C∞(T2), together with the decoupling ϕn◦πΛξ0(s,y) =χ(y)eink·s, n∈N,(s,y)∈TΛ⊥ ξ0×TΛξ0. This yields −/parenleftbig TΛξ0∆T∗ Λξ0/parenrightbig ϕn◦πΛξ0(s,y) =−/parenleftbig ∆Λξ0+∆Λ⊥ ξ0/parenrightbig ϕn◦πΛξ0(s,y) =−eink·s∆Λξ0χ(y)+n2|k|2χ(y)eink·s. Moerover,bϕn= 0, according to their respective supports. Hence, recalling that P(in|k|) =−∆− n2|k|2+in|k|b(x), we have /parenleftbig TΛξ0P(in|k|)T∗ Λξ0/parenrightbig ϕn◦πΛξ0=−eink·s∆Λξ0χ(y), and /ba∇dblP(in|k|)ϕn/ba∇dblL2(T2)=/ba∇dbl/parenleftbig TΛξ0P(in|k|)T∗ Λξ0/parenrightbig ϕn◦πΛξ0/ba∇dblL2(TΛ⊥ ξ0×TΛξ0)=C0/ba∇dbl∆Λξ0χ/ba∇dblL2(TΛξ0). 41Since we also have /ba∇dblϕn/ba∇dblL2(T2)=/ba∇dblTΛξ0ϕn/ba∇dblL2(TΛ⊥ ξ0×TΛξ0)=C0/ba∇dblχ/ba∇dblL2(TΛξ0), we obtain, for all n∈N, /ba∇dblP−1(in|k|)/ba∇dblL(L2(T2))≥/ba∇dblϕn/ba∇dblL2(T2) /ba∇dblP(in|k|)ϕn/ba∇dblL2(T2)=/ba∇dblχ/ba∇dblL2(TΛξ0) /ba∇dbl∆Λξ0χ/ba∇dblL2(TΛξ0)=C >0, which concludes the proof of Theorem 2.5. Acknowledgments. The authors would like to thank Nicolas Burq for having found a signific ant error in a previous version of this article, and for advice on how to fix it. The second author wishes to thank Luc Robbiano for several interesting discussions on the s ubject of this article. A Pseudodifferential calculus In the main part of the article, we use the semiclassical Weyl quantiz ation, that associates to a functionaonT∗R2an operator Oph(a) defined by /parenleftbig Oph(a)u/parenrightbig (x) :=1 (2πh)2/integraldisplay R2/integraldisplay R2ei hξ·(x−y)a/parenleftbiggx+y 2,ξ/parenrightbigg u(y)dy dξ. (A.1) For smooth functions awith uniformly bounded derivatives, Oph(a) defines a continuous operator onS(R2), and also by duality on S′(R2). On a manifold, the quantization Ophmay be defined by working in local coordinates with a partition of unity. On the torus, f ormula (A.1) still makes sense : takinga∈C∞(T∗T2) is equivalent to taking a∈C∞(R2×R2), (2πZ)2-periodic with respect to the x-variable. Then the operator defined by (A.1) preserves the spac e of (2πZ)2-periodic distributions onR2, and hence D′(T2). We sometimes write, with D:=1 i∂, a(x,hD) = Oph(a). We also note that Op1(a) is the classical Weyl quantization, and that we have the relation a(x,hD) = Oph(a(x,ξ)) = Op1(a(x,hξ)). Theorem A.1. There exists a constant C >0such that for any a∈C∞(T∗T2)with uniformly bounded derivatives, we have /ba∇dblOp1(a)/ba∇dblL(L2(T2))≤C/summationdisplay α,β∈{0,1,2}2/ba∇dbl∂α x∂β ξa/ba∇dblL∞(T∗T2). Equivalently, this can be rewritten as /ba∇dblOph(a)/ba∇dblL(L2(T2))≤C/summationdisplay α,β∈{0,1,2}2h|β|/ba∇dbl∂α x∂β ξa/ba∇dblL∞(T∗T2). This precised version of the Calder´ on-Vaillancourt theorem is need ed in Section 13, and proved in [Cor75, Theorem Bρ] or [CM78, Th´ eor` eme 3]. Here in dimension two, this means that only |α|= 4 derivations are needed with respect to the space variable x. B Spectrum of P(z)for a piecewise constant damping (by St´ ephane Nonnenmacher) In this Appendix we provide an explicit description of some part of the spectrum of the damped wave equation (1.1) on T2, for a damping function proportional to the characteristic funct ion of a vertical 42strip. We identify the torus T2with the square {−1/2≤x <1/2,0≤y <1}. We choose some half-widthσ∈(0,1/2), and consider a vertical strip of width 2 σ. Due to translation symmetry of T2, we may center this strip on the axis {x= 0}. Choosing a damping strength /tildewideB >0, we then get the damping function b(x,y) =b(x) =/braceleftBigg 0,|x| ≤σ, /tildewideB, σ< |x| ≤1/2.(B.1) The reason for centering the strip at x= 0 is the parity of the problem w.r.t. that axis, which greatly simplifies the computations. We areinterested in the spectrum ofthe operator Ageneratingthe equation(1.1), which amounts to solving the eigenvalue problem P(z)u= 0,forP(z) =−∆+zb(x)+z2, z∈C, u∈L2(T2), u/\e}atio\slash≡0.(B.2) This spectrum consists in a discrete set {zj}, which is symmetric w.r.t. the horizontal axis: indeed, any solution ( z,u) admits a “sister” solution (¯ z,¯u). Furthermore, any solution with Im z/\e}atio\slash= 0 satisfies Rez=−1 2(u,bu)L2(T2) /ba∇dblu/ba∇dbl2 L2(T2),and thus −/tildewideB/2≤Rez≤0. (B.3) We may thus restrict ourselves to the half-strip {−/tildewideB/2≤Rez≤0,Imz>0}. Our aim is to find high frequency eigenvalues (Im z≫1) which are as close as possible to the imaginary axis. We will prove the following Proposition B.1. There exists C0>0such that the spectrum (B.3)for the damping function (B.1) contains an infinite subsequence {zi}such that Imzi→ ∞and|Rezi| ≤C0 (Imzi)3/2. The proof of the proposition will actually give an explicit value for C0, as a function of ˜B,σ. Proof.To study the high frequency limit Im z→ ∞we will change of variables and take z=i(1/h+/tildewideζ), whereh∈(0,1] will be a small parameter, while /tildewideζ∈Cis assumed to be uniformly bounded when h→0. The eigenvalue equation then takes the form (−h2∆+ih(1+h/tildewideζ)b)u=/parenleftbig 1+2h/tildewideζ(1+h/tildewideζ/2)/parenrightbig u. (B.4) Having chosen bindependent of y, we may naturally Fourier transform along this direction, that is look for solutions of the form u(x,y) =e2iπnyv(x),n∈Z. For each n, we now have to solve the 1-dimensional problem (−h2∂2/∂2 x+ih(1+h/tildewideζ)b(x))v=/parenleftbig 1−(2πhn)2+2h/tildewideζ(1+h/tildewideζ/2)/parenrightbig v. (B.5) Let us call Bdef=/tildewideB(1+h/tildewideζ), ζdef=/tildewideζ(1+h/tildewideζ/2). In terms of these parameters, the above equation reads: (−h2∂2/∂2 x+ihB1l{σ<|x|≤1/2}(x))v=Ev, withE= 1−(2πhn)2+2hζ. (B.6) Since we will assume throughout that /tildewideζ=O(1), we will have in the semiclassical limit B=/tildewideB+O(h),/tildewideζ=ζ(1−hζ/2+O(h2)). (B.7) At leading order we may forget that the variables B,ζare not independent from one another, and consider (B.6) as a bona fide linear eigenvalue problem. 43Since the function b(x) is even, we may separately search for even, resp. odd solutions v(x). Let us start with the even solutions. Since b(x) is piecewise constant, any even and periodic solution v(x) takes the following form on [ −1/2,1/2] (up to a global normalization factor): v(x) =/braceleftBigg cos(kx), |x| ≤σ, βcos/parenleftbig k′(1/2−|x|)/parenrightbig , σ<|x| ≤1/2,, (B.8) k=E1/2 h, k′=(E−ihB)1/2 h. (B.9) We notice that k,k′are defined modulo a change of sign, so we may always assume that Re k≥0, Rek′≥0. The factor βis obtained by imposing the continuity of vand of its derivative v′at the discontinuity point x=σ(we use the notation σ′def= 1/2−σ): cos(kσ) =βcos(k′σ′), −ksin(kσ) =βk′sin(k′σ′). The ratio of these two equations provides the quantization conditio n for the even solutions: tan(kσ) =−k′ ktan(k′σ′). (B.10) Similarly, any odd eigenfunction takes the form (modulo a global norm alization factor): v(x) =/braceleftBigg sin(kx), |x| ≤σ, βsgn(x) sin(k′(1/2−|x|)), σ<|x| ≤1/2,, so the associated eigenvalues should satisfy the condition tan(kσ) =−k k′tan(k′σ′). (B.11) We will now study the solutions of the quantization conditions (B.10) a nd (B.11), taking into account the relations (B.9) between the wavevectors k,k′and the energy E. To describe the full spectrum (which we plan to present in a separate publication), we would need to consider several r´ egimes, depending on the relative scales of Eandh. However, since we are only interested here in proving Proposition B.1, we will focus on the r´ egimeleading to the smallest pos sible values of |Im/tildewideζ|=|Rez|. What characterizes the corresponding eigenmodes v(x) ? From (B.3) we see that the mass of v(x) in the damped region, 2/integraltext1/2 σ|v(x)|2dx, should be small compared to its full mass. Intuitively, if such a mode were carrying a large horizontal “momentum” Re( hk) in the undamped region, it would then strongly penetrate the damped region, because the boundary at x=σis not reflecting. As a result, the mass in the damped region would be of the same order of magnitud e as the one in the undamped one. This hand-waving argument explains why we choose to investiga te the eigenmodes for which hk is the smallest possible, namely of order O(h). This implies that E= (hk)2=O(h2), which means that almost all of the energy is carried by the vertical momentum: hn= (2π)−1+O(h). The study of the full spectrum actually confirms that the smallest v alues of Im/tildewideζare obtained in this r´ egime. Eq.(B.9) implies that the wavevector k′in the damped region is then much larger than k: k′=(−ihB+(hk)2)1/2 h=e−iπ/4(B/h)1/2+O(h1/2). Imk′σ′≈ −σ′(B/2h)1/2is negative and large, so that tan( k′σ′) =−i+O(e2Im(k′σ′)), uniformly w.r.t. Re(k′σ′). 44Even eigenmodes In this situation the even quantization condition (B.10) reads tan(kσ) =ik′ k/parenleftbig 1+O(e−σ′(2B/h)1/2)/parenrightbig . (B.12) Since the r.h.s. is large, kσmust be close to a pole of the tangent function. Hence, for each int eger min a bounded interval10≤m≤Mwe look for a solution of the form km+1/2=π(m+1/2) σ+δkm+1/2,with|δkm+1/2| ≪1. The quantization condition (B.12) then reads σδkm+1/2+O((δkm+1/2)2) =ikm+1/2 e−iπ/4(B/h)1/2+O(h1/2)/parenleftbig 1+O(e−σ′(2B/h)1/2)/parenrightbig =⇒km+1/2=π(m+1/2) σ/parenleftBig 1+h1/2ei3π/4 σB1/2+O(h)/parenrightBig . Using (B.6), the corresponding spectral parameter ζis then given by ζn,m+1/2=(hkm+1/2)2+(2πhn)2−1 2h =(2πhn)2−1 2h+h 2/parenleftBigπ(m+1/2) σ/parenrightBig2 +h3/2/parenleftBigπ(m+1/2) σ/parenrightBig2ei3π/4 σB1/2+O(h2). From the assumptions on the quantum numbers n,m, we check that ζn,m+1/2=O(1). We may now go back to the original variables /tildewideζ,/tildewideB, using the relations (B.7). The spectral parameter /tildewideζhas an imaginary part Im/tildewideζn,m+1/2= Imζn,m+1/2(1−hReζn,m+1/2)+O(h2) =h3/2(π(m+1/2))2 σ3(2/tildewideB)1/2+O(h2).(B.13) Returningbacktothespectralvariable z,theaboveexpressiongivesastringofeigenvalues {zn,m+1/2} with Imzn,m+1/2=h−1+O(1), Rezn,m+1/2=−Im/tildewideζn,m+1/2. These even-parity eigenvalues prove Proposition B.1, and one can take for C0any value greater than(π/2)2 σ3(2/tildewideB)1/2. We remark that the leading order of km+1/2corresponds to the even spectrum of the operator −h2∂2/∂2 xon the undamped interval [ −σ,σ], with Dirichlet boundary conditions. The eigenmode vn,m+1/2associated with /tildewideζn,m+1/2is indeed essentially supported on that interval, where it resembles the Dirichlet eigenmode cos/parenleftbig xπ(1/2+m)/σ/parenrightbig . At the boundary of that interval, it takes the value vn,m+1/2(σ) = (−1)m+1ei3π/4h1/2π(m+1/2) σ/tildewideB1/2+O(h), and decays exponentially fast inside the damping region, with a “pene tration length” (Im k′)−1≈ (2h//tildewideB)1/2. From (B.3) we see that the intensity |vn,m+1/2(σ)|2∼Chpenetrating on a distance ∼h1/2exactly accounts for the size ∼h3/2=hh1/2of the Rezn,m+1/2. We notice that the smallest damping occurs for the state vn,1/2resembling the ground state of the Dirichlet Laplacian. 1Recall that we only need to study values Re k≥0. 45Odd eigenmodes For completeness we also investigate the odd-parity eigenmodes wit hk=O(1). The computations are very similar as in the even-parity case. The odd quantization con dition reads in this r´ egime tan(kσ) =ik k′/parenleftbig 1+O(e−(2B/h)1/2)/parenrightbig . (B.14) The r.h.s. is then very small, showing that σkis close to a zero of the tangent, so we may take km=πm/σ+δkmwith|δkm| ≪1 and 0≤m≤M. We easily see that the case m= 0 does not lead to a solution. For the case m>0 we get δkm=e3iπ/4h1/2πm σ2B1/2+O(h), and thus km=πm σ/parenleftBig 1+h1/2e3iπ/4 σB1/2+O(h)/parenrightBig ,1≤m≤M. These values kmapproximately sit on the same “line” {s(1 +h1/2e3iπ/4 σB1/2), s∈R}as the values km+1/2corresponding to the even eigenmodes, both types of eigenvalues appearing successively. The corresponding energy parameter /tildewideζn,msatisfies Im/tildewideζn,m=h3/2(πm)2 σ3(2/tildewideB)1/2+O(h2). (B.15) As in the even parity case, the eigenmodes vn,mare close to the odd eigenmodes sin/parenleftbig xπm/σ/parenrightbig of the semiclassical Dirichlet Laplacian on [ −σ,σ], and penetrate on a length ∼h1/2inside the damped region. The case of the square If the torus is replaced by the square [ −1/2,1/2]×[0,1] with Dirichlet boundary conditions, with the same damping function (B.1), the eigenmodes P(z) can as well be factorized into u(x,y) = sin(2πny)v(x), withn∈1 2N\0, andv(x) must be an eigenmode of the operator (B.6) vanishing at x=±1/2. We notice that the odd-parity eigenstates (B) satisfy this boun dary conditions, so the eigenvalues zn,m(with real parts given by (B.15)) belong to the spectrum of the dam ped Dirichlet problem. Similarly, in the case of Neumann boundary conditions the eigenmodes factorize as u(x,y) = cos(2πny)v(x), withn∈1 2N. 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2302.06402v2.Damping_of_gravitational_waves_in_f_R__gravity.pdf

arXiv:2302.06402v2 [gr-qc] 24 Oct 2023Damping of gravitational waves in f(R)gravity Haiyuan Feng1,∗and LaiYuan Su1,† 1Department of Physics, Southern University of Science and T echnology, Shenzhen 518055, Guangdong, China Abstract We study the damping of f(R) gravitational waves by matter in flat spacetime and in expan ding universe. Intheformercase, we findthat theLandaudampingo f scalar modein f(R) theoryexists, whilethat of thetensor modein general relativity does not; wealso present theviscosity coefficients and dispersion relations of the two modes. In the later case, we investigate the evolution of tensor and scalar modes in Friedmann-Robertson-Walker (FRW) cosm ology with a matter distribution; by considering the case of f(R) =R+αR2, we analysis the influence of parameter αon wave damping, and put restrictions on its magnitude. ∗Email address: 406606114@qq.com †12131268@mail.sustech.edu.cn 1I. INTRODUCTION The detection of gravitational waves (GW) in the universe gives a sig nificant promotion to the development of modern astronomy and physics. Continuous observations provide crucialdatastorestrict characteristicsofastrophysical sour ces[1–9], aswell astotestgeneral relativity (GR) [10–15]. The interaction of gravitational waves with m atter, though in most cases ignored, has been investigated throughout the history. Ha wking first calculated the damping rate of gravitational waves as γ= 16πGηby viewing matter as a perfect fluid, with ηthe viscosity [16]. Subsequently, Ehlers et.al proved in general tha t gravitational waves traveling through perfect fluids do not suffer from dispersion or dis sipation [17]. While in the collisionless limit, the damping rate of gravitational waves by non- relativistic particles is shown to be related to the particle velocity and the number density [18]. By linearizing the Boltzmann equation and taking into account the collision term, a u nified treatment for damping from collision and the Landau damping is given [19]. Landau damp ing, firstly introduced to investigate the dispersion relationship in the plasma sy stem [20–22], and then generalized to the research of large-scale galaxy clusters [23, 24], is proved to be vanish for gravitational waves in flat spacetime; while in some modified gravity th is is not always the case for the extra modes, as we will shown in f(R) gravity. On the other hand, gravitational waves played a significant role in the evolution of early universe. The o bservation of cosmic tensor fluctuations by measurements of microwave background p olarization is the finest approach to check an inflationary universe. Weinberg has already s ketched out the main approach for calculating the influence of collisionless three massless neutrinos [25]. The perturbations of neutrinos by GW was estimated using the collisionles s Boltzmann equation; the result of the analysis showed that the damping effect of neutrin os on the GW’s spectrum can be rather large. This approach was used to study cosmic gravit ational waves in the radiation-dominatedera. BasedonWeinberg’sconclusions, asetof analyticalsolutionsusing modal expansion with spherical Bessel functions as bases were de veloped [26]. Following that, the damping effect of gravitational waves in cold dark matter was studied when mass- relativistic particles were included [27]. As an attempt to restore the non-renormalization of GR, as well as to alleviate the cosmological constant problem, f(R) theory is proposed. The model has two significant advantages: the actions are sufficiently general to encompass so me of the fundamental prop- 2erties of higher-order gravity while remaining simple enough to be eas ily handled; it appears to be the only ones capable of averting the long-known and catastr ophic Ostrogradski insta- bility [28]. Especially, choosing the simplest f(R) =R+αR2(α >0) alone can provide an explanation for the universe’s accelerated expansion [29], and can b eregarded as a candidate for an inflationary field. The action of f(R) theory has the following form, S[gµν] =1 2κ2/integraldisplay d4x√−gf(R)+/integraldisplay d4x√−gLm, (1) withκ2= 8πG,Lmthe Lagrangian of matter. The field equations can be obtained by varying the above action [30], F(R)Rµν−1 2f(R)gµν−∇µ∇νF(R)+gµν/squareF(R) =κ2T(M) µν, (2) whereF(R) =df(R) dR, and/squareis the d’Alembertian operator. The energy-momentum ten- sorT(M) µν≡ −2√−gδLm δgµνsatisfies the continuity equation ∇µT(M)µν= 0. We rearrange the preceding equations to get Gµν=κ2/parenleftbig T(eff) µν+T(M) µν/parenrightbig κ2T(eff) µν≡gµν 2(f(R)−R)+∇µ∇νF(R)−gµν/squareF(R)+(1−F(R))Rµν,(3) with Einstein tensor Gµν=Rµν−1 2gµνR, and fulfills the Bianchi identity ∇µGµν= 0. It can be proved that the contribution of curvature T(eff) µνalso obey ∇µT(eff)µν= 0. It is possible to get the trace of (3) as 3/squareF(R)+F(R)R−2f(R) =κ2T(M), (4) which clearly shows the difference from Einstein’s trace formula R=−κ2T. The presence of terms /squareF(R) leads to additional propagation degrees of freedom in the f(R) theory. To investigate the equation of f(R) gravitational waves, the metric gµνand Riemann curvature scalar Rin Minkowski spacetime is perturbed as gµν=ηµν+hµν R=R0+δR,(5) where tensor perturbation hµνis restricted by |hµν| ≪ |ηµν|. Background curvature (Minkowski) and scalar perturbation are denoted by R0andδR, respectively. As can 3be shown, different from gravitational waves in GR, the perturbat ion has the form hµν= ¯hTT µν+hS µν, where¯hTT µνrepresents the transverse-traceless (TT) part of the pertur bation. It satisfies∂i¯hTT ij= 0,¯hTT ii= 0, and hS µν=−φηµν(φ≡F′(R0)δR F(R0)) represents the scalar degrees of freedom [30]. The linearized field equations [31–35] is given by /square¯hTT ij=−2κ′2Π(1) ij /squareφ−M2φ=κ′2 3T(1),(6) where M2≡1 3/parenleftbiggF(R0) F′(R0)−R0/parenrightbigg =F(0) 3F′(0)(7) is the square of the effective mass, κ′2≡κ2 F(0), and Π(1) ijis the linear part of the anisotropic part of the spatial components of energy-momentum tensor Tij/parenleftbig Ti j= Πi j+1 3δi j/summationtext3 k=1Tk k/parenrightbig . It couples with gravitational waves and satisfies Π ii= 0,∂iΠij= 0. However, in f(R) gravity, since the emergence of the degrees of freedom R, there is the presence of an extra scalar mode, commonly known as the breathing mode. It is obvious fr om equation (6) that when the effective mass Mapproaches infinity, the system no longer has the excitation of the scalar mode and returns to the tensor mode of GR gravity. As a result, the number of polarizations in f(R) gravity is three [36, 37]. In this paper, by taking into account contributions from the collision term, we investigate the damping of f(R) gravitational waves in the presence of medium matter and determ ine the dispersion relation. In Section II, we introduces linearized f(R) theory and provides wave equations for tensor and scalar modes in Minkowski spacetime . In section III, we ap- ply kinetic theory to investigate the first-order approximation of t he relativistic Boltzmann equation. We calculate the anisotropic part of the spatial compone nts of energy-momentum tensor and derive the dispersion relation of two modes using the rela xation time approxima- tion. We also derive damping coefficients in collision-dominant case and L andau damping in the collisionless limit. In section IV, within the context of FRW inflation ary cosmology, we present the wave equation of tensor and scalar disturbances in the early universe by usingf(R) theory, and investigate the specific case of f(R) =R+αR2. The damping of gravitational waves by neutrinos with mass is also investigated, and the collision term is proved to be negligible. 4II. DISPERSION AND DAMPING OF GRAVITATIONAL WAVES BY RELA- TIVISTIC BOLTZMANN GASES IN FLAT SPACETIME To calculate the anisotropic stress Π ijand the energy-momentum tensor trace Tinduced by gravitational waves, we consider the relativistic Boltzmann equa tion [38–42] pµ∂f ∂xm−gijΓi µν∂f ∂pj=C[f], (8) where distribution function f(xi,pj) as the particle positions and canonical momenta de- scribestheprobabilityofthespatialdistribution. Γi µνistheconnectioncoefficientand pµrep- resents the four-momentum of a single particle with on-shell condit iongµνpµpν=−m2.C[f] is the collision term. To analyse the dynamics of the Boltzmann equatio n, we consider the relaxation time approximation [43], therefore C[f] can be represented as C[f] =−pµuµ τc(fh−f), (9) τcis the particle’s average collision time, and uµdenotes themacroscopic fluid’s four-velocity [44]. Therefore, four-velocity could currently be written as uµ= (1,0,0,0) in the fluid’s rest reference frame. The distribution function of the local equilib rium in the presence of gravitational waves, denoted by fh, is given by fh=g e−pµuµ T±1, (10) where±corresponds to fermions or bosons, gis the number of degrees of freedom for the varieties of single particles, and Tis the temperature. Using geodesic equation of particles, (8) can be simplified by ∂f ∂t+pm pt∂f ∂xm+dpm dt∂f ∂pm=1 τc(fh−f). (11) with dpm dt=1 2∂gµν ∂xmpµpν p0. (12) We will apply the dynamic perturbation approach to determine the fo rmulation of the induced energy-momentum tensor. Firstly, starting with hµν=¯hTT µν−φηµν, the perturbation on-shell condition can be expressed as ǫ=ǫ0+δǫ δǫ=hµνpµpν 2ǫ0=−¯hTT ijpipj−m2φ 2ǫ0,(13) 5with p0≡ǫ0=/radicalbig m2+pipi, (14) we have adopted the first-order perturbation hµνηµαηνβ=−hαβ. Substituting (13) into (12) to derive dpm dt=1 2p0/parenleftbigg pkpl∂¯hTT kl ∂xm+m2∂φ ∂xm/parenrightbigg . (15) Secondly, we handletheperturbeddistributionfunction f=f0(p)+δf(xi,pj,t)according to[19]. By ignoring all higher order terms, the linearized Boltzmann eq uation is obtained ∂δf ∂t+pm p0∂δf ∂xm+1 2p0/parenleftbigg pkpl∂¯hTT kl ∂xm+m2∂φ ∂xm/parenrightbigg∂f0(p) ∂pm=−1 τc(δf−δfh),(16) it is worth emphasizing that δfhrepresents the deviation between the distribution function after local equilibrium and the absence of gravitational waves, whic h could be expanded into a first-order small quantity as δfh=∂f0 ∂ǫδǫby Taylor formula. By Fourier transforming ¯hTT ij(/vector r,t) =ei/vectork·/vector r−iωt¯hTT ij(ω,/vectork) andφ(/vector r,t) =ei/vectork·/vector r−iωtφ(ω,/vectork), we derive the solution of (16). δf(ω,/vectork) =f′(p) 2p¯hTT ijpipj/parenleftBig −1 τc−i/vector p·/vectork p0/parenrightBig −m2φ/parenleftBig i/vector p·/vectork p0+1 τc/parenrightBig /parenleftBig −iω+i/vector p·/vectork p0+1 τc/parenrightBig , (17) wheref′(p)meanstoderivationwithrespectto p. Conclusively, sincetheinducedanisotropic stress tensor is assessed in terms of the distribution function f, the dynamical system is fully characterised [19, 27, 45], Π(1) ij=/integraldisplayd3p (2π)3pipj ǫ0¯δf T(1)=−m2/integraldisplayd3p (2π)31 ǫ0¯δf,(18) where¯δf≡δf−δfhshould be interpreted as the effect of the distribution function’s own variation, since the total shift is the sum of the distribution fun ction’s own and the transformation caused by gravitational waves. We can determine the expression by inserting (17) into (18), which follows Π(1) ij=¯hTT kl/integraldisplayd3p (2π)3pkplpipjf′ 0(p) 2pǫ0 −iω −iω+i/vector p·/vectork p0+1 τc , (19) 6and T(1)=−m2/integraldisplayd3p (2π)3f′ 0(p) 2pǫ0 ¯hTT klpkpl −iω −iω+i/vector p·/vectork p0+1 τc −m2φ iω −iω+i/vector p·/vectork p0+1 τc =m4φ(ω,/vectork)/integraldisplayd3p (2π)3f′ 0(p) 2pǫ0 iω −iω+i/vector p·/vectork p0+1 τc ,(20) based on the angular integration, the contribution of gravitationa l waves in (20) is zero. The first term on the right side of (19) can be shown to be proportional to¯hTT ij, it follows Π(1) ij=¯hTT ij/integraldisplayd3p (2π)3(pipj)2f′ 0(p) pǫ0 −iω −iω+i/vector p·/vectork p0+1 τc , (21) wherei/negationslash=jand only x,ycan be taken. We could produce the dispersion relation of gravitational waves in relativistic particle flow by replacing (20) and ( 21) into (6). ω2−k2+2κ′2/integraldisplayd3p (2π)3(pipj)2f′ 0(p) pǫ0 −iω −iω+i/vector p·/vectork p0+1 τc = 0 ω2−k2−M2+m4κ′2 6/integraldisplayd3p (2π)3f′ 0(p) pǫ0 −iω −iω+i/vector p·/vectork p0+1 τc = 0.(22) Two damping mechanisms must be addressed in order to get mode dam ping from dis- persion. Landau damping and collision-dominated hydrodynamic damp ing are two different typesofgravitationalwavedampingthataredeterminedbytheima ginarypartofthesource. Landau damping is the excitation of two real particle-hole pairs caus ed by decay of the mode without considering thecollisioninto account. Fromthe(20), (21), andthecollisionless limit 1 τc→0 can be derived, ℑ/parenleftBigg Π(1) ij ¯hTT ij/parenrightBigg =−πω/integraldisplayd3p (2π)3(pipj)2f′ 0(p) pǫ0δ/parenleftBigg ω−/vector p·/vectork p0/parenrightBigg ℑ/parenleftbiggT(1) φ/parenrightbigg =πωm4/integraldisplayd3p (2π)3f′ 0(p) 2pǫ0δ/parenleftBigg ω−/vector p·/vectork p0/parenrightBigg ,(23) The preceding formula shows that the Landau damping phenomenon happens only when p0=|p|cosθ, implying that the particles must be massless and move along the wave direc- tion to contribute. However, In the flat spacetime, gravitational waves will not encounter Landau damping because ( pipj)2= (pxpy)2. It is worth noting that the Landau damping 7of scalar modes only contributes when the particle motion direction c orresponds with the wave propagation direction. To investigate another damping mecha nism, we focus at the collision-dominated region ( ω≪1 τc). (20) and (21) will be wirtten as ℑΠ(1) ij=¯hTT kl(ω,/vectork)/integraldisplayd3p (2π)3pkplpipjf′ 0(p) 2pǫ0−ω τc/parenleftBig ω−/vector p·/vectork p0/parenrightBig2 +1 τ2c ≈ −ωτc¯hTT ij(ω,/vectork) 15/integraldisplayd3p (2π)3p3f′ 0(p) ǫ0=−τcωη1¯hTT ij(ω,/vectork),(24) and ℑT(1)=m4φ(ω,/vectork)/integraldisplayd3p (2π)3f′ 0(p) 2pǫ0ω τc/parenleftBig ω−/vector p·/vectork p0/parenrightBig2 +1 τ2c ≈m4ωτcφ(ω,/vectork)/integraldisplayd3p (2π)3f′ 0(p) 2pǫ0=τcωη2φ(ω,/vectork),(25) The collision-dominated viscosity coefficient under the relaxation time approximation are η1 andη2, which follows η1≡τc 15/integraldisplayd3p (2π)3p3f′ 0(p) ǫ0 η2≡m4τc/integraldisplayd3p (2π)3f′ 0(p) 2pǫ0.(26) The viscosity coefficient provided by the given equations is obviously b ased on the dis- tribution function of the equilibrium state and the collision relaxation t ime. These two components are also the primary causes of the damping of tensor a nd scalar modes. In this section, we calculate the solution of the anisotropic stress t ensor induced by gravitational waves by using Fourier transform of the linearized Bo ltzmann equation, and we achieve the coefficient of damping through adopting two mechanis ms of gravitational wave damping. III. DAMPING OF TENSOR AND SCALAR MODES IN COSMOLOGY The detection of cosmic tensor fluctuations by measurements of m icrowave background polarisation is widely expected to provide a uniquely valuable check on t he validity of basic inflationary cosmology. Particularly, gravitational wave spectra g enerated by other nonin- flationary sources have also been proposed [46]. As a result, any fin ding of gravitational wave spectrum would be a tremendously valuable tool in studying the early cosmology. The f(R) wave equation will be discussed in this section. 8A. The wave equations of tensor and scalar mode We will use conformal coordinates to investigate the evolution of te nsor modes in the spatially flat FRW universe. The line element with a perturbation metric is represented by ds2=a2(τ)/bracketleftbig −(1−φ)dτ2+/parenleftbig δij−δijφ+¯hTT ij/parenrightbig dxidxj/bracketrightbig , (27) The cosmological equation satisfied by the gravitational wave tens or mode could be deter- mined by[47, 48], ¨¯hTT ij+(2+aM)H(τ)˙¯hTT ij−∇2¯hTT ij=2κ2 F(R0)a2(τ)Π(1) ij, (28) and the scalar mode ¨φ+/parenleftBigg 2H(τ)+˙F F/parenrightBigg ˙φ−∇2φ+/parenleftBigg a2(τ)M2+4H(τ)˙F F+2¨F F/parenrightBigg φ=−κ2 3F(R0)a2(τ)T(1).(29) a(τ) represents the cosmic evolution factor,˙¯hTT ijand˙φdenote the derivative with respect to the conformal time τ, andHindicates the Hubble constant. aMis defined asF′(R0)˙R F(R0)H, and the scenario of aM=φ= 0 is accompanied by GR gravitational waves. We concentrate on the conclusions on the right-hand side of the above equation, as we did in the flat spacetime. The disturbance of Boltzmann equation (16) is 1 a(τ)∂δf ∂τ+pm∂mδf a(τ)pτ+1 a(τ)dpm dτ∂f0 ∂pm=−1 τc(δf−δfh), (30) which can be simplify to /parenleftbigg∂ ∂τ+vm∂m+1 ¯τc/parenrightbigg δf=δfh ¯τc−dpm dτ∂f0 ∂pm, (31) wherevm≡pm pτ=pm√ pipi+m2a2corresponds to the three-velocity of particles. ¯ τc≡τc a(τ)is collision time in cosmology. Thedpm dτis expressed by dpm dτ=1 2∂mgµνpµpν pτ =∂m¯hTT ijpipj+m2a2(τ)∂mφ 2pτa2(τ).(32) Similarly, the on-shell condition and its perturbation are denoted by ǫ=ǫ0+δǫ ǫ0≡pτ=/radicalBigg m2 a2(τ)+pipi a4(τ) δǫ=δgµνpµpν 2a2(τ)ǫ0=−¯hTT ijpipj−m2a2(τ)φ 2a4(τ)ǫ0,(33) 9The spatial Fourier transform is used to reduce the ultimate Boltzm ann equation (30) to /parenleftbigg∂ ∂τ+Q(τ)/parenrightbigg δf(τ,/vectork) =−f′ 0(p)Q(τ) 2p/bracketleftBig ¯hTT ij(τ,/vectork)pipj+m2a2(τ)φ(τ,/vectork)/bracketrightBig ,(34) with Q(τ)≡i/vector v·/vectork+1 ¯τc(35) we can derive the particular solution of the first-order differential equation (34) δf(τ) =−/integraldisplayτ τ0/parenleftBig ¯hTT ij(τ′,/vectork)pipj+m2a2(τ′)φ(τ′,/vectork)/parenrightBigf′ 0(p) 2p∂e−Λ(τ,τ′) ∂τ′dτ′ Λ(τ,τ′) = Λ1(τ,τ′)+icosθkΛ2(τ,τ′)≡/integraldisplayτ τ′1 ¯τc(τ′′)dτ′′+icosθk/integraldisplayτ τ′v(τ′′)dτ′′,(36) whereτ0depicts the initial assertion at which the system is in equilibrium, and f0is the partial function in the equilibrium. f0=g ept T±1=g e−pτ a0T0±1, (37) withpτ=−/radicalbig m2a2+p2, and the second equality holds for our normalization that the present day scale factor is a0= 1.T0is the current background radiation temperature. The perturbed anisotropic part is a generalization of (18), which can be defined as [27] Π(1) ij=/integraldisplayd3p (2π)3pipj√−g(−pτ)¯δf=/integraldisplayd3p (2π)3pipj a4/radicalbig m2a2+p2¯δf T(1)=−m2/integraldisplayd3p (2π)31√−gǫ0¯δf=−m2/integraldisplayd3p (2π)31 a2/radicalbig m2a2+p2¯δf,(38) with ¯δf=/integraldisplayτ τ0e−Λ(τ,τ′)∂ ∂τ′/bracketleftbiggf′ 0(p) 2p/parenleftBig ¯hTT ij(τ′,/vectork)pipj+m2a2(τ′)φ(τ′,/vectork)/parenrightBig/bracketrightbigg dτ′.(39) The anisotropic sress tensor Π(1) ijandT(1), which follows Π(1) ij=/integraldisplayd3p (2π)3pipjpkpl 2pa4(τ)/radicalbig m2a2+p2/integraldisplayτ τ0e−Λ(τ,τ′)∂ ∂τ′/bracketleftbig f′ 0(p)¯hTT kl(τ′)/bracketrightbig dτ′ T(1)=−m4/integraldisplayd3p (2π)31 2pa2/radicalbig m2a2+p2/integraldisplayτ τ0e−Λ(τ,τ′)∂ ∂τ′/bracketleftbig f′ 0(p)a2(τ′)φ(τ′)/bracketrightbig dτ′,(40) The integral formula would be used to simplify the expression[26, 27], /integraldisplay2π 0dφpipjpkpl p4=π(1−cos2θ)2 4(δijδkl+δikδjl+δilδjk) K(x)≡1 16/integraldisplay1 −1/parenleftbig 1−cos2θ/parenrightbig2eixcosθdcosθ,(41) 10As a consequence, Π(1) ijandT(1)are ultimate able to represented as Π(1) ij=/integraldisplayp5dp 2π2a4/radicalbig m2a2+p2/integraldisplayτ τ0K(kΛ2(τ,τ′))e−Λ1(τ,τ′)∂ ∂τ′/bracketleftbig f′ 0(p)¯hTT ij(τ′)/bracketrightbig dτ′ T(1)=−m4/integraldisplaypdp 4π2a2/radicalbig m2a2+p2/integraldisplayτ τ0sinkΛ2(τ,τ′) kΛ2(τ,τ′)e−Λ1(τ,τ′)∂ ∂τ′/bracketleftbig f′ 0(p)a2(τ′)φ(τ′)/bracketrightbig dτ′. (42) K(x) =j0(x) 15+2j2(x) 21+j4(x) 35is a linear combination of spherical Bessel functions [26]. When m= 0, the right side of the wave equation returns to the previous res ult, although with additionalcollisioncontributions. Sincetheexistenceof (40),twom odesinvolvestheLandau damping phenomena induced by the anisotropic tensor of matter in t he universe’s evolution. We primarily figure out the contribution of the matter term on the rig ht-hand side of the wave equation by linearizing the Boltzmann equation in the FRW metric t o derive a general solution. In the subsequent sections, we will go over the specific f(R) model and investgate the numerical results of two modes travel through decoupled neu trino systems. B. Numerical solution of Damping from neutrinos Neutrinos are one of the fundamental Fermi particles participate d the original Big Bang’s weakandgravitationalinteractions. Sincethepositiveandnegativ eprocessesoftheneutrino interaction before decoupling approach chemical equilibrium, neutr inos satisfy the equilib- rium state distribution function in earlier universe. Weinberg’s origina l study, as well as the majority of the early research, concentrated on the effect o f three massless neutrinos. Recent cosmology investigations, nevertheless, have shown indica tions of departures from the traditional cosmological value of three effective neutrino degr ees of freedom [49]. Ex- periments on neutrino oscillations demonstrate that it has mass, wh ich will have an effect on gravitational wave damping [50]. The Landau damping phenomenon occurs when the k of two modes are longer than the cosmic horizon keq≡a(eq)H(eq) (τeqrepresents the time when the proportion of radiation and matter is the same). This sect ion we focuses on the evolution of two modes as f(R) =R+αR2enters the period of matter-radiation dominance. After the Fourier transform of the spatial part, (28)(29) beco mes 11 ¨¯hTT ij(u)+2H(u)˙¯hTT ij(u)+2α˙R 1+2αR˙¯hTT ij(u)+¯hTT ij(u) =2κ2T4 0 k2(1+2αR)a2(u)Π(1) ij Π(1) ij=/integraldisplay∞ 0x5dx 2π2a4/radicalBig m2a2(u) T2 0+x2/integraldisplayu 0K(Λ2(u,u′))e−Λ1(u,u′)∂ ∂u′/bracketleftbiggdf0(x,u′) dx¯hTT kl(u′)/bracketrightbigg du′, (43) and ¨φ(u)+/parenleftBigg 2H(u)+2α˙R 1+2αR/parenrightBigg ˙φ(u)+/parenleftBigg 1+a2(u) 6αk2+8αH(u)˙R 1+2αR+4α¨R 1+2αR/parenrightBigg φ(u) =−κ2a2(u) 3k2(1+2αR)T(1) T(1)=−m4/integraldisplayxdx 4π2a2/radicalBig m2a2 T2 0+x2/integraldisplayu 0sinΛ2(u,u′) Λ2(u,u′)e−Λ1(u,u′)∂ ∂u′/bracketleftbiggdf0(x,u′) dxa2(u′)φ(u′)/bracketrightbigg du′, (44) with Λ1(u,u′) =/integraldisplayu u′1 k¯τc(u′′)du′′ Λ2(u,u′) =/integraldisplayu u′v(u′′)du′′,(45) where the dimensionless independent variable u≡kτ,x≡p T0. We record the initial moment kτ0as 0 when the wave enters the matter and the radiation. In the per iod dominated by matter and radiation, a(u) is a(u) =u2 u2 0+2u u0√aeq, (46) with u0≡2k√ΩMH0. (47) In standard cosmic evolution, there are three generations of neu trinos corresponding to aeq=1 3600, ΩM= 0.3 [51, 52]. According to the above equation, (45) can be stated as Λ1(u,u′) =u3−u′3 3u2 0kτc+(u2−u′2)√aeq u0kτc Λ2(u,u′)≈(u−u′)/parenleftbigg 1−m2 2x2T2 0/parenrightbigg .(48) The neutrino mass is Taylor extended as a small quantity, reverting to previous gravita- tional wave damping conclusions by m= 0. Nonzero neutrino masses will add an additional k dependence to damping since free streaming, and thus damping will be lowered when the 12temperature is of order of the mass. While neutrinos are relativistic , the k modes that come inside the horizon and contribute considerably to the overall energ y density will be damped more. The following figure depicts the numerical results of (43) and (44). It is seen that there is a major difference in the varying trend of the χ(u)≡¯hTT ij(u) ¯hTT ij(0)with respect to kτ. Askτgoes up, the χ(u) decreases. Note that the m=0 scenario drops faster than the m = 1ev situation in the right panel of the figure; however, The figure o n the left displays the influence on τwhen the model parameters are scaled downwards by orders of ma gnitude, with the oscillation being rapid shown by the blue line ( α= 0). When the magnitude of α >1018m2, there is no oscillation decay solution for this waveform, so the best limit range for the parameters does not exceed 1018m2. In particular, we can clearly see from the third figurethat theevolution of thescalar mode decays almost complete ly at u=1. When m=1ev, the influence of neutrinos on it slows down its oscillation frequency. 0 2 4 6 8 10 12 14 16 18 u=k-0.200.20.40.60.81 =0m2 =1015m2 =1018m2 0 2 4 6 8 10 12 14 16 18 20 u=k-0.200.20.40.60.81 m=0ev m=1ev no-damping FIG. 1. The two diagrams depict the waveform’s evolution ove r time. We fix the parameters kτc= 100,u0= 100 and discuss the decay of gravitational waves. We find fro m the left figure that the phase of the wave is translate and the attenuation of the wave is slowed down as the parameter αincreases. The figure on the right shows that m=0 ev decays fas ter than m=1ev, and only when the temperature and mass are in the same order of mag nitude, the contribution of mass will increase the decay. 130.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 u=k00.20.40.60.81 m=0ev m=1ev no-damping FIG. 2. The third picture mainly describes the evolution of t he scalar mode at α= 1018m2, it can be seen that neutrinos with a mass of 1ev slow down the oscilla tion frequency of the wave. It also show that the scalar mode is almost completely attenuated at u=1, which also makes the detection of this mode more difficult. IV. CONCLUSION AND DISCUSSION In this work, through the application of kinetic theory, we were able to discover the Landau damping phenomenon of the scalar mode in the f(R) gravity in the flat space-time. We also discuss how the early universe’s inflation dampened primordial gravitational waves, and we impose a restriction as to how the parameter αinf(R) =R+αR2could be. In sectionII, weintroducethelinearized f(R)model andprovides wave equationsforthetensor and scalar modes. We can clearly see that the background metric ha s absolute influence on the mass of the scalar mode. The theory degenerates into genera l relativity when the effect mass term reaches infinity, while there is no excitation of this mode. In section III, we calculate the perturbed form of the Boltzmann e quation, derive the perturbed solution of momentum space, and substitute it into the t ransverse-traceless part of the anisotropic stress tensor to establish the dispersion relatio n of scalar and tensor mode. We also investigated the damping coefficient in the collision dominated re gion and Landau damping in the collisionless limit. We discovered that the contribution of tensor mode’s Landau damping is zero, however, the scalar mode exists only when t he particle motion direction coincides with the direction of wave propagation. 14Finally, we examine the Boltzmann equation satisfied by the perturba tion in the FRW scenariooftheconformalcoordinatesystem. Weobtainthewave equationofthetensormode with damping under the f(R) =R+αR2. Furthermore, after passing through neutrinos decoupling, wenumerically solvethedecayofprimordialgravitationa lwaves createdbyearly inflation. We discover that the case of neutrinos oscillate more rapid ly than the undamped case, but the variation between the two is demonstrated to be neg ligible in the case of neutrino’s mass m=0,1ev. 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1906.07297v2.Controlling_acoustic_waves_using_magnetoelastic_Fano_resonances.pdf

Controlling acoustic waves using magneto-elastic Fano resonances Controlling acoustic waves using magneto-elastic Fano resonances O. S. Latcham,1Y. I. Gusieva,2A. V. Shytov,1O. Y. Gorobets,2and V. V. Kruglyak1,a) 1)University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom 2)Igor Sikorsky Kyiv Polytechnic Institute, 37 Prosp. Peremohy, Kyiv, 03056, Ukraine (Dated: 5 May 2020) We propose and analyze theoretically a class of energy-efficient magneto-elastic devices for analogue signal processing. The signals are carried by transverse acoustic waves while the bias magnetic field controls their scattering from a magneto-elastic slab. By tuning the bias field, one can alter the resonant frequency at which the propagating acoustic waves hybridize with the magnetic modes, and thereby control transmission and reflection coefficients of the acoustic waves. The scattering coefficients exhibit Breit-Wigner/Fano resonant behaviour akin to inelastic scattering in atomic and nuclear physics. Employing oblique incidence geometry, one can effectively enhance the strength of magneto- elastic coupling, and thus countermand the magnetic losses due to the Gilbert damping. We apply our theory to discuss potential benefits and issues in realistic systems and suggest routes to enhance performance of the proposed devices. Optical and, more generally, wave-based computing paradigms gain momentum on a promise to replace and com- plement the traditional semiconductor-based technology.1The energy savings inherent to non-volatile memory devices has spurred the rapid growth of research in magnonics,2,3in which spin waves4are exploited as a signal or data carrier. Yet, the progress is hampered by the magnetic loss (damping).5,6In- deed, the propagation distance of spin waves is rather short in ferromagnetic metals while low-damping magnetic insulators are more difficult to structure into nanoscale devices. In con- trast, the propagation distance of acoustic waves is typically much longer than that of spin waves at the same frequencies.7 Hence, their use as the signal or data carrier could reduce the propagation loss to a tolerable level. Notably, one could con- trol the acoustic waves using a magnetic field by coupling them to spin waves within magnetostrictive materials.8–10To minimize the magnetic loss, the size of such magneto-acoustic functional elements should be kept minimal. This implies coupling propagating acoustic waves to confined spin wave modes of finite-sized magnetic elements. As we show below this design idea opens a route towards hybrid devices combin- ing functional benefits of magnonics2,3with the energy effi- ciency of phononics.7,11,12 The phenomena resulting from interaction between coher- ent spin and acoustic waves have already been addressed in the research literature: the spin wave excitation of prop- agating acoustic waves7,13–15and vice versa,8,16–18acous- tic parametric pumping of spin waves,19–21magnon-phonon coupling in cavities22–24and mode locking,25magnonic- phononic crystals,26,27Bragg scattering of spin waves from a surface acoustic wave induced grating,28–30topological prop- erties of magneto-elastic excitations,15,31acoustically driven spin pumping and spin Seebeck effect,32,33and optical excita- tion and detection of magneto-acoustic waves.34–40However, studies of the interaction between propagating acoustic waves and spin wave modes of finite-sized magnetic elements, which are the most promising for applications, have been relatively scarce to date.10,34,36,39 Here, we explore theoretically the class of magneto- a)Electronic mail: V .V .Kruglyak@exeter.ac.uk θHB M I Tω Rω M NM NM FIG. 1: The prototypical magneto-elastic resonator is a thin magnetic slab (M) of width d, biased by an external field HB, and embedded into a non-magnetic (NM) matrix. The acoustic wave with amplitude Iincident at angle qinduces precession of the magnetisation vector Mvia the magneto-elastic coupling. As a result, the wave is partly transmitted and reflected, with respective amplitudes Tw andRw. acoustic devices in which the signal is carried by acoustic waves while the magnetic field controls its propagation via the magnetoelastic interaction in thin isolated magnetic in- clusions as shown in Fig. 1. By changing the applied mag- netic field, one can alter the frequency at which the incident acoustic waves hybridize with the magnetic modes of the in- clusions. Thereby, one can control the acoustic waves by the resonant behaviour of Breit-Wigner and Fano resonances in the magnetic inclusion.41We find that the strength of the res- onances is suppressed by the ubiquitous magnetic damping in realistic materials, but this can be mitigated by employing oblique incidence geometry. To compare magneto-acoustic materials for such devices, we introduce a figure of merit. The magneto-elastic Fano resonance is identified as most promis- ing in terms of frequency and field tuneability. To enhance res- onant behaviour, we explore the oblique incidence as a means by which to enhance the figure of merit. We consider the simplest geometry in which magneto-arXiv:1906.07297v2 [physics.app-ph] 4 May 2020Controlling acoustic waves using magneto-elastic Fano resonances 2 elastic coupling can affect sound propagation. A ferromag- netic slab ("magnetic inclusion") of thickness d, of the or- der of 10 nm, is embedded within a non-magnetic medium (Fig.1). The slab is infinite in the YZplane, has satu- ration magnetization Ms, and is biased by the applied field HB=HBˆz. Due to the magneto-elastic coupling, this equilib- rium configuration is perturbed by shear stresses in the xz- and yzplanes associated with the incident acoustic wave. To derive the equations of motion, we represent the mag- netic energy density Fof the magnetic material as a sum of the magneto-elastic FMEand purely magnetic FMcontributions.42 Taking into account the Zeeman and demagnetizing energies, we write FM=m0HBM+m0 2(NxM2 x+NyM2 y), where Nx(y) are the demagnetising coefficients, Nx+Ny=1,Mis the magnetization and m0is the magnetic permeability. In a crys- tal of cubic symmetry, the magnetoelastic contribution takes the form43 FME=B M2så i6=jMiMjui j+B0 M2så iM2 iuii;i;j=x;y;z;(1) where B0and Bare the linear isotropic and anisotropic magneto-elastic coupling constants, respectively.44The strain tensor is ujk=1 2(¶jUk+¶kUj), where Ujare the displace- ment vector components. To maximize the effect of the cou- pling B, we consider a transverse acoustic plane wave incident on the slab from the left and polarized along the bias field, so thatUx=Uy=0,Uz=U(x;y;t). The non-vanishing com- ponents of the strain tensor are uxz=1 2¶xUanduyz=1 2¶yU, andFMEis linear in both MandU: FME=B Ms(Mxuxz+Myuyz): (2) The magnetization dynamics in the slab is due to the effec- tive magnetic field, m0Heff=dF=dM. We define mas the small perturbation of the magnetic order, i.e. jmjMs. Linearizing the Landau-Lifshitz-Gilbert equation,4we write ¶mx ¶t=gm0(HB+NyMs)my+gB¶U ¶y+a¶my ¶t;(3) ¶my ¶t=gm0(HB+NxMs)mx+gB¶U ¶x+a¶mx ¶t;(4) where gis the gyromagnetic ratio and ais the Gilbert damping constant. To describe the acoustic wave, we in- clude the magneto-elastic contribution to the stress, s(ME) jk= dFME=dujk, into the momentum balance equation: r¶2U ¶t2=¶ ¶x C¶U ¶x+B Msmx +¶ ¶y C¶U ¶y+B Msmy ; (5) where C=c44is the shear modulus and ris the mass density. The non-magnetic medium is described by Eq.(5) with B=0. Since the values of C,B, and Nx;yare constant within each individual material, we shall seek solutions of the equations in the form of plane waves U;mx(y)µexp[i(kw;xx+kw;yywt)]. From herein, we consider all variables in the Fourier domain.For the magnetization precession in the magnetic layer driven by the acoustic wave, we thus obtain mx=gB(wkw;y+iewykw;x) w2ewxewyU; (6) my=igB(ewxkw;y+iwkw;x) w2ewxewyU; (7) where we have denoted wx(y)=gm0(HB+Nx(y)Ms)and ewx(y)=wx(y)iwa. The complex-valued wave number kw;x is given by the dispersion relation k2 w;x=r Cw2 w2ewxewy
k2 w;y w2ewxewy+gB2 MsCewx h w2ewxewy+gB2 MsCewyi ; (8) where kw;yis equal to that of the incident wave, and the branch with Im kw;x>0 describes a forward wave decaying into the slab. Eq. (8) describes the hybridization between acoustic waves and magnetic precession at frequencies close to ferromagnetic resonance (FMR) at frequency wFMR, with linewidth GFMR. The frequency at which the precession am- plitudes (Eqs. (6) and (7)) diverge is given by the condition (wFMR+iGFMR=2)2=ewxewy. In the limit of small a, this yields wFMR=wxwyandGFMR=a(wx+wy). Away from the resonance, Eq. (8) gives the linear dispersion of acous- tic waves. In the non-magnetic medium ( B=0), one finds k2 0=w2r0=C0. Here and below, the subscript ’0’ is used to mark quantities pertaining to the non-magnetic matrix. To calculate the reflection and transmission coefficients, Rw andTw, for a magnetic inclusion, we introduce the mechanical impedance as Z=isxz=wUw. Solution of the wave matching problem can then be expressed via the ratio of load ( ZME) and source ( Z0) impedances. For impedances in the forward (F) and backward (B) directions in the magnetic slab, we find Z(F=B) w;ME=Ckw;x w0 @1+gB2 CM sewyiwkw;y kw;x w2ewxewy1 A: (9) Here, the ‘-’ and ‘+’ signs correspond to (F) and (B), re- spectively. For the non-magnetic material, Eq. (9) recov- ers the usual acoustic impedance45Z0=cosqpr0C0. Due to magnon-phonon hybridization, Re Z(F=B) w;MEdiverges at wFMR and vanishes at a nearby frequency wME. For a=0, the latter is given by wME=s wxwygB2 MsCwy: (10) Reflection Rwand transmission Twcoefficients are then found via the well-known relations45as Rw=(ehw+1)(1hw)sin(kw;xd) (ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd); (11) Tw=i(hw+ehw) (ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd); (12)Controlling acoustic waves using magneto-elastic Fano resonances 3 where dis the thickness of the magnetic inclusion, hw= Z(F) ME=Z0andehw=Z(B) ME=Z0.46In close proximity to the res- onance, the impedances changes rapidly. Expanding Eq. (11) nearwMEin the limit kwd1, we obtain Rw=iGR=2 (wwME)+iGR=2eif+R0; (13) f=2 arctanC C0rwx wytanq ; where R0represents a smooth non-resonant contribution due to elastic mismatch at the interfaces, while frepresents a res- onant phase, which is non-zero for finite qand approaches p rapidly. In a system with no magnetic damping, the hybridiza- tion yields a resonance of finite linewidth GR, GR=gB2 2MsC2cosqp r0C0 wycos2q+C2 C2 0wxsin2q d: (14) The origin of this linewidth can be explained as follows. Due to the magneto-elastic coupling incident propagating acoustic modes can be converted into localised magnon modes. These modes in turn either decay due to the Gilbert damping or are re-emitted as phonons. The rates of these transitions are pro- portional to GFMR andGR, respectively, and the total decay rate is G=GR+GFMR. This is similar to resonant scattering in quantum theory47, such that GRandGFMRare analogous to the the elastic (Ge)and inelastic (Gi)linewidths respectively. When a=0,GFMRvanishes, and G=GR. Acoustic waves in the geometry of Fig. 1 can be scattered via several channels. E.g. in a non-magnetic system ( B=0), elastic mismatch can yield Fabry-Pérot resonance due to the quarter wavelength matching of dand the acoustic wave- length. However, this occurs at very high frequencies, which we do not consider here. To understand the resonant magneto- elastic response, it is instructive to consider first the case of normal incidence ( q=0), when the demagnetising energy takes a simplified form due to the lack of immediate inter- faces to form surface poles in ythe direction, so that Nx=1 andNy=0. Including magneto-elastic coupling ( B6=0), we plot the frequency dependence of RwandTwusing Eq. (11) and (12) in Fig.2. To gain a quantitative insight, we analysed a magnetic inclusion made of cobalt ( r=8900kgm3,B= 10MPa, C=80GPa, g=176GHzT1,M=1MAm1), em- bedded into a non-magnetic matrix ( r0=3192kgm3;C0= 298GPa). To highlight the resonant behaviour, we first sup- press ato 104. The reflection coefficient exhibits an asym- metric non-monotonic dependence, shown as a black curve in Fig.2(a), characteristic of Fano resonance.27,41This line shape can be attributed to coupling between the discrete FMR mode of the magnetic inclusion and the continuum of propagating acoustic modes in the surrounding non-magnetic material.41 If the two materials had matching elastic properties, Rwwould exhibit a symmetric Breit-Wigner lineshape.47The transmis- sion shown in Fig.2(b) exhibits an approximately symmetric dip near the resonance.48The absorbancejAwj2=1jRwj2 jTwj2, shown in Fig.2(c) exhibits a symmetric peak, since theacoustic waves are damped in our model only due to the cou- pling with spin waves. To consider how the magneto-elastic resonance is affected by the damping, we also plot the response for aof 103and 102, red and blue curves in Fig.2, respectively. An increase ofafrom 104to 103significantly suppresses and broad- ens the resonant peak. For a more common, realistic value of 102the resonance is quenched entirely. A stronger magne- toelastic coupling (i.e. high values of B) could, in principle, countermand this suppression. This, however, is also likely to enhance the phonon contribution to the magnetic damping, leading to a correlation between Bandaobserved in realistic magnetic materials.49 To characterise the strength of the Fano resonance, we note that the fate of the magnon excited by the incident acoustic wave is decided by the relation between the emission rate GR, see Eq. (14), and absorption rate GFMR. Hence, we introduce the respective figure of merit as ¡=GR=GFMR. This quan- tity depends upon the material parameters, device geometry, and bias field. As seen from the first terms on the l.h.s. of Eqs. (6) and (7), the relation between the dynamic magnetisa- tion components mx;yare determined by the quantities wxand wy. Equating these terms, one finds mxµmyp wy=wx, i.e. the precession of mis highly elliptical,50due to the demagnetis- ing field along x. This negatively affects the phonon-magnon coupling for normal incidence ( ky=0): the acoustic wave couples only to mx, as given by the second term in Eqs. (6) and (7). One way to mitigate this is to increase HB, mov- ing the ratio wy=wxcloser to 1 and thus improving the figure of merit. To compare different magneto-elastic materials, the dependence on the layer thickness dand elastic properties of the non-magnetic matrix (i.e. r0andC0) can be eliminated by calculating a ratio of the figures of merit for the compared ma- terials. The comparison can be performed either at the same value of the bias field, or at the same operating frequency. The latter situation is more appropriate for a device application, but to avoid unphysical parameters, we present our results for the same m0HB. An example of such comparisons for yttrium iron garnet (YIG), cobalt (Co) and permalloy (Py) is offered in Table I. Another way to improve ¡is to employ the oblique inci- dence ( q6=0), in which the acoustic mode is also coupled to the magnetisation component my. The latter is not suppressed by the demagnetisation effects if Ny1. The resulting en- hancement in ¡is reflected in the full equation by the inclu- sion of wxandwyfromGR, ¡=GR GFMR=gdB2 2p r0C0 HBcos2q+C2 C2 0Mssin2q aC2M2scosq;(15) where wxwyandHBMsis assumed. For small q, the approximation Nx'1 and Ny'0 still holds. As a result, non- zeroqincreases peak reflectivity, as seen in Fig.3. The evolu- tion of the curves in Fig.3 with qis explained by the variation of the phase fof the resonant scattering relative to that of the non-resonant contribution R0. The latter changes its sign at in- cidence angle of about 30, which yields a nearly symmetric curve (blue), and an inverted Fano resonance at larger anglesControlling acoustic waves using magneto-elastic Fano resonances 4 7.10 7.12 7.14 7.16 7.180.000.050.100.150.200.25|R(f)|(a)Damping, α: 10−4 10−3 10−2 7.10 7.12 7.14 7.16 7.18 Frequency, f (GHz)0.750.800.850.900.951.00|T(f)|(b) 7.10 7.12 7.14 7.16 7.180.00.10.20.3|A(f)|2(c) FIG. 2: The frequency dependence of the absolute values of (a) reflection and (b) transmission coefficients and (c) absorbance is shown for a 20nm thick magnetic inclusion. The vertical dashed and solid black lines represent the ferromagnetic resonance frequency wFMRand magneto-elastic resonance frequency wMErespectively. The non-magnetic and magnetic materials are assumed to be silicon nitride and cobalt, respectively, with parameters given in the text. The bias field is m0HB=50mT, which leads to fME7:138 GHz. (green). Although larger incidence angles may be hard to im- plement in a practical device, the resonant scattering is still enhanced at smaller angles. Above, we have focused on the simplest geometry that ad- mits full analytic treatment. To implement our idea exper- imentally, particular care should be taken about the acous- tic waves polarization and propagation direction relative to the direction of the magnetization. Indeed, our choice max- imises magnetoelastic response. If however, the polariza- tion is orthogonal to the bias field HB, i.e. Uz=0, the cou- pling would be second-order in magnetization components mx;y, and would not contribute to the linearized LLG equation. Furthermore, we have neglected the exchange and magneto- dipolar fields that could arise due to the non-uniformity of the magnetization. To assess the accuracy of this approximation, we note that the length scale of this non-uniformity is set by the acoustic wavelength l, of about 420nm for our parame- ters rather than by the magnetic slab thickness d. The asso- 7.00 7.05 7.10 7.15 7.20 7.25 Frequency, f (GHz)0.010.020.030.040.050.06|R(f)|0◦ 15◦ 30◦45◦ FIG. 3: Peak R(f)is enhanced and slightly shifted to the left in the oblique incidence geometry ( q>0). Coloured curves represent specific incidence angles sweeping from 0to 45. Moderate Gilbert damping of a=103is assumed. The dashed vertical line corresponds to the magnetoelastic resonance frequency. 0 10 20 30 40 Angle, θ (deg.)0.000.020.040.06Figure of Merit, Υ 0.050.100.150.200.25 ΓR(10−1)/FMR(ns)−1ΓFMR ΓRΥFIG. 4: Figure of merit ¡and radiative linewidth GRare both enhanced in the oblique incidence geometry ( q>0). Ferromagnetic linewidth GFMRremains unchanged. Co is assumed with a=103: ciated exchange field is m0Ms(klex)2'9mT. The k-dependent contributions to the magneto-dipole field vanish at normal in- TABLE I: Comparison of the figure of merit ¡for different materials, assuming d=20nm, m0HB=50mT and C0=298GPa. Parameters YIG Co Py ¡(q=0) 4:3x1021:7x1032:7x104 GR(ns1) 1 :9x1047:5x1032:0x104 GFMR (ns1) 4 :4x1034.3 0.74 ¡(q=30) 4:1x1022:5x1032:8x104 GR(ns1) 1 :8x1041:1x1022:1x104 GFMR (ns1) 4 :4x1034.3 0.74 fME=wME=2p(GHz) 2.97 7.14 6.26 B(MJm3) 0.55 10 -0.9 C(GPa) 74 80 50 r(kgm3) 5170 8900 8720 a 0:9x1041:8x1024:0x103 Ms(kAm1) 140 1000 760Controlling acoustic waves using magneto-elastic Fano resonances 5 cidence but may become significant at oblique incidence, giv- ingm0Mskyd'98mT at q=15. In principle, these could increase the resonant frequency of the slab by a few GHz but would complicate the theory significantly. The detailed analy- sis of the associated effects is beyond the scope of this report. In summary, we have demonstrated that the coupling be- tween the magnetisation and strain fields can be used to con- trol acoustic waves by magnetic inclusions. We show that the frequency dependence of the waves’ reflection coefficient from the inclusions has a Fano-like lineshape, which is partic- ularly sensitive to the magnetic damping. Figure of merit is introduced to compare magnetoelastic materials and to char- acterize device performance. In particular, the figure of merit is significantly enhanced for oblique incidence of acoustic waves, which enhances their coupling to the magnetic modes. We envision that further routes may be taken to transform our prototype designs into working devices, such as forming a magneto-acoustic metamaterial to take advantage of spatial resonance. 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1908.03194v5.Annihilation_of_topological_solitons_in_magnetism_with_spin_wave_burst_finale__The_role_of_nonequilibrium_electrons_causing_nonlocal_damping_and_spin_pumping_over_ultrabroadband_frequency_range.pdf

Annihilation of topological solitons in magnetism with spin wave burst nale: The role of nonequilibrium electrons causing nonlocal damping and spin pumping over ultrabroadband frequency range Marko D. Petrovi c,1Utkarsh Bajpai,1Petr Plech a c,2and Branislav K. Nikoli c1, 1Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA 2Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA We not only reproduce burst of short-wavelength spin waves (SWs) observed in recent experiment [S. Woo et al. , Nat. Phys. 13, 448 (2017)] on magnetic-eld-driven annihilation of two magnetic domain walls (DWs) but, furthermore, we predict that this setup additionally generates highly un- usual pumping of electronic spin currents in the absence of any bias voltage. Prior to the instant of annihilation, their power spectrum is ultrabroadband , so they can be converted into rapidly changing in time charge currents, via the inverse spin Hall eect, as a source of THz radiation of bandwidth '27 THz where the lowest frequency is controlled by the applied magnetic eld. The spin pump- ing stems from time-dependent elds introduced into the quantum Hamiltonian of electrons by the classical dynamics of localized magnetic moments (LMMs) comprising the domains. The pumped currents carry spin-polarized electrons which, in turn, exert backaction on LMMs in the form of nonlocal damping which is more than twice as large as conventional local Gilbert damping. The nonlocal damping can substantially modify the spectrum of emitted SWs when compared to widely- used micromagnetic simulations where conduction electrons are completely absent . Since we use fully microscopic (i.e., Hamiltonian-based) framework, self-consistently combining time-dependent elec- tronic nonequilibrium Green functions with the Landau-Lifshitz-Gilbert equation, we also demon- strate that previously derived phenomenological formulas miss ultrabroadband spin pumping while underestimating the magnitude of nonlocal damping due to nonequilibrium electrons . Introduction .|The control of the domain wall (DW) motion1{3within magnetic nanowires by magnetic eld or current pulses is both a fundamental problem for nonequilibrium quantum many-body physics and a build- ing block of envisaged applications in digital memories.4 logic5and articial neural networks.6Since DWs will be closely packed in such devices, understanding interaction between them is a problem of great interest.7For ex- ample, head-to-head or tail-to-tail DWs|illustrated as the left (L) or right (R) noncollinear texture of local- ized magnetic moments (LMMs), respectively, in Fig. 1| behave as free magnetic monopoles carrying topological charge.8The topological charge (or the winding number) Q1 R dx@x, associated with winding of LMMs as they interpolate between two uniform degenerate ground states with = 0 or=, is opposite for adjacent DWs, such as QL=1 andQR= +1 for DWs in Fig. 1. Thus, long-range attractive interaction between DWs can lead to their annihilation , resulting in the ground state without any DWs.9{12This is possible because to- tal topological charge remains conserved, QL+QR= 0. The nonequilibrium dynamics13triggered by annihilation of topological solitons is also of great interest in many other elds of physics, such as cosmology,14gravitational waves,15quantum13and string eld16theories, liquid crystals17and Bose-Einstein condensates.18,19 The recent experiment20has monitored annihilation of two DWs within a metallic ferromagnetic nanowire by observing intense burst of spin waves (SWs) at the mo- ment of annihilation. Thus generated large-amplitude SWs are dominated by exchange, rather than dipolar, interaction between LMMs and are, therefore, of short wavelength. The SWs of 10 nm wavelength are cru-cial for scalability of magnonics-based technologies,21,22 like signal transmission or memory-in-logic and logic- in-memory low-power digital computing architectures. However, they are dicult to excite by other methods due to the requirement for high magnetic elds.23,24 The computational simulations of DW annihila- tion,9,10,20together with theoretical analysis of generic features of such a phenomenon,11have been based exclu- sively on classical micromagnetics where one solves cou- pled Landau-Lifshitz-Gilbert (LLG) equations25for the dynamics of LMMs viewed as rotating classical vectors of xed length. On the other hand, the dynamics of LMMs comprising two DWs also generates time-dependent elds which will push the surrounding conduction electrons out of equilibrium. The nonequilibrium electrons comprise pumped spin current26{28(as well as charge currents if the left-right symmetry of the device is broken28,29) in the absence of any externally applied bias voltage. The pumped spin currents ow out of the DW region into the external circuit, and since they carry away excess an- gular momentum of precessing LMMs, the backaction of nonequilibrium electrons on LMMs emerges26as an ad- ditional damping-like (DL) spin-transfer torque (STT). The STT, as a phenomenon in which spin angu- lar momentum of conduction electrons is transferred to LMMs when they are not aligned with electronic spin-polarization, is usually discussed for externally in- jected spin current.30But here it is the result of compli- cated many-body nonequilibrium state in which LMMs drive electrons out of equilibrium which, in turn, ex- ertbackaction in the form of STT onto LMMs to modify their dynamics in a self-consistent fashion.27,31 Such eects are absent from classical micromagneticsarXiv:1908.03194v5 [cond-mat.mes-hall] 24 Jun 20212 FIG. 1. Schematic view of a ferromagnetic nanowire modeled as a 1D tight-binding chain whose sites host classical LMMs (red arrows) interacting with spins (blue arrow) of conduc- tion electrons. The nanowire is attached to two NM leads terminating into the macroscopic reservoirs kept at the same chemical potential. The two DWs within the nanowire carry opposite topological charge,8QL=1 for the L one and QR= +1 for the R one. They collide with the opposite ve- locities VL DWandVR DWand annihilate, upon application of an external magnetic eld Bextparallel to the nanowire, thereby mimicking the setup of the experiment in Ref. 20. or atomistic spin dynamics25because they do not in- clude conduction electrons. This has prompted deriva- tion of a multitude of phenomenological expressions32{39 for the so-called nonlocal (i.e., magnetization-texture- dependent) and spatially nonuniform (i.e., position- dependent) Gilbert damping that could be added into the LLG equation and micromagnetics codes40{42to cap- ture the backaction of nonequilibrium electrons while not simulating them explicitly. Such expressions do not re- quire spin-orbit (SO) or magnetic disorder scattering, which are necessary for conventional local Gilbert damp- ing,43{45but they were estimated33,36to be usually a small eect unless additional conditions (such as narrow DWs or intrinsic SO coupling splitting the band struc- ture33) are present. On the other hand, a surprising result40of Gilbert damping extracted from experiments on magnetic-eld-driven DW being several times larger than the value obtained from standard ferromagnetic res- onance measurements can only be accounted by including additional nonlocal damping. In this Letter, we unravel complicated many-body nonequilibrium state of LMMs and conduction elec- trons created by DW annihilation using recently de- veloped27,46{49quantum-classical formalism which com- bines time-dependent nonequilibrium Green function (TDNEGF)50,51description of quantum dynamics of con- duction electrons with the LLG equation description of classical dynamics of LMMs on each atom.25Such TD- NEGF+LLG formalism is fully microscopic, since it re- quires only the quantum Hamiltonian of electrons and the classical Hamiltonian of LMMs as input, and numerically exact . We apply it to a setup depicted in Fig. 1 where two DWs reside at time t= 0 within a one-dimensional (1D) magnetic nanowire attached to two normal metal (NM) leads, terminating into the macroscopic reservoirs without any bias voltage. Our principal results are: ( i) annihilation of two DWs [Fig. 2] pumps highly unusual electronic spin currents whose power spectrum is ultrabroadband prior to the in- FIG. 2. (a) Sequence of snapshots of two DWs, in the course of their collision and annihilation in the setup of Fig. 1; and (b) the corresponding time-dependence of the z-component of LMMs where blue and orange line mark t= 6:9 ps (when two DWs start vanishing) and t= 7:2 ps (when all LMMs become nearly parallel to the x-axis) from panel (a). A movie animating panels (a) and (b) is provided in the SM.58Spatio- temporal prole of: (c) angle eq iand (d) \nonadiabaticity" angleneq ieq i, with the meaning of neq iandeq iillustrated in the inset above panel (c); (e) DL STT [Eq. (3)] as electronic backaction on LMMs; (f) ratio of DL STT to conventional local Gilbert damping [Eq. (2)]; and (g) ratio of the sum of DL STT to the sum of conventional local Gilbert damping over all LMMs. stant of annihilation [Fig. 3(d)], unlike the narrow peak around a single frequency for standard spin pumping;26 (ii) because pumped spin currents carry away excess angular momentum of precessing LMMs, this acts as DL STT on LMMs which is spatially [Figs. 2(e) and 4(b)] and time [Fig. 2(g)] dependent, as well as '2:4 times larger [Fig. 2(f)] than conventional local Gilbert damping [Eq. (2)]. This turns out to be remarkably similar to'2:3 ratio of nonlocal and local Gilbert damping measured experimentally in permalloy,40but it is severely underestimated by phenomenological the- ories32,33[Fig. 4(a),(b)]. Models and methods .|The classical Hamiltonian for3 ≃ 27 THz FIG. 3. Time dependence of: (a){(c) electronic spin currents pumped into the right NM lead during DW collision and annihila- tion; (e){(g) SW-generated contribution to spin currents in panels (a){(c), respectively, after spin current carried by SW from Fig. 2(b) is stopped at the magnetic-nanowire/nonmagnetic-NM-lead interface and converted (as observed experimentally20,61) into electronic spin current in the right NM lead. Vertical dashed lines mark times t= 6:9 ps andt= 7:2 ps whose snapshots of LMMs are shown in Fig. 2(a). For easy comparison, gray curves in panels (f) and (g) are the same as the signal in panels (b) and (c), respectively, for post-annihilation times t7:2 ps. Panels (d) and (h) plot FFT power spectrum of signals in panels (c) and (g), respectively, before (red curve) and after (brown curves) completed annihilation at t= 7:2 ps. LMMs, described by unit vectors Mi(t) at each site iof 1D lattice, is chosen as H=JX hijiMi
MjKX i(Mx i)2 +DX i(My i)2BX iMi
Bext; (1) whereJ= 0:1 eV is the Heisenberg exchange coupling between the nearest-neighbor LMMs; K= 0:05 eV is the magnetic anisotropy along the x-axis; andD= 0:007 eV is the demagnetizing eld along the y-axis. The last term in Eq. (1) is the Zeeman energy ( Bis the Bohr magne- ton) describing the interaction of LMMs with an external magnetic eld Bextparallel to the nanowire in Fig. 1 driv- ing the DW dynamics, as employed in the experiment.20 The classical dynamics of LMMs is described by a system of coupled LLG equations25(using notation @t@=@t) @tMi=gMiBe i+Mi@tMi +g M Tih IS exti +Ti[Mi(t)] : (2) where Be i=1 M@H=@Miis the eective magnetic eld (Mis the magnitude of LMMs); gis the gy- romagnetic ratio; and the magnitude of conventional local Gilbert damping is specied by spatially- and time-independent , set as= 0:01 as the typi- cal value measured40in metallic ferromagnets. The spatial prole of a single DW in equilibrium, i.e., at timet= 0 as the initial condition, is given by Mi(Q;X DW) = cosi(Q;X DW);0;sini(Q;X DW)
, wherei(Q;X DW) =Qarccos [tanh ( xiXDW)];Q is the topological charge; and XDW is the positionof the DW. The initial conguration of two DWs, Mi(t= 0) = Mi(QL;XL) +Mi(QR;XR), positioned at sitesXL= 15 andXR= 30 harbors opposite topological chargesQR=QL= 1 around these sites. In general, two additional terms32,33,52in Eq. (2) ex- tend the original LLG equation|STT due to externally injected electronic spin current,30which is actually ab- sentTih IS exti 0 in the setup of Fig. 1; and STT due to backaction of electrons Ti[Mi(t)] =Jsd(h^siineq(t)h^siieq t)Mi(t); (3) driven out of equilibrium by Mi(t). HereJsd= 0:1 eV is chosen as the s-dexchange coupling (as mea- sured in permalloy53) between LMMs and electron spin. We obtain \adiabatic"54,55electronic spin density, h^siieq t= Tr [ eq tjiihij ], from grand canonical equilib- rium density matrix (DM) for instantaneous congura- tion of Mi(t) at timet[see Eq. (5)]. Here = (^x;^y;^z) is the vector of the Pauli matrices. The nonequilibrium electronic spin density, h^siineq(t) = Tr [ neq(t)jiihij ], requires to compute time-dependent nonequilibrium DM, neq(t) =~G<(t;t)=i, which we construct using TD- NEGF algorithms explained in Refs. 56 and 57 and com- bine27with the classical LLG equations [Eq. (2)] using time stept= 0:1 fs. The TDNEGF calculations require as an input a quantum Hamiltonian for electrons, which is chosen as the tight-binding one ^H(t) = X hiji^cy i^ciJsdX i^cy i
Mi(t)^ci: (4) Here ^cy i= (^cy i";^cy i#) is a row vector containing operators ^cy iwhich create an electron of spin =";#at the sitei,4 and ^ciis a column vector that contains the correspond- ing annihilation operators; and = 1 eV is the nearest- neighbor hopping. The magnetic nanowire in the setup in Fig. 1 consists of 45 sites and it is attached to semi- innite NM leads modeled by the rst term in ^H. The Fermi energy of the reservoirs is set at EF= 0 eV. Due to the computational complexity of TDNEGF calcula- tions,51we use magnetic eld jBextj= 100 T to complete DW annihilation on ps time scale (in the experiment20 this happens within 2 ns). Results .|Figure 2(a) demonstrates that TDNEGF+LLG-computed snapshots of Mi(t)fully reproduce annihilation in the experiment,20including - nalewhen SW burst is emitted at t'7:2 ps in Fig. 2(b). The corresponding complete spatio-temporal proles are animated as a movie provided in the Supplemental Material (SM).58However, in contrast to micromagnetic simulations of Ref. 20 where electrons are absent, Fig. 2(d) shows that they generate spin density h^siineq(t) which is noncollinear with either Mi(t) orh^siieq t. This leads to \nonadiabaticity" angle ( neq ieq i)6= 0 in Fig. 2(d) and nonzero STT [Eq. (3) and Fig. 2(e)] as self-consistent backaction of conduction electrons onto LMMs driven out of equilibrium by the dynamics of LMMs themselves. The STT vector, Ti=TFL i+TDL i, can be decomposed [see inset above Fig. 2(e)] into: ( i) even under time-reversal or eld-like (FL) torque, which aects precession of LMM around Be i; and ( ii) odd under time-reversal or DL torque, which either enhances Gilbert term [Eq. (2)] by pushing LMM toward Be ior competes with it as antidamping. Figure 2(f) shows that TDL i[Mi(t)] acts like an additional nonlocal damping while being'2:4 times larger than conventional local Gilbert damping Mi@tMi[Eq. (2)]. The quantum transport signature of DW vanishing within the time interval t= 6:9{7:2 ps in Fig. 2(a) is the reduction in the magnitude of pumped electronic spin currents [Fig. 3(a){(c)]. In fact, ISx R(t)!0 becomes zero [Fig. 3(a)] at t= 7:2 ps at which LMMs in Fig. 2(a) turn nearly parallel to the x-axis while precessing around it. The frequency power spectrum [red curve in Fig. 3(d)] obtained from fast Fourier transform (FFT) of ISz R(t), for times prior to completed annihilation and SW burst at t= 7:2 ps, reveal highly unusual spin pumping over an ultrabroadband frequency range. This can be contrasted with the usual spin pumping26whose power spectrum is just a peak around a single frequency,59as also ob- tained [brown curve in Fig. 3(d)] by FFT of ISz R(t) at post-annihilation times t>7:2 ps. The spin current in Fig. 3(a){(c) has contributions from both electrons moved by time-dependent Mi(t) and SW hitting the magnetic-nanowire/NM-lead interface. At this interface, SW spin current is stopped and trans- muted47,48,60into an electronic spin current owing into the NM lead. The transmutation is often employed ex- perimentally for direct electrical detection of SWs, where an electronic spin current on the NM side is converted into a voltage signal via the inverse spin Hall eect.20,61 ~10-7FIG. 4. Spatial prole at t= 6:9 ps of: (a) locally pumped spin current ISx i!j47between sites iandj; and nonlocal damp- ing due to backaction of nonequilibrium electrons . Solid lines in (a) and (b) are obtained from TDNEGF+LLG calcula- tions, and dashed lines are obtained from SMF theory phe- nomenological formulas.32,33,69(c){(e) FFT power spectra22 ofMz i(t) where (c) and (d) are TDNEGF+LLG-computed with= 0:01 and= 0, respectively, while (e) is LLG- computed with backaction of nonequilibrium electrons re- moved, Ti[Mi(t)]0, in Eq. (2). The dashed horizontal lines in panels (c){(e) mark frequencies of peaks in Fig. 3(d). Within the TDNEGF+LLG picture, SW reaching the last LMM of the magnetic nanowire, at the sites i= 1 ori= 45 in our setup, initiates their dynamics whose coupling to conduction electrons in the neighboring left and right NM leads, respectively, leads to pumping47of the electronic spin current into the NM leads. The prop- erly isolated electronic spin current due to transmutation of SW burst, which we denote by IS;SW p , is either zero or very small until the burst is generated in Fig. 3(e){ (g), as expected. We note that detected spin current in the NM leads was attributed in the experiment20solely to SWs, which corresponds in our picture to considering onlyIS;SW p while disregarding ISpIS;SW p . Discussion .|A computationally simpler alternative to our numerical self-consistent TDNEGF+LLG is to \in- tegrate out electrons"31,62{65and derive eective expres- sions solely in terms of Mi(t), which can then be added into the LLG Eq. (2) and micromagnetics codes.40{42 For example, spin motive force (SMF) theory69gives ISx SMF(x) =gB~G0 4e2[@M(x;t)=@t@M(x;t)=@x]xfor the spin current pumped by dynamical magnetic texture, so thatMD
@tMis the corresponding nonlocal Gilbert damping.32,33Here M(x;t) is local magnetization (as- suming our 1D system); D=P (M@M)(M @M)(using notation ;;2 fx;y;zg) is 33 spatially-dependent damping tensor; and =gB~G0 4e2 withG0=G"+G#being the total conductivity. We compare in Fig. 4: ( i) spatial prole of ISx SMF(x) to locally pumped spin current ISx i!j47from TDNEGF+LLG calcu-5 lations [Fig. 4(a)] to nd that the former predicts negli- gible spin current owing into the leads, thereby missing ultrabroadband spin pumping predicted in Fig. 3(d); ( ii) spatial prole of MD
@tMto DL STT TDL ifrom TD- NEGF+LLG calculations, to nd that the former has comparable magnitude only within the DW region but with substantially diering proles. Note also that47 [P iTi(t)]=~ 2eh IS L(t) +IS R(t)i +P i~ 2@h^ s iineq @t, which makes the sum of DL STT plotted in Fig. 2(g) time- dependent during collision, in contrast to the sum of lo- cal Gilbert damping shown in Fig. 2(g). The backaction of nonequilibrium electrons viaTi[Mi(t)] can strongly aect the dynamics of LMMs, especially for the case of short wavelength SWs and narrow DWs,32,33,41,42as con- rmed by comparing FFT power spectra of Mz i(t) com- puted by TDNEGF+LLG [Fig. 4(c),(d)] with those from LLG calculations [Fig. 4(e)] without any backaction . We note that SMF theory69is derived in the \adi- abatic" limit,2,54which assumes that electron spin re- mains in the the lowest energy state at each time. \Adi- abaticity" is used in two dierent contexts in spintron- ics with noncollinear magnetic textures|temporal and spatial.2In the former case, such as when electrons in- teract with classical macrospin due to collinear LMMs, one assumes that classical spins are slow and h^siineq(t) can \perfectly lock"2to the direction Mi(t) of LMMs. In the latter case, such as for electrons traversing thick DW, one assumes that electron spin keeps the lowest en- ergy state by rotating according to the orientation of Mi(t) at each spatial point, thereby evading re ection from the texture.2The concept of \adiabatic" limit is made a bit more quantitative by considering2ratio of relevant energy scales, Jsd=~!1 orJsd=BjBextj1, in the former case; or combination of energy and spa- tial scales,JsddDW=~vF=JsddDW= a1, in the latter case (where vFis the Fermi velocity, ais the lattice spac- ing anddDWis the DW thickness). In our simulations, Jsd=BjBextj10 andJsddDW= a1 fordDW10a in Fig. 2(a). Thus, it seems that fair comparison of our results to SMF theory requires to substantially increase Jsd. However, Jsd= 0:1 eV (i.e., =Jsd=10, for typical 1 eV which controls how fast is quantum dynam- ics of electrons) in our simulations is xed by measured properties of permalloy.53 Let us recall that rigorous denition of \adiabaticity" assumes that conduction electrons within a closed quan- tum system54at timetare in the ground state j 0i for the given conguration of LMMs Mi(t),j (t)i= j 0[Mi(t)]i; or in open system55their quantum state is specied by grand canonical DM eq t=1 Z dEImGr tf(E): (5) where the retarded GF, Gr t= EH[Mi(t)]LR1, and eq tdepend parametrically66{68(or implic- itly, so we put tin the subscript) on time via instanta- neous conguration of Mi(t), thereby eectively assum- ing@tMi(t) = 0. Here Im Gr t= (Gr t[Gr t]y)=2i;L;R are self-energies due to the leads; and f(E) is the Fermi function. For example, the analysis of Ref. 69 assumes h^siineq(t)kh^siieq tto reveal the origin of spin and charge pumping in SMF theory|nonzero angle eq ibetween h^siieq tandMi(t) with the transverse component scaling jh^siieq tMi(t)j= h^siieq t
Mi(t)
/1=Jsdas the signature of \adiabatic" limit. Note that our eq i.4[Fig. 2(c)] in the region of two DWs (and eq i!0 elsewhere). Ad- ditional Figs. S1{S3 in the SM,58where we isolate two neighboring LMMs from the right DW in Fig. 1 and put them in steady precession with frequency !for sim- plicity of analysis, demonstrate that entering such \adi- abatic" limit requires unrealistically large Jsd&2 eV. Also, our exact55result [Figs. S1(b), S2(b) and S3(b) in the SM58] showsjh^siieq tMi(t)j= h^siieq t
Mi(t)
/1=J2 sd (instead of/1=Jsdof Ref. 69). Changing ~!|which, according to Fig. 3(c), is eectively increased by the dynamics of annihilation from ~!'0:01 eV, set ini- tially by Bext, toward ~!'0:1 eV|only modies scal- ing of the transverse component of h^siineq(t) withJsd [Figs. S1(a), S2(a), S3(a), S4(b) and S4(d) in the SM58]. The nonadiabatic corrections55,66{68take into account @tMi(t)6= 0. We note that only in the limit Jsd!1 , h^siineq(t)h^siieq t
!0. Nevertheless, multiplication of these two limits within Eq. (3) yields nonzero geo- metric STT,54,55as signied by Jsd-independent STT [Figs. S1(c), S2(c) and S3(c) in the SM58]. Otherwise, \nonadiabaticity" angle is always present ( neq ieq i)6= 0 [Fig. 2(d)], even in the absence of spin relaxation due to magnetic impurities or SO coupling,70and it can be di- rectly related to additional spin and charge pumping48,70 [see also Figs. S1(f), S2(f) and S3(f) in the SM58]. Conclusions and outlook .|The pumped spin current over ultrabroadband frequency range [Fig. 3(d)], as our central prediction, can be converted into rapidly chang- ing transient charge current via the inverse spin Hall ef- fect.71{73Such charge current will, in turn, emit electro- magnetic radiation covering 0:03{27 THz range (for jBextj1 T) or0:3{27:3 THz range (for jBextj10 T), which is highly sought range of frequencies for variety of applications.72,73 ACKNOWLEDGMENTS M. D. P., U. B., and B. K. 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1508.07118v3.The_inviscid_limit_for_the_Landau_Lifshitz_Gilbert_equation_in_the_critical_Besov_space.pdf

arXiv:1508.07118v3 [math.AP] 25 Aug 2016THE INVISCID LIMIT FOR THE LANDAU-LIFSHITZ-GILBERT EQUATION IN THE CRITICAL BESOV SPACE ZIHUA GUO AND CHUNYAN HUANG Abstract. We prove that in dimensions three and higher the Landau-Lifshitz- Gilbert equation with small initial data in the critical Besov space is glob ally well- posed in a uniform way with respect to the Gilbert damping parameter . Then we show that the global solution converges to that of the Schr¨ oding er maps in the natural space as the Gilbert damping term vanishes. The proof is ba sed on some studies on the derivative Ginzburg-Landau equations. 1.Introduction Inthis paperwe study theCauchy problemfortheLandau-Lifshitz -Gilbert (LLG) equation ∂ts=as×∆s−εs×(s×∆s), s(x,0) =s0(x), (1.1) wheres(x,t) :Rn×R→S2⊂R3,×denotes the wedge product in R3,a∈R andε >0 is the Gilbert damping parameter. The equation (1.1) is one of the equations of ferromagnetic spin chain, which was proposed by Land au-Lifshitz [19] in studying the dispersive theory of magnetisation of ferromagnet s. Later on, such equations were also found in the condensed matter physics. The LL G equation has been studied extensively, see [17, 7] for an introduction on the equ ation. Formally, if a= 0, then (1.1) reduces to the heat flow equations for harmonic maps ∂ts=−εs×(s×∆s), s(x,0) =s0(x), (1.2) and ifε= 0, then (1.1) reduces to the Schr¨ odinger maps ∂ts=as×∆s, s(x,0) =s0(x). (1.3) Both special cases have been objects of intense research. The p urpose of this paper is to study the inviscid limit of (1.1), namely, to prove rigorously that t he solutions of (1.1) converges to the solutions of (1.3) as ε→0 under optimal conditions on the initial data. The inviscid limit is an important topic in mathematical physics, and has b een studied in various settings, e.g. for hyperbolic-dissipative equation s such as Navier- Stokes equation to Euler equation (see [11] and references ther ein), for dispersive- dissipative equations such as KdV-Burgers equation to KdV equatio n (see [9]) and Ginzburg-Landau equation to Schr¨ odinger equations (see [23, 12 ]). The LLG equa- tion (1.1) is an equation with both dispersive and dissipative effects. T his can be 2010Mathematics Subject Classification. 35Q55. Key words and phrases. Landau-Lifshitz-Gilbert equation, Schr¨ odinger maps, Inviscid limit , Critical Besov Space. 12 Z. GUO AND C. HUANG seen from the stereographic projection transform. It was know n that (see [18]) let u=P(s) =s1+is2 1+s3, (1.4) wheres= (s1,s2,s3) is a solution to (1.1), then usolves the following complex derivative Ginzburg-Landau type equation (i∂t+∆−iε∆)u=2a¯u 1+|u|2n/summationdisplay j=1(∂xju)2−2iε¯u 1+|u|2n/summationdisplay j=1(∂xju)2 u(x,0) =u0.(1.5) On the other hand, the projection transform has an inverse P−1(u) =/parenleftbiggu+ ¯u 1+|u|2,−i(u−¯u) 1+|u|2,1−|u|2 1+|u|2/parenrightbigg . (1.6) Therefore, (1.1) is equivalent to (1.5) assuming PandP−1is well-defined, and we will focus on (1.5). The previous works [13, 14, 1, 2, 3] on the Schr¨ odinger maps (ε= 0) were also based on this transform. Note that (1.5) is invariant u nder the following scaling transform: for λ >0 u(x,t)→u(λx,λ2t), u0(x)→u0(λx). Thus the critical Besov space is ˙Bn/2 2,1in the sense of scaling. To study the inviscid limit, the crucial task is to obtain uniform well-pos edness with respect to the inviscid parameter. Energy method was used in [1 1]. For dispersive-dissipative equations, one needs to exploit the dispersiv e effect uniformly. Strichartz estimates and energy estimates were used in [23] for Gin zburg-Landau equations, andBourgainspacewasused in[9]forKdV-Burgersequ ations. In[24,10] the inviscid limit for the derivative Ginzburg-Landau equations were s tudied by us- ing the Strichartz estimates, local smoothing estimates and maxima l function esti- mates. However, these results requires high regularities when app lied to equation (1.5). In this paper we will use Bourgain-type space and exploit the n ull structure that are inspired by the latest development for the Schr¨ odinger m aps (ε= 0) (see [4, 3, 1, 2, 13, 14, 8]) to study (1.5) with small initial data in the critica l Besov space. In [2] and [14] it was proved independently that global well-p osedness for (1.3) holds for small data in the critical Besov space. We will extend t heir results to (1.1) uniformly with respect to ε. We exploit the Bourgain space in a differ- ent way from both [2] and [14]. One of the novelties is the use of X0,1-structure that results in many simplifications even for the Schr¨ odinger maps. The presence of dissipative term brings many technical difficulties, e.g. the lack of s ymmetry in time and incompatibility with Xs,bstructure. We need to overcome these dif- ficulties when extending the linear estimates for the Schr¨ odinger e quation to the Schr¨ odinger-dissipative equation uniformly with respect to ε. By scaling we may assume a=±1. From now on, we assume a= 1 since the other case a=−1 is similar. For Q∈S2, the space ˙Bs Qis defined by ˙Bs Q=˙Bs Q(Rn;S2) ={f:Rn→R3;f−Q∈˙Bs 2,1,|f(x)| ≡1 a.e. in Rn}, where˙Bs 2,1is the standard Besov space. It was known the critical space is ˙Bn/2 Q. The main result of this paper isLANDAU-LIFSHITZ EQUATION 3 Theorem 1.1. Assumen≥3. The LLG equation (1.1)is globally well-posed for small data s0∈˙Bn/2 Q(Rn;S2),Q∈S2in a uniform way with respect to ε∈(0,1]. Moreover, for any T >0, the solution converges to that of Schr¨ odinger map (1.3) inC([−T,T] :˙Bn/2 Q)asε→0. As we consider the inviscid limit in the strongest topology (same space as the initial data), no convergence rate is expected. This can be seen fr om linear solutions for (1.5). However, if assuming initial data has higher regularity, on e can have convergence rate O(εT) (see (5.3) below). 2.Definitions and Notations Forx,y∈R,x/lessorsimilarymeans that there exists a constant Csuch that x≤Cy, and x∼ymeans that x/lessorsimilaryandy/lessorsimilarx. We use F(f),ˆfto denote the space-time Fourier transform of f, andFxi,tfto denote the Fourier transform with respect to xi,t. Letη:R→[0,1] be an even, non-negative, radially decreasing smooth function such that: a) ηis compactly supported in {ξ:|ξ| ≤8/5}; b)η≡1 for|ξ| ≤5/4. For k∈Zletχk(ξ) =η(ξ/2k)−η(ξ/2k−1),χ≤k(ξ) =η(ξ/2k),/tildewideχk(ξ) =/summationtext9n l=−9nχk+l(ξ), and then define the Littlewood-Paley projectors Pk,P≤k,P≥konL2(Rn) by /hatwidestPku(ξ) =χk(|ξ|)/hatwideu(ξ),/hatwideP≤ku(ξ) =χ≤k(|ξ|)/hatwideu(ξ), andP≥k=I−P≤k−1,P[k1,k2]=/summationtextk2 j=k1Pj. We also define /tildewidePku=F−1/tildewideχk(|ξ|)/hatwideu(ξ) LetSn−1be the unit sphere in Rn. Fore∈Sn−1, define /hatwidePk,eu(ξ) =/tildewideχk(|ξ· e|)χk(|ξ|)/hatwideu(ξ). Since for |ξ| ∼2kwe have 5n/summationdisplay l=−5nχk+l(ξ1)+···+5n/summationdisplay l=−5nχk+l(ξn)∼1, then let βj k(ξ) =/summationtext5n l=−5nχk+l(ξj) /summationtextn j=1/summationtext5n l=−5nχk+l(ξj)·1/summationdisplay l=−1χk+l(|ξ|), j= 1,···,n. Define the operator Θj konL2(Rn) by/hatwidestΘj kf(ξ) =βj k(ξ)ˆf(ξ), 1≤j≤n. Lete1= (1,0,···,0),···,en= (0,···,0,1). Then we have Pk=n/summationdisplay j=1Pk,ejΘj k. (2.1) For anyk∈Z, we define the modulation projectors Qk,Q≤k,Q≥konL2(Rn×R) by /hatwidestQku(ξ,τ) =χk(τ+|ξ|2)/hatwideu(ξ,τ),/hatwideQ≤ku(ξ,τ) =χ≤k(τ+|ξ|2)/hatwideu(ξ,τ), andQ≥k=I−Q≤k−1,Q[k1,k2]=/summationtextk2 j=k1Qj. For anye∈Sn−1, we can decompose Rn=λe⊕He, whereHeis the hyperplane with normal vector e, endowed with the induced measure. For 1 ≤p,q <∞, we defineLp,q ethe anisotropic Lebesgue space by /ba∇dblf/ba∇dblLp,q e=/parenleftBigg/integraldisplay R/parenleftbigg/integraldisplay He×R|f(λe+y,t)|qdydt/parenrightbiggp/q dλ/parenrightBigg1/p4 Z. GUO AND C. HUANG with the usual definition if p=∞orq=∞. We write Lp,q ej=Lp xjLq ¯xj,t. We use ˙Bs p,qto denote the homogeneous Besov spaces on Rnwhich is the completion of the Schwartz functions under the norm /ba∇dblf/ba∇dbl˙Bsp,q= (/summationdisplay k∈Z2qsk/ba∇dblPkf/ba∇dblq Lp)1/q. To exploit the null-structure we also need the Bourgain-type space associated to theSchr¨ odinger equation. Inthis paperwe use themodulation-ho mogeneousversion as in [2, 8]. We define X0,b,qto be the completion of the space of Schwartz functions with the norm /ba∇dblf/ba∇dblX0,b,q= (/summationdisplay k∈Z2kbq/ba∇dblQkf/ba∇dblq L2 t,x)1/q. (2.2) Ifq= 2 we simply write X0,b=X0,b,2. By the Plancherel’s equality we have /ba∇dblf/ba∇dblX0,1=/ba∇dbl(i∂t+ ∆)f/ba∇dblL2 t,x. SinceX0,b,qis not closed under conjugation, we also define the space ¯X0,b,qby the norm /ba∇dblf/ba∇dbl¯X0,b,q=/ba∇dbl¯f/ba∇dblX0,b,q, and similarly write ¯X0,b= ¯X0,b,2. It’s easy to see that X0,b,qfunction is unique modulo solutions of the homo- geneous Schr¨ odinger equation. For a more detailed description of theX0,b,pspaces we refer the readers to [21] and [20]. We use X0,b,p +to denote the space restricted to the interval [0 ,∞): /ba∇dblf/ba∇dblX0,b,p += inf ˜f:˜f=font∈[0,∞)/ba∇dbl˜f/ba∇dblX0,b,p. In particular, we have /ba∇dblf/ba∇dblX0,1 +∼ /ba∇dbl˜f/ba∇dblX0,1 (2.3) where˜f=f(t,x)1t≥0+f(−t,x)1t<0. LetL=∂t−i∆ and¯L=∂t+i∆. We define the main dyadic function space. If f(x,t)∈L2(Rn×R+) has spatial frequency localized in {|ξ| ∼2k}, define /ba∇dblf/ba∇dblFk=/ba∇dblf/ba∇dblX0,1/2,∞ ++/ba∇dblf/ba∇dblL∞ tL2x+/ba∇dblf/ba∇dbl L2 tL2n n−2 x +2−(n−1)k/2sup e∈Sn−1/ba∇dblf/ba∇dblL2,∞ e+2k/2sup |j−k|≤20sup e∈Sn−1/ba∇dblPj,ef/ba∇dblL∞,2 e, /ba∇dblf/ba∇dblYk=/ba∇dblf/ba∇dblL∞ tL2x+/ba∇dblf/ba∇dbl L2 tL2n n−2 x+2−(n−1)k/2sup e∈Sn−1/ba∇dblf/ba∇dblL2,∞ e +2−kinf f=f1+f2(/ba∇dblLf1/ba∇dblL2 t,x+/ba∇dbl¯Lf2/ba∇dblL2 t,x), /ba∇dblf/ba∇dblZk=2−k/ba∇dblLf/ba∇dblL2 t,x /ba∇dblf/ba∇dblNk= inf f=f1+f2+f3(/ba∇dblf1/ba∇dblL1 tL2x+2−k/2sup e∈Sn−1/ba∇dblf2/ba∇dblL1,2 e+/ba∇dblf3/ba∇dblX0,−1/2,1 +)+2−k/ba∇dblf/ba∇dblL2 t,x. Then we define the space Fs,Ys,Zs,Nswith the following norm /ba∇dblu/ba∇dblFs=/summationdisplay k∈Z2ks/ba∇dblPku/ba∇dblFk,/ba∇dblu/ba∇dblYs=/summationdisplay k∈Z2ks/ba∇dblPku/ba∇dblYk, /ba∇dblu/ba∇dblZs=/summationdisplay k∈Z2ks/ba∇dblPku/ba∇dblZk,/ba∇dblu/ba∇dblNs=/summationdisplay k∈Z2ks/ba∇dblPku/ba∇dblNk. Obviously, Fk∩Zk⊂Yk, and thus Fs∩Zs⊂Ys. In the end of this section, we present a standard extension lemma (See Lemma 5.4 in [22]) giving the r elation between Xs,band other space-time norm.LANDAU-LIFSHITZ EQUATION 5 Lemma 2.1. Letk∈ZandBbe a space-time norm satisfying with some C(k) /ba∇dbleit0eit∆Pkf/ba∇dblB≤C(k)/ba∇dblPkf/ba∇dbl2 for anyt0∈Randf∈L2. Then /ba∇dblPku/ba∇dblB/lessorsimilarC(k)/ba∇dblPku/ba∇dblX0,1/2,1. 3.Uniform linear estimates In this section we prove some uniform linear estimates for the equat ion (1.5) with respect to the dissipative parameter. First we recall the known line ar estimates for the Schr¨ odinger equation, see [16] and [14]. Lemma 3.1. Assumen≥3. For any k∈Zwe have /ba∇dbleit∆Pkf/ba∇dbl L2 tL2n n−2 x∩L∞ tL2x/lessorsimilar/ba∇dblPkf/ba∇dbl2, (3.1) sup e∈Sn−1/ba∇dbleit∆Pkf/ba∇dblL2,∞ e/lessorsimilar2(n−1)k 2/ba∇dblPkf/ba∇dbl2, (3.2) sup e∈Sn−1/ba∇dbleit∆Pk,ef/ba∇dblL∞,2 e/lessorsimilar2−k 2/ba∇dblPkf/ba∇dbl2. (3.3) Lemma 3.2. Assumen≥3,u,Fsolves the equation: for ε >0 ut−i∆u−ε∆u=F(x,t), u(x,0) =u0. Then for b∈[1,∞] /ba∇dblPku/ba∇dblX0,1/2,b +/lessorsimilar/ba∇dblPku0/ba∇dblL2+/ba∇dblPkF/ba∇dblX0,−1/2,b +, (3.4) /ba∇dblPku/ba∇dblZk/lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2 t,x, (3.5) where the implicit constant is independent of ε. Proof.First we show the second inequality. We have u=eit∆+εt∆u0+/integraldisplayt 0ei(t−s)∆+ε(t−s)∆F(s)ds (3.6) and thus /ba∇dblPku/ba∇dblZk/lessorsimilarε2k/ba∇dblPku/ba∇dblL2 t,x+2−k/ba∇dblPkF/ba∇dblL2 t,x /lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2 t,x. Now we show the first inequality. We only prove the case b=∞since the other cases aresimilar. First we assume F= 0. Then u=eit∆+εt∆u0. Let ˜u=eit∆+ε|t|∆u0, then ˜uis an extension of u. Then /ba∇dblPku/ba∇dblX0,1/2,∞ +/lessorsimilarsup j2j/2/ba∇dblFt(e−it|ξ|2−ε|t|·|ξ|2)(τ)ˆu0(ξ)χk(ξ)χj(τ+|ξ|2)/ba∇dblL2 τ,ξ /lessorsimilarsup j2j/2/ba∇dbl(ε|ξ|2)−1/parenleftbigg 1+|τ+|ξ|2|2 (ε|ξ|2)2/parenrightbigg−1 ˆu0(ξ)χk(ξ)χj(τ+|ξ|2)/ba∇dblL2 τ,ξ /lessorsimilar/ba∇dblPku0/ba∇dbl2. Next we assume u0= 0. Fix an extension ˜FofFsuch that /ba∇dbl˜F/ba∇dblX0,−1/2,∞≤2/ba∇dblF/ba∇dblX0,−1/2,∞ +.6 Z. GUO AND C. HUANG Then define ˜ u=F−11 τ+|ξ|2+iε|ξ|2F˜F. We see ˜ uis an extension of uand then /ba∇dblPku/ba∇dblX0,1/2,∞ +/lessorsimilar/ba∇dbl˜u/ba∇dblX0,1/2,∞ /lessorsimilarsup j2j/2/ba∇dblχk(ξ)χj(τ+|ξ|2)1 τ+|ξ|2+iε|ξ|2F˜F/ba∇dblL2 τ,ξ /lessorsimilar/ba∇dbl˜F/ba∇dblX0,−1/2,∞/lessorsimilar/ba∇dblF/ba∇dblX0,−1/2,∞ +. Thus we complete the proof. /square Lemma 3.3. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation ut−i∆u−ε∆u=F(x,t), u(x,0) =u0. Then for any e∈Sn−1we have /ba∇dblPku/ba∇dblL2,∞ e/lessorsimilar2k(n−1)/2/ba∇dblu0/ba∇dblL2+2k(n−2)/2sup e∈Sn−1/ba∇dblF/ba∇dblL1,2 e, (3.7) /ba∇dblPk,eu/ba∇dblL∞,2 e/lessorsimilar2−k/2/ba∇dblu0/ba∇dblL2+2−ksup e∈Sn−1/ba∇dblF/ba∇dblL1,2 e, (3.8) where the implicit constant is independent of ε. Proof.Ifε= 0, then the inequalities were proved in [14]. By the scaling and rotational invariance, we may assume k= 0 and e= (1,0,···,0). Then the second inequality follows from Proposition 2.5, 2.7 in [10]. We prove the first ineq uality by the following two steps. Step 1: F= 0. From the fact |eεt∆f(·,t)(x)| ≤(εt)−n/2/integraldisplay e−|x−y|2 2εt|f(y,t)|dy /lessorsimilar(εt)−n/2/integraldisplay e−|x−y|2 2εt/ba∇dblf(y,t)/ba∇dblL∞ tdy we get /ba∇dbleit∆+εt∆P0u0/ba∇dblL2x1L∞ ¯x1,t/lessorsimilar/ba∇dbleit∆P0u0/ba∇dblL2x1L∞ ¯x1,t/lessorsimilar/ba∇dblu0/ba∇dbl2. Step 2: u0= 0. Decompose P0u=U1+···Unsuch that FxUiis supported in {|ξ| ∼1 :|ξi| ∼ 1}×R. Thus it suffices to show /ba∇dblUi/ba∇dblL2x1L∞ ¯x,t/lessorsimilar/ba∇dblF/ba∇dblL1,2 ei. (3.9) We only show the estimate for U1. We still write u=U1. We assume FxFis supported in {|ξ| ∼1 :ξ1∼1}×R. We have u(t,x) =/integraldisplay Rn+1eitτeixξ τ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)dξdτ =/integraldisplay Rn+1eitτeixξ τ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)(1{−τ−|¯ξ|2∼1}c+1−τ−|¯ξ|2∼1,|τ+|ξ|2|/lessorsimilarε +1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫ε)dξdτ :=u1+u2+u3. Foru1, we simply use the Plancherel equality and get /ba∇dbl∆u1/ba∇dblL2+/ba∇dbl∂tu1/ba∇dbl2≤ /ba∇dblF/ba∇dbl2,LANDAU-LIFSHITZ EQUATION 7 and thus by Sobolev embedding and Bernstein’s inequality we obtain th e desired estimate. For u2, using the Lemma 2.1, Lemma 3.1 and Bernstein’s inequality we get /ba∇dblu2/ba∇dblL2x1L∞ ¯x1,t/lessorsimilar/ba∇dblu2/ba∇dbl˙X0,1/2,1/lessorsimilarε−1/2/ba∇dbl/hatwideF/ba∇dblL2/lessorsimilar/ba∇dblF/ba∇dblL1,2 e1. Now we estimate u3. Since|τ+|ξ|2| ≫ε, we have 1 τ+|ξ|2+iε|ξ|2=1 τ+|ξ|2+∞/summationdisplay k=1(−iε|ξ|2)k (τ+|ξ|2)k+1. Moreover, let s= (−τ−|¯ξ|2)1/2. Thenτ+|ξ|2=−(s−ξ1)(s+ξ1), and thus we get |s−ξ1| ≫ε,|s+ξ1| ∼1 (τ+|ξ|2)−1=−1 2s(1 s−ξ1+1 s+ξ1) =−1 2s(s−ξ1)(1+s−ξ1 s+ξ1). Hence 1 τ+|ξ|2+iε|ξ|2=1 τ+|ξ|2+∞/summationdisplay k=1(−iε|ξ|2)k (2s(ξ1−s))k+1 +∞/summationdisplay k=1(−iε|ξ|2)k (2s(ξ1−s))k+1[(1+s−ξ1 s+ξ1)k+1−1] :=a1(ξ,τ)+a2(ξ,τ)+a3(ξ,τ). Inserting this identity into the expression of u3, then we have u3=u1 3+u2 3+u3 3, where uj 3=/integraldisplay Rn+1eitτeixξaj(ξ,τ)/hatwideF(ξ,τ)1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫εdξdτ, j = 1,2,3. Foru1 3, this corresponds to the case ε= 0 which is proved in [14]. For u3 3, we can control it similarly as u1, since |a3(ξ,τ)|/lessorsimilar∞/summationdisplay k=1εk|ξ|2k (2|s(s−ξ1)|)k+1(k+1)|s−ξ1| |s+ξ1|/lessorsimilar1. It remains to control u2 3. LetGk(x1,¯ξ,τ) =F−1 x11−τ−|¯ξ|2∼11|ξ|∼1|ξ|2k/hatwideF. Note that /ba∇dblGk/ba∇dblL1,2 e1/lessorsimilarck/ba∇dblF/ba∇dblL1,2 e1. Then u2 3=∞/summationdisplay k=1(−iε)k/integraldisplay R/integraldisplay Rn+1eitτeixξ1|s−ξ1|≫ε (2s(ξ1−s))k+1[e−iy1ξ1Gk(y1,¯ξ,τ)]dξdτdy 1 =∞/summationdisplay k=1(−iε)k/integraldisplay RTk y1(G(y1,·))(t,x)dy1 where Tk y1(f)(t,x) =/integraldisplay Rn+1eitτeixξ1|s−ξ1|≫ε (2s(ξ1−s))k+1[e−iy1ξ1f(y1,¯ξ,τ)]dξdτ.8 Z. GUO AND C. HUANG We have Tk y1(f)(t,x) =/integraldisplay Rn/integraldisplay Rei(x1−y1)ξ11|s−ξ1|≫ε (ξ1−s)k+1dξ1·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ =/integraldisplay Rei(x1−y1)ξ11|ξ1|≫ε (ξ1)k+1dξ1·/integraldisplay Rnei(x1−y1)s(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ. Then we get |Tk y1(f)(t,x)|/lessorsimilarM−kε−k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Rnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Making a change of variable η1=s=/radicalbig −τ−|¯ξ|2,dτ=−2η1dη1, we obtain /integraldisplay Rnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ =/integraldisplay Rnei(x1−y1)η1·(2η1)−keit(η2 1+|¯ξ|2)ei¯x·¯ξ[f(y1,¯ξ,η2 1+|¯ξ|2)]d¯ξdτ. Thus, by the linear estimate (see Lemma 3.1) we get /ba∇dblTk y1(f)/ba∇dbl L2x1L∞ ¯x,t/lessorsimilarM−kε−k/ba∇dblf/ba∇dbl2, which suffices to give the estimate for u2 3. We complete the proof of the lemma. /square Lemma 3.4. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation ut−i∆u−ε∆u=F(x,t), u(x,0) =u0. Then for any e∈Sn−1we have /ba∇dblPku/ba∇dbl L∞ tL2x∩L2 tL2n n−2 x/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblNk, (3.10) where the implicit constant is independent of ε. Proof.Since we have for t >0 /ba∇dbleit∆+εt∆u0/ba∇dblL∞x/lessorsimilart−n/2/ba∇dblu0/ba∇dblL1x /ba∇dbleit∆+εt∆u0/ba∇dblL2x/lessorsimilar/ba∇dblu0/ba∇dblL2x where the implicit constant is independent of ε, then by the abstract framework of Keel-Tao [16] we get the Strichartz estimates /ba∇dblPku/ba∇dbl L∞ tL2x∩L2 tL2n n−2 x/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblL1 tL2x, with the implicit constant independent of ε. By the same argument as in Step 2 of the proof of Lemma 3.3, we get /ba∇dblPku/ba∇dbl L∞ tL2x∩L2 tL2n n−2 x/lessorsimilar/ba∇dblu0/ba∇dblL2+2−k/2sup e∈Sn−1/ba∇dblF/ba∇dblL1,2 e. On the other hand, by Lemma 2.1, Lemma 3.1 and Lemma 3.2, we get /ba∇dblPku/ba∇dbl L∞ tL2x∩L2 tL2n n−2 x/lessorsimilar/ba∇dblPku/ba∇dblX0,1/2,1 +/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblX0,−1/2,1 +. Thus we complete the proof. /square Gathering the above lemmas, we can get the following linear estimates :LANDAU-LIFSHITZ EQUATION 9 Lemma 3.5 (Linear estimates) .Assumen≥3,u,Fsolves the equation: for ε >0 ut−i∆u−ε∆u=F(x,t), u(x,0) =u0. Then for s∈R /ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs 2,1+/ba∇dblF/ba∇dblNs, (3.11) where the implicit constant is independent of ε. 4.Nonlinear estimates In this section we prove some nonlinear estimates. The nonlinear ter m in the Landau-Lifshitz equation is G(u) =¯u 1+|u|2n/summationdisplay j=1(∂xju)2. By Taylor’s expansion, if /ba∇dblu/ba∇dbl∞<1 we have G(u) =∞/summationdisplay k=0¯u(−1)k|u|2kn/summationdisplay j=1(∂xju)2. So we will need to do multilinear estimates. Lemma 4.1. (1) Ifj≥2k−100andXis a space-time translation invariant Banach space, then Q≤jPkis bounded on Xwith bound independent of j,k. (2) For any j,k,Q≤jPk,eis bounded on Lp,2 eandQ≤jis bounded on Lp tL2 xfor 1≤p≤ ∞, with bound independent of j,k. Proof.See the proof of Lemma 5.4 in [8]. /square Lemma 4.2. Assumen≥3,k1,k2,k3∈Z. Then /ba∇dblPk1uPk2v/ba∇dblL2 t,x/lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2,(4.1) /ba∇dblPk3(Pk1uPk2v)/ba∇dblL2 t,x/lessorsimilar2(n−2)min( k1,k2,k3) 2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblYk2. (4.2) Proof.For the first inequality, we have /ba∇dblPk1uPk2v/ba∇dblL2 t,x/lessorsimilarn/summationdisplay j=1/ba∇dblPk1uPk2,ejΘj k2v/ba∇dblL2 t,x/lessorsimilarn/summationdisplay j=1/ba∇dblPk1u/ba∇dblL2,∞ ej/ba∇dblPk2,ejv/ba∇dblL∞,2 ej /lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2. For the second inequality, if k3≤min(k1,k2), then /ba∇dblPk3(Pk1uPk2v)/ba∇dblL2 t,x/lessorsimilar2k3(n−2)/2/ba∇dblPk1uPk2v/ba∇dbl L2 tL2n 2n−2 x /lessorsimilar2k3(n−2)/2/ba∇dblPk1u/ba∇dbl L2 tL2n n−2 x/ba∇dblPk2v/ba∇dblL∞ tL2x. Ifk1≤min(k2,k3), then /ba∇dblPk3(Pk1uPk2v)/ba∇dblL2 t,x/lessorsimilar/ba∇dblPk1u/ba∇dblL2 tL∞x/ba∇dblPk2v/ba∇dblL∞ tL2x /lessorsimilar2k1(n−2)/2/ba∇dblPk1u/ba∇dbl L2 tL2n n−2 x/ba∇dblPk2v/ba∇dblL∞ tL2x. Ifk2≤min(k1,k3), the proof is identical to the above case. /square10 Z. GUO AND C. HUANG Lemma 4.3 (Algebra properties) .Ifs≥n/2, then we have /ba∇dbluv/ba∇dblYs/lessorsimilar/ba∇dblu/ba∇dblYs/ba∇dblv/ba∇dblYn/2+/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblYs. Proof.We only show the case s=n/2. By the embedding ˙Bn/2 2,1⊂L∞ xwe get /ba∇dblu/ba∇dblL∞ x,t≤ /ba∇dblu/ba∇dblL∞ t˙Bn/2 2,1/lessorsimilar/ba∇dblu/ba∇dblYn/2. The Lebesgue component can be easily handled by para-product de composition and H¨ older’s inequality. Now we deal with Xs,b-type space. By (2.3) it suffices to show /summationdisplay k2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2, (4.3) We have /summationdisplay k2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1 /lessorsimilar/summationdisplay ki2nk3/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1 /lessorsimilar(/summationdisplay ki:k1≤k2+/summationdisplay ki:k1>k2)2k3n/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1 :=I+II. By symmetry, we only need to estimate the term I. AssumePk1f=Pk1f1+Pk1f2,Pk2g=Pk2g1+Pk2g2such that /ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk1f/ba∇dblX0,1+¯X0,1, /ba∇dblPk2g1/ba∇dblX0,1+/ba∇dblPk2g2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk2g/ba∇dblX0,1+¯X0,1. Then we have I/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1 +/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g2)/ba∇dbl¯X0,1 :=I1+I2. We only estimate the term I1since the term I2can be estimated in a similar way. We have I1/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1. (4.4) First we assume k3≤k1+5 in the summation of (4.4). We have I1/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1 +/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1) :=I11+I12.LANDAU-LIFSHITZ EQUATION 11 For the term I11we have I11/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12k3n/22−k32k1+k2/ba∇dblPk1fj/ba∇dblL∞ tLnx/ba∇dblPk2g1/ba∇dbl L2 tL2n n−2 x /lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12k3n/22−k32k22k1n/2/ba∇dblPk1fj/ba∇dblL∞ tL2x/ba∇dblPk2g1/ba∇dbl L2 tL2n n−2 x /lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2. For the term I12, we need to exploit the nonlinear interactions as in [8]. We have FPk3Q≥k1+k2+10(Pk1fjPk2g1) =χk3(ξ3)χ≥k1+k2+10(τ3+|ξ3|2)/integraldisplay ξ3=ξ1+ξ2,τ3=τ1+τ2χk1(ξ1)/hatwidefj(τ1,ξ1)χk2(ξ2)/hatwideg1(τ2,ξ2). We assume j= 1 since j= 2 is similar. On the plane {ξ3=ξ1+ξ2,τ3=τ1+τ2}we have τ3+|ξ3|2=τ1+|ξ1|2+τ2+|ξ2|2−H(ξ1,ξ2) (4.5) whereHis the resonance function in the product Pk3(Pk1fjPk2g1) H(ξ1,ξ2) =|ξ1|2+|ξ2|2−|ξ1+ξ2|2. (4.6) Since|H|/lessorsimilar2k1+k2, then one of Pk1fj,Pk2g1has modulation larger than the output modulation, namely max(|τ1+|ξ1|2|,|τ2+|ξ2|2|)/greaterorsimilar|τ3+|ξ3|2|. IfPk1fjhas larger modulation, then I12/lessorsimilar/summationdisplay ki:k1≤k22nk3/22−k3/ba∇dbl2j3/ba∇dblPk3Qj3(Pk1fjPk2g1)/ba∇dblL2 t,x/ba∇dbll2 j3≥k1+k2+10 /lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22nk3/22−k3(/summationdisplay j3≥k1+k2+1022j3/ba∇dblQ≥j3Pk1fj/ba∇dbl2 L2 t,x)1/2/ba∇dblPk2g1/ba∇dblL∞ tL2x /lessorsimilar/summationdisplay ki:k1≤k22nk32−k3(/ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1)/ba∇dblPk2g1/ba∇dblYk2 /lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2. IfPk2g1has larger modulation, then I12/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞ t,x(/summationdisplay j3≥k1+k222j3/ba∇dblPk2Q≥j3g1/ba∇dbl2 L2 tL2x)1/2 /lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k32nk1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblX0,1 /lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.12 Z. GUO AND C. HUANG Now we assume k3≥k1+6 in the summation of (4.4) and thus |k2−k3| ≤4. We have I1/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1 +/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1) :=˜I11+˜I12. By Lemma 4.2 we get ˜I11/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1 /lessorsimilar/summationdisplay ki:k1≤k22nk3/22k12(n−2)k1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblYk2/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2. For the term ˜I12, similarly as the term I12, one ofPk1fj,Pk2g1has modulation larger than the output modulation. If Pk1fjhas larger modulation, then ˜I12/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3(/summationdisplay j322j3/ba∇dblPk1Q≥j3fj/ba∇dbl2 L2 tL∞x)1/2/ba∇dblPk2g1/ba∇dblL∞ tL2x /lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2. IfPk2g1has larger modulation, then ˜I12/lessorsimilar/summationdisplay ki:k1≤k22/summationdisplay j=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞ tL∞x(/summationdisplay j322j3/ba∇dblPk2Q≥j3g1/ba∇dbl2 L2 tL2x)1/2 /lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2. Thus, we complete the proof. /square Lemma 4.4. We have /summationdisplay kj2k3(n−2)/2/ba∇dblPk3[un/summationdisplay i=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2 t,x/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.(4.7) Proof.We have LHS of (4.7) /lessorsimilar/summationdisplay kj2k3(n−2)/2/ba∇dblPk3[P≥k3−10un/summationdisplay i=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2 t,x +/summationdisplay kj2k3(n−2)/2/ba∇dblPk3[P≤k3−10un/summationdisplay i=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2 t,x :=I+II.LANDAU-LIFSHITZ EQUATION 13 By symmetry, we may assume k1≤k2in the above summation. For the term II, sincen≥3, then we have II/lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay kj2k3(n−2)/2/ba∇dbl˜Pk3n/summationdisplay i=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2 t,x /lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay k1,k3≤k2+52k3(n−2)/22k1+k22[(n−1)k1−k2]/2/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. For the term I, ifk3≤k2+20, then we get from Lemma 4.2 that I/lessorsimilar/summationdisplay kj2nk3/22k3(n−2)/2/ba∇dblP≥k3−10u/ba∇dblL∞ tL2x/ba∇dbln/summationdisplay i=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2 t,x /lessorsimilar/summationdisplay kj2nk3/22k3(n−2)/22(n+1)k1/22k2/2/ba∇dblP≥k3−10u/ba∇dblL∞ tL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. Ifk3≥k2+20, then uhas frequency ∼2k3, and thus we get I/lessorsimilar/summationdisplay kj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞ tL2x/ba∇dbln/summationdisplay i=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2 tL∞x /lessorsimilar/summationdisplay kj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞ tL2x2nk2/2/ba∇dbln/summationdisplay i=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2 tL2x /lessorsimilar/summationdisplay kj2k3(n−2)/22k1+k22(n−1)k1/22(n−1)k2/2/ba∇dblPk3u/ba∇dblL∞ tL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. Therefore we complete the proof. /square Lemma 4.5. We have /ba∇dblun/summationdisplay i=1(∂xiv∂xiw)/ba∇dblNn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2∩Zn/2/ba∇dblw/ba∇dblFn/2∩Zn/2. (4.8) Proof.By the definition of Nn/2, theL2component was handled by the previous lemma. We only need to control /summationdisplay ki2k4n/2/ba∇dblPk4[Pk1un/summationdisplay i=1(Pk2∂xiv∂xiPk3w)]/ba∇dblNk4. (4.9)14 Z. GUO AND C. HUANG By symmetry we may assume k2≤k3in the above summation. If in the above summation we assume k4≤k1+40, then (4.9)/lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1 tL2x /lessorsimilar/summationdisplay ki2k1n/2/ba∇dblPk1u/ba∇dblL∞ tL2x2k2/ba∇dblPk2v/ba∇dblL2 tL∞x2k3/ba∇dblPk3w/ba∇dblL2 tL∞x /lessorsimilar/summationdisplay ki2k1n/2/ba∇dblPk1u/ba∇dblL∞ tL2x2k2n/2/ba∇dblPk2v/ba∇dbl L2 tL2n n−2 x2k3n/2/ba∇dblPk3w/ba∇dbl L2 tL2n n−2 x /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. Thus from now on we assume k4≥k1+ 40 in the summation of (4.9). We bound the summation case by case. Case 1: k2≤k1+20 In this case we have k4≥k2+20 and hence |k4−k3| ≤5. By Lemma 4.2 we get (4.9)/lessorsimilar/summationdisplay ki2k4n/22−k4/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1,2 e /lessorsimilar/summationdisplay ki2k4n/22−k4/2/ba∇dblPk1uPk3∂xiw/ba∇dblL2 x,t/ba∇dblPk2∂xiv/ba∇dblL2,∞ e /lessorsimilar/summationdisplay ki2k4n/22−k4/22(n−1)k1/22−k3/22(n−1)k2/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk3∂xiw/ba∇dblFk3/ba∇dblPk2∂xiv/ba∇dblFk2 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. Case 2: k2≥k1+21 In this case we have k4≤k3+40. Let g=/summationtextn i=1(Pk2∂xiv·Pk3∂xiw). Then we have (4.9)/lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1uQ≤k2+k3g]/ba∇dblNk4+/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1uQ≥k2+k3g]/ba∇dblNk4 :=I+II. First we estimate the term II. We have II/lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4 +/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4 :=II1+II2.LANDAU-LIFSHITZ EQUATION 15 For the term II1we have II1/lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblL1 tL2x /lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2 tL∞x/ba∇dblQ≥k2+k3g]/ba∇dblL2 tL2x /lessorsimilar/summationdisplay ki2k4n/22k1n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2 tL2x/ba∇dblQ≥k2+k3g]/ba∇dblL2 tL2x /lessorsimilar/summationdisplay ki2k4n/22k1n/22−(k2+k3)/ba∇dblPk1Q≥2k1+10u/ba∇dblX0,1 ·2[(n−1)k2−k3]/22k2+k3/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. For the term II2, sincek4≥k1+ 40, then we may assume ghas frequency of size 2k4. The resonance function in the product Pk1u·Pk4gis of size /lessorsimilar2k1+k4. Thus the output modulation is of size /greaterorsimilar2k2+k3. Then we get II2/lessorsimilar/summationdisplay ki2k4n/22−(k2+k3)/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblL2 t,x /lessorsimilar/summationdisplay ki2k4n/22−(k2+k3)/22k1n/2/ba∇dblPk1u/ba∇dblL∞ tL2x·/ba∇dblg/ba∇dblL2 t,x /lessorsimilar/summationdisplay ki2k4n/22−(k2+k3)/22k1n/22[(n−1)k2−k3]/22k2+k3/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. Now we estimate the term I. We have I/lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay i=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiw)]/ba∇dblNk4 +/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay i=1(Pk2∂xiQ≤k2+k3+39v·Pk3∂xiw)]/ba∇dblNk4 :=I1+I2. For the term I1, since the resonance function in the product Pk2v·Pk3wis of size /lessorsimilar2k2+k3, then we may assume Pk3whas modulation of size /greaterorsimilar2k2+k3. Then we get I1/lessorsimilar/summationdisplay ki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay i=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiQ≥k2+k3−5w)]/ba∇dblL1,2 e /lessorsimilar/summationdisplay ki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL∞ t,x2k2+k3/ba∇dblPk2v/ba∇dblL2,∞ e/ba∇dblPk3Q≥k2+k3−5w/ba∇dblL2 t,x /lessorsimilar/summationdisplay ki2k4n/22−k4/22(k2+k3)/22(n−1)k2/22k1n/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.16 Z. GUO AND C. HUANG Finally, we estimate the term I2. For this term, we need to use the null structure observed by Bejenaru [1]. We can rewrite −2∇u·∇v= (i∂t+∆)u·v+u·(i∂t+∆)v−(i∂t+∆)(u·v).(4.10) Then we have I2=/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblNk4 +/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Lw)]/ba∇dblNk4 +/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4 :=I21+I22+I23. For the term I21, we have I21/lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblL1 tL2x /lessorsimilar/summationdisplay ki2k4n/22k1n/2/ba∇dblPk1u/ba∇dblL∞ tL2x2k2(n−2)/2/ba∇dblPk2Lv/ba∇dblL2 t,x/ba∇dblPk3w/ba∇dbl L2 tL2n n−2 x /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblZn/2/ba∇dblw/ba∇dblFn/2. For the term I22, we may assume whas modulation /lessorsimilar2k2+k3. Then we get I22/lessorsimilar/summationdisplay ki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Q≤k2+k3+100Lw)]/ba∇dblL1,2 e /lessorsimilar/summationdisplay ki2k4n/22−k4/22k1n/2/ba∇dblPk1u/ba∇dblL∞ tL2x/ba∇dblPk2v/ba∇dblL2,∞ e/ba∇dblPk3Q≤k2+k3+100Lw)]/ba∇dblL2 t,x /lessorsimilar/summationdisplay ki2k4n/22−k4/22nk1/22(n−1)k2/22(k2+k3)/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblX0,1/2,∞ /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. Next we estimate the term I23. We have I23/lessorsimilar/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q[k1+k4+100,k2+k3]L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4 +/summationdisplay ki2k4n/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4 :=I231+I232. For the term I232we have I232/lessorsimilar/summationdisplay ki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1,2 e /lessorsimilar/summationdisplay ki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL2,∞ e2k1+k42[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.LANDAU-LIFSHITZ EQUATION 17 For the term I231we have I231/lessorsimilar/summationdisplay kik2+k3/summationdisplay j2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4 +/summationdisplay kik2+k3/summationdisplay j2=k1+k4+1002k4n/2/ba∇dblPk4Q≥j2−9[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4 :=I2311+I2312. For the term I2312we have I2312/lessorsimilar/summationdisplay kik2+k3/summationdisplay j2=k1+k4+100/summationdisplay j3≥k2−92k4n/22−j3/2 ·/ba∇dblPk4Qj3[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL2 t,x /lessorsimilar/summationdisplay ki2k4n/22k1n/22(k2+k3)/22[(n−1)k2−k3]/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. For the term I2311we have I2311/lessorsimilar/summationdisplay kik2+k3/summationdisplay j2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1˜Qj2u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1 tL2x /lessorsimilar/summationdisplay kik2+k3/summationdisplay j2=k1+k4+1002k4n/22k1n/2/ba∇dblPk1˜Qj2u/ba∇dblL2 t,x2j22[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3 /lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2. Therefore, we complete the proof. /square Combining all the estimates above we get Lemma 4.6 (Nonlinear estimates) .Assumeu∈Fn/2∩Zn/2with/ba∇dblu/ba∇dblYn/2≪1. Then/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble¯u 1+|u|2n/summationdisplay j=1(∂xju)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble Nn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2 1−/ba∇dblu/ba∇dbl2 Yn/2/ba∇dblu/ba∇dblFn/2∩Zn/2/ba∇dblu/ba∇dblFn/2∩Zn/2. Proof.SinceYn/2⊂L∞, then ¯u 1+|u|2n/summationdisplay j=1(∂xju)2=∞/summationdisplay k=0¯u(−1)k|u|2kn/summationdisplay j=1(∂xju)2. The lemma follows from Lemma 4.5, Lemma 4.4 and Lemma 4.3. /square 5.The limit behaviour In this section we prove Theorem 1.1. It is equivalent to prove Theorem 5.1. Assumen≥3,ε∈[0,1]. There exists 0< δ≪1such for any φ∈˙Bn/2 2,1with/ba∇dblφ/ba∇dbl˙Bn/2 2,1≤δ, there exists a unique global solution uεto(1.5)such that /ba∇dbluε/ba∇dblFn/2∩Zn/2/lessorsimilarδ,18 Z. GUO AND C. HUANG where the implicit constant is independent of ε. The map φ→uεis Lipshitz from Bδ(˙Bn/2 2,1)toC(R;˙Bn/2 2,1)and the Lipshitz constant is independent of ε. Moreover, for anyT >0, lim ε→0+/ba∇dbluε−u0/ba∇dblC([0,T];˙Bn/2 2,1)= 0. Fortheuniformglobalwell-posedness, wecanproveitbystandard Picarditeration argument by using the linear and nonlinear estimates proved in the pr evious section. Indeed, define Φu0(u) :=eit∆+εt∆u0 −i/integraldisplayt 0ei(t−s)∆+ε(t−s)∆/bracketleftbigg2¯u 1+|u|2n/summationdisplay j=1(∂xju)2−2iε¯u 1+|u|2n/summationdisplay j=1(∂xju)2/bracketrightbigg ds. ThenusingtheLemma3.5andLemma4.6wecanshowΦ u0isacontractionmapping in the set {u:/ba∇dblu/ba∇dblFn/2∩Zn/2≤Cδ} if/ba∇dblu0/ba∇dbl˙Bn/2 2,1≤δwithδ >0sufficiently small. Thus wehaveexistence anduniqueness. Moreover, by standard arguments we immediately have the persist ence of regularity, namely if u0∈˙Bs 2,1for some s > n/2, thenu∈Fs∩Zsand /ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs 2,1(5.1) uniformly with respect to ε∈(0,1]. Now we prove the limit behaviour. Assume uεis a solution to the Landau-Lifshitz equation with small initial data φ1∈˙Bn/2 2,1, anduis a solution to the Schr¨ odinger map with small initial data φ2∈˙Bn/2 2,1. Letw=uε−u,φ=φ1−φ2, thenwsolves (i∂t+∆)w=iε∆uε+/bracketleftbigg2¯uε 1+|uε|2n/summationdisplay j=1(∂xjuε)2−2¯u 1+|u|2n/summationdisplay j=1(∂xju)2/bracketrightbigg −2iε¯uε 1+|uε|2n/summationdisplay j=1(∂xjuε)2, (5.2) w(0) =φ. First we assume in addition φ1∈˙B(n+4)/2 2,1. By the linear and nonlinear estimates, for anyT >0 we get /ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2 2,1+εT/ba∇dbluε/ba∇dblL∞ t˙B(n+4)/2 2,1+δ2/ba∇dblw/ba∇dblFn/2∩Zn/2+ε/ba∇dbluε/ba∇dbl3 Fn/2∩Zn/2. Then we get by (5.1) /ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2 2,1+εT/ba∇dblφ1/ba∇dbl˙B(n+4)/2 2,1+εδ3. (5.3) Now we assume φ1=φ2=ϕ∈˙Bn/2 2,1with small norm. For fixed T >0, we need to prove that ∀η >0, there exists σ >0 such that if 0 < ε < σthen /ba∇dblSε T(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2 2,1)< η (5.4)LANDAU-LIFSHITZ EQUATION 19 whereSε Tis the solution map corresponding to (5.2) and ST=S0 T. We denote ϕK=P≤Kϕ. Then we get /ba∇dblSε T(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2 2,1) ≤/ba∇dblSε T(ϕ)−Sε T(ϕK)/ba∇dblC([0,T];˙Bn/2 2,1) +/ba∇dblSε T(ϕK)−ST(ϕK)/ba∇dblC([0,T];˙Bn/2 2,1)+/ba∇dblST(ϕK)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2 2,1). From the uniform global well-posedness and (5.3), we get /ba∇dblSǫ T(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2 2,1)/lessorsimilar/ba∇dblϕK−ϕ/ba∇dbl˙Bn/2 2,1+εC(T,K,/ba∇dblϕ/ba∇dbl˙Bn/2 2,1).(5.5) We first fix Klarge enough, then let εgo to zero, therefore (5.4) holds. Acknowledgment. Z. Guo is supported in part by NNSF of China (No.11371037), and C. Huang is supported in part by NNSF of China (No. 11201498). References [1] I. Bejenaru, On Schr¨ odinger maps, Amer. J. Math. 130 (2008 ), 1033-1065. [2] I. Bejenaru, Global results for Schr¨ odinger maps in dimensions n≥3, Comm. Partial Differ- ential Equations 33 (2008), 451-477. [3] I. Bejenaru, A. D. Ionescu, and C. E. Kenig, Global existence a nd uniqueness of Schr¨ odinger maps in dimensions d≥4, Adv. Math. 215 (2007), 263-291. [4] I. Bejenaru, A. D. Ionescu, C. E. Kenigand D. Tataru, Global S chr¨ odingermapsin dimensions d≥2: small data in the critical Sobolevspaces, Annals ofMathematics1 73(2011), 1443-1506. [5] N.H. Chang,J.Shatah, K.Uhlenbeck, Schr¨ odingermaps, Com m.PureAppl. Math.53(2000), 590-602. [6] W. Ding and Y. Wang, Local Schr¨ odinger flow into K¨ ahler manifold s, Sci. China Ser. A, 44 (2001), 1446-1464. [7] B. Guo and S. Ding. Landau-Lifshitz equations, volume 1 of Front iers of Research with the Chinese Academy of Sciences. World Scientific Publishing Co. Pte. Ltd ., Hackensack, NJ, 2008. [8] Z. Guo, Spherically averaged maximal function and scattering fo r the 2D cubic derivative Schr¨ odinger equation, to appear Int. Math. Res. Notices. [9] Z. Guo and B. Wang, Global well posedness and inviscid limit for the K orteweg-de Vries- Burgers equation, J. Differential Equations 246 (2009) 3864-390 1. [10] L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzbu rg-Landau equation with small data in modulation and Sobolev spaces, Appl. Comput. Harmon. Anal. 32(2012),no.2, 197-222. [11] T. Hmidi and S. Keraani, Inviscid limit for the two-dimensional N-S system in a critical Besov space, Asymptot. Anal., 53 (3) (2007), 125-138. [12] C. Huang, B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation, J. Funct. Anal., 255 (2008), 681-725. [13] A. D. Ionescu and C. E. Kenig, Low-regularity Schr¨ odinger ma ps, Differential Integral Equa- tions 19 (2006), 1271-1300. [14] A. D. Ionescu and C. E. Kenig, Low-regularity Schr¨ odinger ma ps, II: global well-posedness in dimensions d≥3, Comm. Math. Phys., 271 (2007), 523-559. [15] C. E. Kenig, G. Ponce, and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schr¨ odinger equations, Invent. Math., 134 (1998), 489-545. [16] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. M ath., 120 (1998), 360–413. [17] M. Lakshmanan. The fascinating world of the Landau-Lifshitz- Gilbert equation: an overview. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (193 9):1280-1300, 2011. [18] M. Lakshmanan and K. Nakamura, Landau-Lifshitz Equation of Ferromagnetism: Exact Treatment of the Gilbert Damping, Phy. Rev. Let. 53 (1984), NO. 2 6, 2497-2499. [19] L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sovietunion. 8 (1935) 153-169.20 Z. GUO AND C. HUANG [20] T. Tao, Global regularity of wave maps II. Small energy in two dim ensions, Commun. Math. Phys. 224 (2001), 443-544. [21] D. Tataru, Local andglobalresults forthe wavemaps I, Comm . PartialDifferential Equations, 23 (1998), no. 9-10, 1781-1793. [22] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method fo r Nonlinear Evolution Equations, I, World Scientific Press, 2011. [23] B. Wang, The limit behavior of solutions for the Cauchy problem of the Complex Ginzburg- Landau equation, Commu. Pure. Appl. Math., 55 (2002), 0481-050 8. [24] B. Wang and Y. Wang, The inviscid limit for the derivative Ginzburg- Landau equations, J. Math. Pures Appl., 83 (2004), 477-502. School of Mathematical Sciences, Monash University, VIC 38 00, Australia & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China E-mail address :zihua.guo@monash.edu School of Statistics and Mathematics, Central University o f Finance and Eco- nomics, Beijing 100081, China E-mail address :hcy@cufe.edu.cn

1709.04911v2.Intrinsic_Damping_Phenomena_from_Quantum_to_Classical_Magnets_An_ab_initio_Study_of_Gilbert_Damping_in_Pt_Co_Bilayer.pdf

Intrinsic Damping Phenomena from Quantum to Classical Magnets: An ab-initio Study of Gilbert Damping in Pt/Co Bilayer Farzad Mahfouzi,1,Jinwoong Kim,1, 2and Nicholas Kioussis1,y 1Department of Physics and Astronomy, California State University, Northridge, CA, USA 2Department of Physics and Astronomy, Rutgers University, NJ, USA A fully quantum mechanical description of the precessional damping of Pt/Co bilayer is presented in the framework of the Keldysh Green function approach using ab initio electronic structure cal- culations. In contrast to previous calculations of classical Gilbert damping ( GD), we demonstrate thatGDin the quantum case does not diverge in the ballistic regime due to the nite size of the total spin, S. In the limit of S!1 we show that the formalism recovers the torque correla- tion expression for GDwhich we decompose into spin-pumping and spin-orbital torque correlation contributions. The formalism is generalized to take into account a self consistently determined de- phasing mechanism which preserves the conservation laws and allows the investigation of the eect of disorder. The dependence of GDon Pt thickness and disorder strength is calculated and the spin diusion length of Pt and spin mixing conductance of the bilayer are determined and compared with experiments. PACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg I. INTRODUCTION Magnetic materials provide an intellectually rich arena for fundamental scientic discovery and for the invention of faster, smaller and more energy-ecient technologies. The intimate relationship of charge transport and mag- netic structure in metallic systems on one hand, and the rich physics occurring at the interface between dierent materials in layered structures on the other hand, are the hallmark of the ourishing research eld of spintronics.1{5 Recently, intense focus has been placed on the signi- cant role played by spin-orbit coupling (SOC) and the ef- fect of interfacial inversion symmetry breaking on the dy- namics of the magnetization in ferromagnet (FM)-normal metal (NM) bilayer systems. Of prime importance to this eld is the (precessional) magnetization damping phe- nomena, usually treated phenomenologically by means of a parameter referred to as Gilbert damping constant, GD, in the LandauLifshitzGilbert (LLG) equation of motiond~ m=dt = ~ m~B+GD~ md~ m=dt , which de- scribes the rate of the angular momentum loss of the FM.6Here,~ mis the unit vector along the magnetization direction and ~Bis an eective magnetic eld. In FM/NM bilayer devices the eect of the NM on the Gilbert damping of the FM is typically considered as an additive eect, where the total Gilbert damping can be separated into an intrinsic bulk contribution and an inter- facial component due to the presence of the NM.7,8While the interfacial Gilbert damping is usually attributed to the loss of angular momentum due to pumped spin cur- rent into the NM,9,10in metallic bulk FMs the intrin- sic Gilbert damping constant is described by the cou- pling between the conduction electrons and the (time- dependent) magnetization degree of freedom.11 The conventional approach to determine the Gilbert damping constant involves calculating the imaginary part of the time-dependent susceptibility of the FM in thepresence of conduction electrons in the linear response regime.12{14In this case, the time-dependent magneti- zation term in the electronic Hamiltonian leads to the excitation of electrons close to the Fermi surface trans- ferring angular momentum to the conduction electrons. The excited electrons in turn relax to the ground state by interacting with their environment, namely through phonons, photons and/or collective spin/charge excita- tions. These interactions are typically parameterized phenomenologically by the broadening of the energy lev- els,=~=2, whereis the relaxation time of the elec- trons close to the Fermi surface. The phenomenological treatment of the electronic relaxation is valid when the energy broadening is small which corresponds to clean systems, i.e., D(EF)1, whereD(EF) is the den- sity of states per atom at the Fermi energy. In the case of large[D(EF)&1)] however, this approach vio- lates the conservation laws and a more accurate descrip- tion of the relaxation mechanism that preserves the en- ergy, charge and angular momentum conservation laws are required.15The importance of including the vertex corrections has already been pointed out in the literature when the Gilbert damping is dominated by the interband contribution,16{18i.e.,when there is a signicant number of states available within the energy window of around the Fermi energy. In this paper we investigate the magnetic damping phe- nomena through a dierent Lens in which the FM is as- sumed to be small and quantum mechanical. We show that in the limit of large magnetic moments we recover dierent conventional expressions for the Gilbert damp- ing of a classical FM. We calculate the Gilbert damping for a Pt/Co bilayer system versus the energy broaden- ing,and show that in the limit of clean systems and small magnetic moments the FM damping is governed by a coherent dynamics. We show that in the limit of large broadening  > 1meV which is typically the case at room temperature, the relaxation time approximationarXiv:1709.04911v2 [cond-mat.mes-hall] 14 Nov 20172 fails. Hence, we employ a self consistent approach pre- serving the conservation laws. We calculate the Gilbert damping versus the Pt and Co thicknesses and by tting the results to spin diusion model we calculate the spin diusion length and spin mixing conductance of Pt. II. THEORETICAL FORMALISM OF MAGNETIZATION DAMPING For a metallic FM the magnetization degree of freedom is inherently coupled to the electronic degrees of freedom of the conduction electrons. It is usually convenient to treat each degree of freedom separately with the corre- sponding time-dependent Hamiltonians that do not con- serve the energy. However, since the total energy of the system is conserved, it is possible to consider the total Hamiltonian of the combined system and solve the corre- sponding stationary equations of motion. For an isolated metallic FM the wave function of the coupled electron- magnetic moment conguration system is of the form, jm~ki=jS;mi j~ki, where the parameter Sdenotes the total spin of the nano-FM ( S! 1 in the classi- cal limit),m=S:::; +S, are the eigenvalues of the total Szof the nano-FM, refers to the Kronecker prod- uct, anddenotes the atomic orbitals and spin of the electron Bloch states. The single-quasi-particle retarded Green function and the corresponding density matrix can be obtained from,19  Ei^H~kHM1 2S^
~k^~ 
~S ^Gr ~k(E) =^1;(1) and ^~k=ZdE ^Gr ~k(E)f(EHM)^Ga ~k(E): (2) Here,HM= ~B
~S, is the Hamiltonian of the nano- FM in the presence of an external magnetic eld ~Bwith eigenstates,jS;mi, is the gyromagnetic ratio, f(E) is the Fermi-Dirac distribution function, ^~ is the vector of the Pauli matrices, ^H~kis the non-spin-polarized Hamilto- nian matrix in the presence of spin orbit coupling (SOC), and^
~kis the~k-dependent exchange splitting matrix, dis- cussed in detail in Sec. III. We employ the notation that bold symbols operate on jS;mibasis set and symbols with hat operate on the j~kis. Here, for simplicity we ignore explicitly writing the identity matrices ^1 and 1as well as the Kronecker product symbol in the expressions. A schematic description of the FM-Bloch electron en- tangled system and the damping process of the nano-FM is shown in Fig. 1. The presence of the magnetic Hamil- tonian in the Fermi distribution function in Eq. (2) act- ing as a chemical potential leads to transition between magnetic statesjS;mialong the direction in which the magnetic energy is minimized19. The transition rate of the FM from the excited states, jS;mi, to states with FIG. 1: (Color online) Schematic representation of the com- bined FM-Bloch electron system. The horizontal planes de- note the eigenstates, jS;miof the total Szof the nano-FM with eigenvalues m=S;S+ 1;:::; +S. For more details see Fig. 2 in Ref.19 . lower energy ( i.e.the damping rate) can be calculated from19, Tm=1 2=(T mT+ m); (3) where, T m=1 2SNX ~kTrel[^
~k^S m^~k;m;m1]: (4) Here,Nis the number of ~k-points in the rst Brillouin zone,Trel, is the trace over the Bloch electron degrees of freedom,S m=p S(S+ 1)m(m1), and ^ ^xi^y. The precessional Gilbert damping constant can be de- termined from conservation of the total angular mo- mentum by equating the change of angular momen- tum per unit cell for the Bloch electrons, Tm, and the magnetic moment obtained from LLG equation, GDMtotsin2()=2, which leads to, GD(m) =2 Mtot!sin2(m)Tm S2 Mtot!(S(S+ 1)m2)Tm: (5) Here, cos(m) =mp S(S+1), is the cone angle of precession andMtotis the total magnetic moment per unit cell in units of1 2gBwithgandBbeing the Land e factor and magneton Bohr respectively. The Larmor frequency, !, can be obtained from the eective magnetic eld along the precession axis, ~!= Bz. The exact treatment of the magnetic degree of freedom within the single domain dynamical regime oers a more accurate description of the damping phenomena that can be used even when the classical equation of motion LLG is not applicable. However, since in most cases of in- terest the FM behaves as a classical magnetic moment, where the adiabatic approximation can be employed to describe the magnetization dynamics, in the following two sections we consider the S!1 limit and close to adiabatic regime for the FM dynamics.3 A. Classical Regime: Relaxation Time Approximation The dissipative component of the nonequilibrium elec- tronic density matrix, to lowest order in @=@t, can be determined by expanding the Fermi-Dirac distribution in Eq. (2) to lowest order in [ HM]mm0=mm0m~!. Performing a Fourier transformation with respect to the discrete Larmor frequency modes, m!i@=@t , we nd that, ^dis neq(t) =1 ~^Gri@^Ga=@t, where ^Gr= EFi ^H(t)1and ^Ga= (^Gr)yare the retarded and advanced Green functions calculated at the Fermi energy, EF, and a xed time t. The energy absorption rate of the electrons can be determined from the expectation value of the time derivative of the electronic Hamiltonian, E0 e= <(Tr(^dis neq(t)@^H=@t )), where<() refers to the real part. Calculating the time-derivative of the Green function and using the identity, ^Gr^Ga=^Ga^Gr==(^Gr), where,=() refers to the anti-Hermitian part of the matrix, the torque correlation (TC) expression for the energy excitation rate of the electrons is of the form, E0 e=~ NX kTrh =(^Gr)@^H @t=(^Gr)@^H @ti : (6) In the case of semi-innite NM leads attached to the FM, using,=(^Gr) =^Gr^^Ga=^Ga^^Gr, Eq.(6) can be written as E0 e=~ NX kTrh ^@^Gr @t^@^Ga @ti (7) where, ^ =^1 + ( ^r^a)=2i, with ^r=abeing the retarded=advanced self energy due to the NM lead at- tached to the FM which describes the escape rate of electrons from/to the reservoir. It is useful to separate the dissipation phenomena into local andnonlocal compo- nents as follows. Applying the unitary operator, ^U(t) = ei!^zt=2ei^x=2ei!^zt=2= cos( 2)^1 +isin( 2)(^+ei!t+ ^ei!t), to x the magnetization orientation along z we nd, @(^U^Gr 0^Uy) @t! 2sin() ^G0ei!t+^G0yei!t ;(8) where we have ignored higher order terms in and, ^G0= [^Gr 0;^+]^Gr 0[^H0;^+]^Gr 0: (9) Here, [;] refers to the commutation relation, ^H0is the time independent terms of the Hamiltonian, and ^Gr=a 0 refers to the Green function corresponding to magnetiza- tion alongz-axis. Using Eq. (7) for the average energyabsorption rate we obtain, E0 e=~!2 2Nsin2()X kTr ^^G0^^G0y =~!2 2Nsin2()X k< Tr ^[^Gr 0;^+]^[^Ga 0;^] +=(^Gr)[^H0;^+]=(^Gr)[^H0;^] 2 [=(^Gr 0);^+]^^Ga 0[^H0;^] : (10) In the absence of the SOC, the rst term in Eq. (10) is the only non-vanishing term which corresponds to the pumped spin current into the reservoir [i.e. ISz=~Tr(^z^^dis neq)=2] dissipated in the NM (no back ow). This spin pumping component is conventionally formulated in terms of the spin mixing conductance20, ISz=~g"#sin2()=4, which acts as a nonlocal dissi- pation mechanism. The second term, referred to as the spin-orbital torque correlation11,21(SOTC) expression for damping, is commonly used to calculate the intrinsic con- tribution to the Gilbert damping constant for bulk metal- lic FMs. The third term arises when both SOC and the reservoir are present. It is important to note that the formalism presented above is valid only in the limit of small(ballistic regime). On the other hand, in the case of large, typical in experiments at room temperature, the results may not be reliable due to the fact that in the absence of metallic leads a nite acts as a ctitious reservoir that yields a nonzero dissipation of spin cur- rent even in the absence of SOC. A simple approach to rectify the problem is to ignore the eect of nite in the spin pumping term in calculating the Gilbert damp- ing constant. A more accurate approach is to employ a dephasing mechanism that preserves the conservation laws, which we refer it to as conserving torque correlation approach discussed in the following subsection. B. Classical Regime: Conserving Dephasing Mechanism Rather than using the broadening parameter, , as a phenomenological parameter, we determine the self en- ergy of the Bloch electrons interacting with a dephas- ing bath associated with phonons, disorder, etc. using a self-consistent Green function approach22. Assuming a momentum-relaxing self energy given by, ^r=a int(E;t) =1 NX k^k^Gr=a k(E;t)^y k; (11) where ^kis the interaction coupling matrix, the dressed Green function, ^Gr=a k(E;t) , and corresponding self en- ergy, ^r=a int(E;t), are calculated self-consistently. This will in turn yield a renormalized broadening matrix, ^int==(^r int), which is the vertex correction modi- cation of the innitesimal initial broadening 0.4 The nonequilibrium density matrix is calculated from ^dis neq(k;t) =~ ^Gr k^int^Ga k@^Hk(t) @t+^Saa t ^Ga k;(12) where the time derivative vertex correction term is ^Saa t=1 NX k^k^Ga k@^Hk(t) @t+^Saa t ^Ga k^y k: (13) The energy excitation rate for the Bloch electrons then reads, E0 e=~ NX k<h Tr@^Hk(t) @t+^Sar t ^dis neq(k;t)i ;(14) where ^Sar t=1 NX k^k^Ga k@^Hk(t) @t+^Sar t ^Gr k^y k: (15) The vertex correction Eqs. (13) and (15) can be solved either exactly by transforming them into a system of lin- ear equations or by solving them self consistently. Due to the large number of orbitals and atoms per unit cell for the Co/Pt bilayer the latter approach is computationally more ecient. In the following numerical calculations we assume ^k=int^1 to be a constant independent of kand of orbitals, which can be viewed as the root mean square value of a random on-site potential, int=p hV2 randi, whereh:::idenotes an ensemble averaging, where the self energy in Eq. (11) corresponds to the self consistent Born approximation. C. Gilbert Damping Calculation Having determined the energy absorption rate of the Bloch electrons due to the precessing FM, from conserva- tion of energy one can deduce the energy dissipation rate of the FM from, E0 M=E0 e. Using the LLG equation of motion the energy dissipation rate per unit cell of the precessing FM can be obtained from E0 M=1 2Mtot~!@mz @t=1 2GDMtot~!2sin2();(16) where~ mis the unit vector along the magnetization of the FM. The Gilbert damping parameter can then be obtained from GD=2E0 e Mtot~!2sin2(): (17) III. COMPUTATIONAL SCHEME The spin-polarized density functional theory calcula- tions for the hcp Co(0001)/fcc Pt(111) bilayer were car- ried out using the Vienna ab initio simulation package(VASP)23,24. The pseudopotential and wave functions are treated within the projector-augmented wave (PAW) method25,26. Structural relaxations were carried using the generalized gradient approximation as parameter- ized by Perdew et al.27when the largest atomic force is smaller than 0.01 eV/ A. The plane wave cuto energy is 500 eV and a 14 141kpoint mesh is used in the 2D BZ sampling. The Pt( m)/Co(n) bilayer is modeled employing the slab supercell approach along the [111] consisting of mfcc Pt monolayers (MLs) (ABC stacking) (m=1, 2,:::, 6),nhcp Co MLs (AB stacking) Co ( n= 6), and a 25 A thick vacuum region separating the periodic slabs. The in-plane lattice constant of the hexagonal unit cell was set to the experimental value of 2.505 A for bulk Co. The Gilbert damping constant was calculated us- ing the tight-binding parameters obtained from VASP- Wannier90 calculations28with a 250250k-mesh for the bilayer and 250 250250k-mesh for bulk Co. The electron Hamiltonian, ^H~k, and exchange splitting, ^
~k, matrices in Eq. (1) in the Wannier basis have the form ^H~k=^HSOC+1 2X ~ n1 D~ n(^H"" ~ n+^H## ~ n)e2i~ n
~k(18) ^
~k=1 2X ~ n1 D~ n(^H"" ~ n^H## ~ n)e2i~ n
~k; (19) where ^HSOC is the SOC Hamiltonian matrix, ^H"" ~ nand ^H## ~ nare the spin-majority and spin-minority matrices, ~ n= (n1;n2;n3) are integers denoting the lattice vectors, D~ nis the degeneracy of the Wigner-Seitz grid point, and ki2[0;1]. The ^H"" ~ nand ^H## ~ nare determined from spin-polarized VASP-Wannier90 calculations without SOC. On the other hand, ^HSOC, is determined from VASP-Wannier90 non-spin-polarized calculations with SOC as the follow- ing. Using the identity, Tr[^Li^Lj] =ijl(l+1)(2l+1) 3, where ^Liis the angular momentum operator of orbital land i;j=x;y;z , the SOC strength of the Ith atom can be calculated from I l=3Tr[^HP ll;II^Li^i] l(l+ 1)(2l+ 1): (20) Here, the superscript Pdenotes the paramagnetic Hamil- tonian,IIare the on-site Hamiltonian matrix elements for atomIat~ n= (0;0;0), andllis the block Hamiltonian matrix corresponding to orbital l. The result is indepen- dent of the direction of the angular momentum operator. We nd that Pt d=0.5 eV and Co d=70 meV for the d- orbitals of Pt and Co, respectively, that are somewhat smaller than the values considered in the literature7(i.e. Pt d=0.65 eV,Co d=85 meV). The SOC Hamiltonian can in turn be written as, hI;lmsj^HSOCjI0;l0m0s0i=1 2ll0II0I lX ihlmj^Lijlm0i^i ss0: (21)5 0 45 90 135 1800.0280.02850.0290.02950.030.03050.0310.03150.032 Cone Angle (deg)Gilbert Damping Student Version of MATLAB FIG. 2: (Color online). Gilbert damping versus precessional cone angle calculated from Eq. (5) for Pt(1 ML)/Co(6 ML) bilayer system for S=60 and=1 meV, respectively. IV. RESULTS AND DISCUSSION The Gilbert damping constant (calculated from Eq. (5))versus the precessional cone angle, m= cos1 mp S(S+1) , for the Pt(1 ML)/Co(6 ML) bilayer is shown in Fig. 2 with =1 meV and S=60. We nd that the Gilbert damping is relatively independent of the cone angle. The small angular dependence of the Gilbert damping is material dependent and it could increase or decrease upon increasing the cone angle, depending on the material. In order to see the transition from quantum mechanical to classical dynamical regimes, in Fig. 3 we present the Gilbert damping constant of the Pt(1 ML)/Co(6 ML) bilayer versus the broadening parameter, , for dier- ent values of the total spin Sof the FM. For the case of niteSwe used Eq. (5) while for S=1we used the TC expression Eq. (6). We nd that for nite S the Gilbert damping value exhibits a peak in the small  regime (clean system) where the peak value increases lin- early withSand shifts to smaller broadening value with increasingS. The underlying origin of the GD() behav- ior withSin coherent regime can be understood in terms of the coherent transport of quasi-particles along the aux- iliary direction min Fig. 1, where the auxiliary current ow (damping rate) depends linearly on the chemical po- tential dierence between the rst ( m= +S) and last layers (m=S) which is simply 2 S. This suggests that in the limit of innite Sand ballistic regime !0 the intrinsic Gilbert damping diverges. It was shown that the problem of innite Gilbert damping in the ballistic regime can be removed by taking into account the collec- tive excitations.29,30 As we discussed in Sec. II A, the relaxation time ap- proximation is valid only in the small limit and it vi- olates the conservation law when is large. In order 10−410−210010−310−210−1100 Broadening, η, (eV)Gilbert Damping S=10S=30S=60S=∞ Student Version of MATLABFIG. 3: (Color online) Gilbert damping versus broadening parameter for the Pt(1 ML)/Co(6 ML) bilayer for dierent values of the total spin Sof the FM nanocluster. Eqs. (5) and (6) were used to calculate the Gilbert damping for nite and innite S, respectively. to quantify the validity of the relaxation time approxi- mation, in Fig. 4 we display the Gilbert damping ver- sus broadening, ;or interaction parameter, int, for the Pt(1 ML)/Co(6 ML) bilayer with S=1, using (i)torque correlation(TC) expression (Eq. (6)); (ii)the spin-orbital torque correlation (SOTC) expression (sec- ond term in Eq. (10);) and (iii)the conserving TC expres- sion (Eq. (14)). The upper horizontal-axis refers to the interaction strength intof the conserving TC method and the lower one refers to the broadening parameter, . The calculations show that for  >20 meV the TC results deviate substantially from those of the conserv- ing TC method. Ignoring the spin pumping contribu- tion to the Gilbert damping in Eq. (10) and considering only the SOTC component increases the range of the va- lidity of the relaxation time approximation. Therefore, the overestimation of the Gilbert damping using the TC method can be attributed to the disappearance of elec- trons (pumped spin current) in the presence of the nite non-Hermitian term, i^1, in the Hamiltonian. We have used the conserving TC approach to calculate the eect of inton the Gilbert damping as a function of the Pt layer thickness for the Pt( m)/Co(6 ML) bilayer. As an example, we display in Fig. 5 the results of Gilbert damping versus Pt thickness for int= 1eVwhich yields a Gilbert damping value of 0.005 for bulk Co ( m= 0 ML) and is in the range of 0.00531,32to 0.01133{35reported experimentally. Note that this large intvalue describes the Gilbert damping in the resistivity-like regime which might not be appropriate to experiment, where the bulk Gilbert damping decreases with temperature, suggesting that it is in the conductivity regime.36 For a given intwe tted the ab initio calculated Gilbert damping versus Pt thickness to the spin diu-6 10−410−210010−310−210−1100 Broadening, η, (eV)Gilbert Damping10−1100Interaction Strength, λint, (eV) Conserving TC MethodSOTC MethodTC Method Student Version of MATLAB FIG. 4: (Color online). Gilbert damping of Pt(1 ML)/Co(6 ML) bilayer versus the broadening parameter (lower ab- scissa) and interaction strength, int, (upper abscissa), using the torque correlation (TC), spin-orbital torque correlation (SOTC), and conserving TC expressions given by Eqs. (6), (10) and (14), respectively. sion model,37{39 Pt=Co =Co+ge "#VCo 2MCodCo(1e2dPt=Lsf Pt):(22) Here,ge "#is the eective spin mixing conductance, dCo (dPt) is the thickness of Co (Pt), VCo= 10:5A3 (MCo= 1:6B) is the volume (magnetic moment) per atom in bulk Co, and Lsf Ptis the spin diusion length of Pt. The inset of Fig. 5 shows the variation of the ef- fective spin mixing conductance and spin diusion length with the interaction strength int. In the diusive regime int>0:2eV,Lsf Ptranges between 1 to 6 nm in agree- ment with experiment ndings which are between 0.5 and 10 nm33,40. Moreover, the eective spin mixing conduc- tance is relatively independent of intoscillating around 20 nm2, which is approximately half of the experimen- tal value of35 - 40 nm2.33,41On the other hand, in the ballistic regime ( int<0.2 eV), although the er- rorbar in tting to the diusion model is relatively large, the value of Lsf Pt0.5 nm is in agreement with Ref.7and experimental observation40. V. CONCLUDING REMARKS We have developed an ab initio -based electronic struc- ture framework to study the magnetization dynamics ofa nano-FM where its magnetization is treated quantum mechanically. The formalism was applied to investigate the intrinsic Gilbert damping of a Co/Pt bilayer as a 0 1 2 300.0050.010.0150.02 Pt Thickness, dPt (nm)Gilbert Damping 10−210−110002468 Interaction Strength, λint (eV)Spin Diffusion Length (nm)100101102103 g↑↓eff (nm−2) Student Version of MATLAB FIG. 5: (Color online). Ab initio values (circles) of Gilbert damping versus Pt thickness for Pt( mML)/Co(6 ML) bilayer wheremranges between 0 and 6 and int= 1eV. The dashed curve is the t of the Gilbert damping values to Eq. (22). Inset: spin diusion length (left ordinate) and eective spin mixing conductance, ge "#, (right ordinate) versus interaction strength. The errorbar for ge "#is equal to the root mean square deviation of the damping data from the tted curve. function of energy broadening. We showed that in the limit of small Sand ballistic regime the FM damping is governed by coherent dynamics, where the Gilbert damp- ing is proportional to S. In order to study the eect of disorder on the Gilbert damping we used a relaxation scheme within the self-consistent Born approximation. Theab initio calculated Gilbert damping as a function of Pt thickness were tted to the spin diusion model for a wide range of disorder strength. In the limit of large dis- order strength the calculated spin diusion length and eective spin mixing conductance are in relative agree- ment with experimental observations. Acknowledgments The work is supported by NSF ERC-Translational Ap- plications of Nanoscale Multiferroic Systems (TANMS)- Grant No. 1160504 and by NSF-Partnership in Research and Education in Materials (PREM) Grant No. DMR- 1205734.7 Electronic address: Farzad.Mahfouzi@gmail.com yElectronic address: nick.Kioussis@csun.edu 1Ioan Mihai Miron, Gilles Gaudin, Stphane Auret, Bernard Rodmacq, Alain Schuhl, Stefania Pizzini, Jan Vo- gel and Pietro Gambardella, Current-driven spin torque in- duced by the Rashba eect in a ferromagnetic metal layer, Nat. Mater. 9, 230234 (2010). 2Ioan Mihai Miron, Kevin Garello, Gilles Gaudin, Pierre- Jean Zermatten, Marius V. Costache, Stphane Auret, Sbastien Bandiera, Bernard Rodmacq, Alain Schuhl and Pietro Gambardella, Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection, Nature 476, 189193 (2011). 3Luqiao Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. 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2310.08807v1.Unified_framework_of_the_microscopic_Landau_Lifshitz_Gilbert_equation_and_its_application_to_Skyrmion_dynamics.pdf

arXiv:2310.08807v1 [cond-mat.mes-hall] 13 Oct 2023Unified framework of the microscopic Landau-Lifshitz-Gilb ert equation and its application to Skyrmion dynamics Fuming Xu§,1Gaoyang Li§,1Jian Chen,2Zhizhou Yu,3Lei Zhang,4, 5,∗Baigeng Wang,6and Jian Wang1, 2,† 1College of Physics and Optoelectronic Engineering, Shenzh en University, Shenzhen 518060, China 2Department of Physics, The University of Hong Kong, Pokfula m Road, Hong Kong, China 3School of Physics and Technology, Nanjing Normal Universit y, Nanjing 210023, China 4State Key Laboratory of Quantum Optics and Quantum Optics De vices, Institute of Laser Spectroscopy, Shanxi University, Taiyu an 030006, China 5Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China 6National Laboratory of Solid State Microstructures and Dep artment of Physics, Nanjing University, Nanjing 210093, Ch ina The Landau-Lifshitz-Gilbert (LLG) equation is widely used to describe magnetization dynamics. We develop a unified framework of the microscopic LLG equation based on t he nonequilibrium Green’s function formalism. We present a unified treatment for expressing the microscopi c LLG equation in several limiting cases, including the adiabatic, inertial, and nonadiabatic limits with resp ect to the precession frequency for a magnetization with fixed magnitude, as well as the spatial adiabatic limit f or the magnetization with slow variation in both its magnitude and direction. The coefficients of those terms in the microscopic LLG equation are explicitly expressed in terms of nonequilibrium Green’s functions. As a concrete example, this microscopic theory is applied to simulate the dynamics of a magnetic Skyrmion driv en by quantum parametric pumping. Our work provides a practical formalism of the microscopic LLG equat ion for exploring magnetization dynamics. I. INTRODUCTION Single-molecule magnets (SMMs) are mesoscopic mag- nets with permanent magnetization, which show both classic al properties and quantum properties.1–7SMMs are appealing due to their potential applications as memory cells and pre- cessing units in spintronic devices.8,9Transport of SMMs cou- pled with leads has been investigated both experimentally10–13 and theoretically.7,14–20Transport measurements on magnetic molecules such as Mn1210andFe811revealed interesting phe- nomena, including peaks in the differential conductance an d Coulomb blockades. Dc- and ac-driven magnetization switch - ing and noise as well as the influence on I-V characteristics were discussed in a normal metal/ferromagnet/normal metal structure.15Current-induced switching of a SMM junction was theoretically studied in the adiabatic regime within th e Born-Oppenheimer approximation.16It was found that mag- netic exchange interactions between molecular magnets can be tuned by electric voltage or temperature bias.17Transient spin dynamics in a SMM was investigated with generalized spin equation of motion.21A microscopic formalism was re- cently proposed for consistent modeling of coupled atomic magnetization and lattice dynamics.22 For a SMM with magnetization M, its magnetization dynamics can be semiclassically described by the Landau- Lifshitz-Gilbert (LLG) equation of motion23–29 dm dt=−γm×Heff+m×(αdm dt)+τSTT, (1) wherem=M/Mis the unit magnetization vector, γis the gyromagnetic ratio, and Heffis the effective magnetic field around which the magnet precesses. αis the Gilbert damping tensor describing the dissipation of the precession, and τSTT is the spin transfer torque due to the misalignment between the magnetization and the transport electron spin.30–33 The LLG equation is widely adopted to describe magnetiza- tion dynamics in the adiabatic limit, where the magnetizati onprecesses slowly and the typical time scale is in the order of ns. The Gilbert damping term is in general a 3×3tensor,32 which can be deduced from experimental data, scattering ma- trix theory,28,29or first-principles calculation.34–36Later, the LLG equation was generalized to study ultrafast dynamics in - duced bypselectrical pulse37,38orfslaser pulse39–42, which extends the magnetization switching time down to psor even sub-ps. This is refereed as the inertial regime43, where the time scale involved is much shorter than that of the adiabati c limit. In the inertial limit, a nonlinear inertial term was i ntro- duced into the LLG equation44–48, which was applied to sim- ulate ultrafast spin dynamics.21,49,50Direct observation of in- ertial spin dynamics was experimentally realized in ferrom ag- netic thin films in the form of magnetization nutation at a fre - quency of 0.5THz .51When the magnetization varies in both temporal and spatial domains, two adiabatic spin torques we re incorporated into the LLG equation52, which can describe the dynamics of magnetic textures such as Skymions.53 Magnetic Skyrmions (Sk) stabilized by the Dzyaloshinskii- Moriya interaction (DMI) or competing interaction between frustrated magnets are topologically nontrivial spin text ures showing chiral particle-like nature. When an electron tra- verses the Sk, it acquires a Berry phase and experiences a Lorentz-like force, leading to the topological Hall effect54. At the same time, the Magnus force due to the back ac- tion on Sk gives rise to Skyrmion Hall effect55,56. The exis- tence of Sk has been verified in magnetic materials includ- ing MnSi57and PdFe/Ir(111)58. The radius of an Sk can be as small as a few nm59,60and is stable even at room temperatures61,62. Sk can be operated at ultralow current density,63–65which makes it promising in spintronic appli- cations including the magnetic memory and logic gates.66,67 Various investigations show that Sk can be manipulated by spin torque due to the charge/spin current injection63,64, exter- nal electric field,68,69magnetic field gradient70, temperature gradient,71–74and strain,75etc. However, Sk driven by quan- tum parametric pumping has not been explored.2 Quantum parametric pump refers to such a process: in an open system without bias voltages, cyclic variation of syst em parameters can give rise to a net dc current per cycle76–84. In the adiabatic limit, this quantum parametric pump requires at least two pumping parameters with a phase difference and the pumped current is proportional to the area enclosed by the trajectory of pumping parameters in parameter space.76 It was found that the adiabatic pumped current is related to Berry phase.85Beyond the adiabatic limit, the cyclic fre- quency may serve as another dimension in parameter space and hence a single-parameter quantum pump is possible at finite frequences.86,87In general, quantum parametric pump can be formulated in terms of photon-assisted transport.88,89 Quantum parametric pump can also generate heat current90,91, whose lower bound is Joule heating during the pumping pro- cess. This defines an optimal quantum pump92,93that is noise- less and pumps out quantized charge per cycle94–96. Quantum parametric pumping theory has been extended to account for Andreev reflection in the presence of superconducting lead97, correlated charge pump98,99, and parametric spin pump100,101, providing more physical insights. It is interesting to gene ral- ize quantum parametric pump to Skyrmion transport, which may offer new operating paradigms for spintronic devices. In this work, we investigate the microscopic origin of the LLG equation and the Gilbert damping. We focus on sev- eral limiting cases of the LLG equation. For a magnetization with fixed magnitude, the adiabatic, inertial, and nonadiab atic limits with respect to its precession frequency are discuss ed. When both the magnitude and direction of a magnetization vary slowly in space, which is referred as the adiabatic limi t in spatial domain, our formalism can also be extended to cover this limit. We will provide a unified treatment of all these cases and explicitly express each term in the microscopic LL G equation in the language of nonequilibrium Green’s functio ns. As an example, we apply the microscopic LLG equation to simulate the dynamics of a Skyrmion driven by quantum para- metric pumping in a two-dimensional (2D) system. This paper is organized as follows. In Sec. II, a single- molecule magnet (SMM) transport setup and corresponding Hamiltonians are introduced. In Sec. III, a stochastic Langevin equation for magnetization dynamics is derived from the equation of motion by separating fast (electron) and slow (magnetization) degrees of freedom, forming a microscopic version of the LLG equation. In Sec. IV, four limiting cases of the microscopic LLG equation are discussed. In Sec. V, we numerically study Sk transport driven by quantum parametri c pumping. Finally, a brief summary is given in Sec. VI. II. MODEL The model system under investigation is shown in Fig. 1, where a noninteracting quantum dot (QD) representing a single-molecule magnet (SMM) with magnetization Mis connected to two leads. A uniform magnetic field B=Bˆez is applied in the central region. In addition, we assume that there is a dc bias or spin bias across the system providing a spin transfer torque or spin orbit torque. FIG. 1. Sketch of the model system. A single-molecule magnet (SMM) represented by the quantum dot (QD) is connected to the left and right leads. A uniform magnetic field is applied in th e cen- tral region, around which the SMM magnetization precesses. The Hamiltonian of this system is given by ( /planckover2pi1= 1) ˆHtotal=ˆHL+ˆHR+ˆHD+ˆHT, with the lead Hamiltonian ( α=L,R ), ˆHα=/summationdisplay kσǫkασˆc† kασˆckασ, (2) and the Hamiltonian of the central region, ˆHD=ˆH0+ˆH′+γˆM·B. (3) HereˆH0is the Hamiltonian of the QD with spin-orbit interac- tion (SOI)102 ˆH0=/summationdisplay nσǫnσˆd† nσˆdnσ+/summationdisplay mn(tSO nmd† m↑dn↓+H.c.),(4) withtSO nm=−tSO mn.ˆH′is the interaction between the electron spin and the magnetization as well as the magnetic field, ˆH′=J/summationdisplay nˆsn·ˆM+γe/summationdisplay nˆsn·B. We can also add uniaxial anisotropy field to ˆH′. The coupling Hamiltonian between the QD and the leads is ˆHT=/summationdisplay kαn,σσ′[tσσ′ kαnˆc† kασˆdnσ′+H.c.]. (5) In the above equations, ˆd† nσ(ˆc† kασ) creates an electron with energyǫnσ(ǫkασ) in the QD (lead α). In general, the leads can be metallic or ferromagnetic. Here ˆsn=1 2ψ† nσψnis the electron spin in the central region, with ψ† n= (d† n↑,d† n↓). The Pauli matrices satisfy [σx,σy] = 2iσz, and the magnetization Mfollows the commutation relation [Mx,My] =i/planckover2pi1Mz.J is the exchange interaction between the magnetization and t he spin of conducting electrons. γ(γe) is the gyromagnetic ratio of the magnet (electron). If we choose the magnetic field in the zdirection as the laboratory frame, and (θ,φ)the polar and azimuthal angles3 of the magnetization, the spin dependent coupling matrix is given by tσσ′ kαn=/bracketleftbigˆRtkαn/bracketrightbig σσ′, (6) withˆRthe rotational operator103 ˆR=e−iθ 2ˆσye−iφ 2ˆσz=/parenleftBigg e−iφ 2cos(θ 2)−eiφ 2sin(θ 2) e−iφ 2sin(θ 2)eiφ 2cos(θ 2)/parenrightBigg .(7) III. MAGNETIZATION DYNAMICS From the Heisenberg equation of motion, the magnetization dynamics in the central region is governed by ˙ˆM=−γˆM×B−JˆM׈sD, (8) whereˆsD=/summationtext nˆsnis the total electron spin. In deriving the above equation, the following relation is used: [ˆσ,ˆσ·A] =−2iˆσ×A. (9) Now we separate an operator into its quantum average and its fluctuation, then ˆsD=/an}b∇acketle{tˆsD/an}b∇acket∇i}ht+δˆsD, andˆM=/an}b∇acketle{tˆM/an}b∇acket∇i}ht+δˆM, whereδˆsD(δˆM) is the fluctuation of the electron (magnet) spin. We can transform Eq. ( 8) into a Langevin equation. For the expectation value M(t) =/an}b∇acketle{tˆM(t)/an}b∇acket∇i}ht,104 ˙M=M×[−γB−JsD+δˆB], (10) or ˙M=−γM×[Heff−δˆB′], where sD=/an}b∇acketle{tˆsD/an}b∇acket∇i}ht=−i 2Tr[σG<(t,t)]. (11) HereG< ijσσ′(t′,t) =i/an}b∇acketle{td† jσ′(t′)diσ(t)/an}b∇acket∇i}htis the lesser Green’s function of electrons, which will be discussed in detail be- low. The effective magnetic field Heffis defined as the vari- ation of the free energy of the system with respect to the magnetization32,105,106 Heff=1 γδHtotal δM. (12) AndδˆB=γδˆB′contributes from the fluctuations M×δˆB=−δ˙ˆM−γδˆM×B−JM×δˆsD −JδˆM×sD−JδˆM×δˆsD. These fluctuations can play an important role in determining the motion of the magnetization, such as reducing or enhanc- ing the threshold bias of magnetization switching.16 To transform Eq. ( 10) into the usual LLG equation, we fur- ther separate sDin Eq. ( 11) into the time-reversal symmetricand antisymmetric components, ss Dandsa D. ThenM×ss D andM×sa Dcorrespond to the dissipative and dissipativeless terms, respectively. Thus, Eq. ( 10) is rewritten as ˙M=−γM×B−JM×sa D−JM×ss D. (13) Note that the last term in Eq. ( 13),M×ss D, corresponds to the damping of magnetization. As will be discussed below that in the adiabatic approximation, it assumes the form M×(α˙M) whereαis the Gilbert damping tensor which is expressed in terms of nonequilibrium Green’s function (see Eq. ( 19)). The second term in Eq. ( 13),M×sa D, corresponds to the spin transfer torque. In the presence of SOI, M×sa Dis the spin or- bit torque in collinear ferromagnetic systems, which has fie ld- like and damping-like components, respectively, along the di- rectionsM×uandM×(M×u)withu·M= 0. Hereu is the unit vector of the spin current.31 IV . MICROSCOPIC LLG EQUATION IN DIFFERENT LIMITS In this section, we will drive the LLG equation and express the Gilbert damping tensor in terms of the nonequilibrium Green’s functions. We also discuss the fluctuation in the equ a- tion of motion and the spin continuity equation, showing tha t the spin transfer torque is insufficient to describe magneti za- tion dynamics in general conditions. We focus on several limiting cases of the microscopic LLG equation (Eq. ( 13)). These cases correspond to different lim- its: (1) Adiabatic limit in temporal domain where the preces s- ing frequency of the magnet is low and sDcan be expanded up to the first order in frequency; (2) Inertial regime where t he time scale is much shorter than that of the adiabatic limit, e .g., magnetization switching in psor even sub- psrange37–42; (3) Nonadiabatic regime where adiabatic approximation in tem- poral domain is removed. We will work on the linear coupling between the magnetization and the environment,107and derive the Gilbert damping coefficient as a function of the precess- ing frequency; (4) In the above situations, we have assumed that the magnetization has fixed magnitude and only its direc - tion varies in space. Our theory can be easily extended to ad- dress the motion of domain walls where the magnetization is nonuniform. In the simplest case, we assume that the magne- tization varies slowly in space so that adiabatic approxima tion in spatial domain can be taken. In this spatial adiabatic lim it, two additional toques are incorporated into the LLG equatio n which are naturally obtained in our theory. A. Adiabatic limit As the magnetization precesses, the electron spin and hence spin-orbit energy of each state changes32,106, which drives the system out of equilibrium. In the language of frozen Green’s functions (Eqs. ( 41) and ( 44)), total spin of the QD (Eqs. ( 11)) can be expanded in terms of the precession frequency, which consists of two parts: the quasi-static part s(0) D, and the adia-4 batic change s(1) Dto the first order in frequency sD=s(0) D+s(1) D, where s(0) D=−i 2/integraldisplaydE 2πTr[σG< f], (14) and s(1) D=−1 4/integraldisplaydE 2πTr[G< fσGr fGr fσ−Ga fGa fσG< fσ]˙b,(15) with˙b=JM˙m. Concerning the magnetization dynamics, the effective field Heff(t)(Eq. ( 12)) can be separated into two contributions: an anisotropy field and a damping field106, Heff(t) =Hani eff(t)+Hdamp eff(t), (16) with Hani eff(t) =B+(J/γ)s(0) D,Hdamp eff(t) = (J/γ)s(1) D.(17) Substituting Eqs. ( 14) and ( 15) into Eq. ( 13), ignoring the fluctuation, and noting that ˙b=JM˙m, we obtain the deter- ministic Landau-Lifshitz-Gilbert equation, dm dt=−γm×Hani eff−m×(α˙m), (18) where m=M/M=/parenleftbig sinθcosφ,sinθsinφ,cosθ/parenrightbig , is the unit vector in the magnetization direction. αis the3×3 Gilbert damping tensor28,43, which is defined in terms of the frozen Green’s functions: αij=(JM)2 4/integraldisplaydE 2πRe{Tr[G< fσiGr fGr fσj]}. (19) As shown in Appendix D, this damping tensor recovers that obtained in Ref. [ 28] via the scattering matrix theory in the limit of zero temperature and in the absence of external bias . In general, the Gilbert damping tensor depends on m(t)and bias voltage through the frozen Green’s functions Gr fandG< f. This agrees with the observation in Ref. [ 108] using the effec- tive field theory of breathing Fermi surface mode. B. Inertial regime In this regime, the magnetization has both precessional and nutational motions. We focus on the linear coupling between the magnetization and the environment so that an additional “inertial” term enters the LLG equation, which describes th e nutation of the magnet. In this case, the adiabatic approxim a- tion is not good enough. One has to expand the spin density sDat least to the second order in frequency. In the inertial regime, we assume that the magnitude of the magnetization isfixed while only its direction varies. Iterating Eq. ( 44) to the second order in frequency, we have Gr=Gr f−iGr f˙Gr f+Gr f(˙Gr f)2+(Gr f)2¨Gr f, from which we find the contribution s(2) Din the inertial limit, s(2) D=−1 8/integraldisplaydE 2πIm{Tr[G< fσ(Gr f)3σ]}·∂2 tb +1 16/integraldisplaydE 2πIm{Tr[G< fσ(Gr f)4(σ·∂tb)2]} −i 8/integraldisplaydE 2πTr[σ(Gr f)2G< f(Ga f)2(σ·∂tb)2].(20) To make comparison with Ref. [ 43], we keep only the linear term∂2 tmand neglect other nonlinear terms such as (∂tm)2. With this new term, the LLG equation in inertial regime is written as43–46,53 dm dt=−γm×Hani eff−m×[αdm dt]−m×[¯αd2m dt2],(21) where the inertial term is a 3×3tensor given by ¯αij=(JM)2 2/integraldisplaydE 2πIm{Tr[G< fσi(Gr f)3σj]}. (22) This additional inertial term has been obtained both phenomenologically44and semiclassically43. Ref. [ 35] pro- posed a first-principles method for calculating the inertia term in the semiclassical limit. Here we derive the quantum inert ial tensor in terms of the frozen Green’s function. C. Nonadiabatic regime Now we consider the magnetization dynamics at finite pre- cession frequency, whose time scale is still much larger tha n that of electrons. Since no analytic solution exists in gene ral conditions, we only focus on the linear coupling in the ex- change interaction J. In this nonadiabatic regime, we treat the coupling Jas a small perturbation, and rewrite the equa- tion determining the Green’s function as /parenleftbig i∂ ∂t−˜H0−H′(t)−Σr/parenrightbig Gr(t,t′) =δ(t−t′), (23) where˜H0=H0+γesD·Bis the unperturbed Hamilto- nian, including the bare Hamiltonian of the QD defined in Eq. ( 4) and the Hamiltonian due to the constant external field. H′(t) =Jσ·M(t)is the perturbative term due to exchange coupling between the magnetization and the electron spin. The unperturbed retarded Green’s function satisfies (i∂ ∂t−˜H0−Σr)Gr 0(t−t′) =δ(t−t′). (24) SinceGr 0(t−t′)only depends on the time difference, it is convenient to work in the energy representation, Gr 0(E) = [E−H0−γe 2σ·B−Σr]−1, (25)5 whereGr 0(t−t′)andGr(t,t′)are related through the Dyson equation Gr(t,t′) =Gr 0(t−t′)+/integraldisplay dt1Gr 0(t−t1)H′(t1)Gr(t1,t′). In the first order perturbation, we have Gr=Gr 0+Gr 0H′Gr 0, and G<=G< 0+Gr 0H′G< 0+G< 0H′Ga 0. Using /integraldisplay dt1Gr 0(t−t1)H′(t1)G< 0(t1−t) =/integraldisplaydE 2π/integraldisplaydω 2πe−iωtGr 0(E+ω)H′(ω)G< 0(E), and /integraldisplay dt1G< 0(t−t1)H′(t1)Ga 0(t1−t) =/integraldisplaydE 2π/integraldisplaydω 2πe−iωtG< 0(E+ω)H′(ω)Ga 0(E), whereH′(ω) =J 2σ·M(ω)withωthe precession frequency, the spin density sDof the quantum dot can be evaluated sD=s(0) D+s(1) D. (26) Heres(0) Dis independent of Mand timet: s(0) D=−i 2/integraldisplaydE 2πTr[σG< 0(E)]. Ands(1) Ddepends linearly on M(t) s(1) D=−i 2/integraldisplaydE 2π/integraldisplaydω 2πe−iωtTr[σGr 0(E+ω)H′(ω)G< 0(E) +σG< 0(E+ω)H′(ω)Ga 0(E)]. Using the anisotropic field Hani eff(t)and the damping field Hdamp eff(t)expressed in Eq. ( 17) and ignoring the fluctuations, we can obtain a deterministic dynamic equation from Eq. ( 13): dm dt=−γm×Hani eff−γm×Hdamp eff, (27) where Hani eff(t) =B−iJ 2γ/integraldisplaydE 2πTr[σG< 0(E)], (28) and Hdamp eff(t) =−/integraldisplaydω 2πe−iωtm(ω)˜α(ω). (29)Here˜α(ω)is the frequency dependent Gilbert damping tensor defined as ˜α(ω) =i 4J2M2/integraldisplaydE 2πTr/bracketleftbigg G< 0(E)σGr 0(E+ω)σ +Ga 0(E)σG< 0(E+ω)σ/bracketrightbigg . (30) It is easy to confirm that when ωgoes to zero, we can recover the results in the adiabatic and inertial limits. D. Adiabatic limit in spatial domain When both the magnitude and direction of the magnetiza- tion vary slowly in space, we refer to this situation as the ad ia- batic limit in spatial domain. In this case, two additional t erms emerge in the LLG equation52, dm dt=−γm×Hani eff−m×[αdm dt] +bJ(je·∇)m−cJm×(je·∇)m, (31) wherebJandcJare constants defined in Ref. [ 52]. Here the term with coefficient bJis related to the adiabatic process of the nonequilibrium conducting electrons.52In contrast, the other term with coefficient cJcorresponds to the nonadiabatic process which changes sign upon time-reversal operation. In this limit, the coupling between the magnetization and the electron spin can be approximated as ˆH′=Jˆsr·ˆM(r,t)+γeˆsr·B. (32) Eqs. ( 13), (46), and ( 47) are still valid except that MandsD are local variables depending on position, where sD(x)is de- fined as sD(x) =−i 2/integraldisplaydE 2πTrs[σG<]xx, (33) where the trace is taken only in spin space. To derive the adiabatic term in Eq. ( 31), we start from Eq. ( 46) and then use Eq. ( 47). From Eq. ( 46), we have109 dsD dt+∇·js=JM×sD, (34) wherejsis the spin current density and the term −γesD×B is neglected. Using the fact that js≈ −b0jem(whereb0= µBP/e andPis the polarization) and neglecting the second order terms such as ∂tδsD, we find from Eq. ( 34)110 JM×δsD=−b0(je·∇)m, (35) whereδsDdenotes the contribution due to the spatial varia- tion of the magnetization ∇m. The nonadiabatic term can be generated by iterating the following equation, dm dt=−γm×Hani eff−m×[αdm dt]+bJ(je·∇)m,(36)6 from which we arrive at105, dm dt=−γ 1+α2m×Hani eff−γα 1+α2m×[m×Hani eff] +bJ 1+α2(je·∇)m+bJα 1+α2m×(je·∇)m,(37) where we have assumed that the Gilbert damping tensor αis diagonal, i.e., αij=αδij. The nonadiabatic term can also be derived explicitly, as shown in Appendix E. V . SKYRMION DYNAMICS DRIVEN BY QUANTUM PARAMETRIC PUMPING In this section, we apply our microscopic theory to inves- tigate Skyrmion dynamics in a 2D system driven by quantum parametric pumping. Initially, an Sk is placed in the centra l re- gion of a two-lead system, as shown in Fig. 2. Then, we apply two time-dependent voltage gates with a phase difference in the system to drive a dc electric current. The electron flow, i n turn, interacts with the Sk, which gives rise to quantum para - metric pumping of the Sk. In the tight-binding representati on, the Sk is described by the following Hamiltonian HSk=−Jex/summationdisplay /angbracketlefti,j/angbracketrightmi·mj+/summationdisplay /angbracketlefti,j/angbracketrightD·(mi×mj) −K/summationdisplay i(mi·ˆz)2−µ/summationdisplay imi·B. (38) HereJexis the Heisenberg exchange interaction. D= D(ri−rj)/|ri−rj|is the Dzyaloshinskii-Moriya interaction (DMI).Kis the perpendicular magnetic anisotropy constant, andµis the magnitude of the magnetic moment. To facilitate parametric pumping, we apply gate voltages in two different regions of the system with the following form, Vp=V1cos(ωpt)+V2cos(ωpt+φ), whereV1=Vδ(x−l1)andV2=Vδ(x−l2)are potential landscapes with Vthe pumping amplitude, ωpis the pumping frequency, and φis the phase difference. The central scatter- ing region is discretized into a 40×40mesh. The positions of gate voltages are l1= 1 andl2= 5, which are displayed in Fig. 2. In the adiabatic pumping regime (small ωplimit), the cyclic variation of two potentials V1andV2can pump out a net current when φ/ne}ationslash=nπ76,82. Thus, the total Hamiltonian of the system consists of Hα,H0,HT,H′(given by Eqs. ( 2), (4), (5), and ( 32), respectively), HSk, andVp. Since the Sk has slow varying spin texture in space, its dy- namics can be approximated by the adiabatic limit in spatial domain. The following LLG equation describing the Sk dy- namics driven by parametric pumping needs to be solved, dmi dt=−γmi×[Heff−(J/γ)sD]−mi×(α˙mi),(39) where the effective field Heffis defined as Heff=1 γδHSk δmi. /s120/s121 /s109 /s122/s49 /s45/s49/s86 /s49/s32/s32/s86 /s50 /s76/s101/s102/s116 /s108/s101/s97/s100/s82/s105/s103/s104/s116 /s108/s101/s97/s100 FIG. 2. Schematic plot of a central region that hosts a Skyrmi on and is connected to two metallic leads. The central region co nsists of a square lattice of size 40 ×40. The arrows denote the in-plane component of the magnetization texture of the Skyrmion. Pum ping potentials V1andV2are applied on the first and fifth column layers of the central region, which are labeled by dark gray bars. TABLE I. Unit conversion table for Jex= 1meV and a= 0.5nm. Distance x ˆx=a = 0.5nm Timet ˆt=/planckover2pi1/Jex ≈0.66ps Current density κˆκ= 2eJex/a2/planckover2pi1≈2×1012A/m2 Velocityv ˆv=Jexa/(h) ≈7.59×102m/s HeresDis defined in terms of Green’s functions in Eq. ( 33). The Gilbert damping tensor αis assumed to be a diagonal ma- trix,αij=αδij. It is worth mentioning that Eq. ( 39) already includes the (je· ∇)mandm×(je· ∇)mterms, which is discussed in Sec. IV D and Appendix E. Initial configuration of the Sk is generated by manually cre- ating a topological unity charge at the center of the system a nd then relaxing the spin texture numerically until the magnet ic energy is stable. Note that mzat the Sk center is negative, while the outside values are positive. In numerical simula- tion, the central region is a 40a×40asquare lattice with a the lattice spacing; the relaxed Sk radius is r0= 10 , which is the minimal distance between the Sk center mz i(0) =−1 andmz i(r0) = 1 . Parameters are set as D= 0.2Jex26, K= 0.07Jex26,J= 2Jex,B= 0, andα= 0.4. The Heisen- berg exchange constant Jex=t= 1 is chosen as the energy unit, wheretis the hopping energy. We set /planckover2pi1=γ=a= 1, and then the coefficients to convert the time t, current den- sityκ, and velocity vto SI units are /planckover2pi1/Jex,2eJex/(a2/planckover2pi1), andJexa/h.111,112Table Ishows the expressions and partic- ular values for Jex= 1meV anda= 0.5nm. Our numerical calculation proceeds as follows. First, with the initial Sk configuration chosen at t=t0, we calculate the total Hamiltonian of the system and then the frozen Green’s function in Eq. ( 45) that determines sDin Eq. ( 33). Second, the LLG equation in Eq. ( 39) is solved by using the fourth-7 /s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48 /s48 /s50 /s52 /s54 /s56/s48/s52/s56/s49/s50/s49/s54/s48/s49/s48/s50/s48/s51/s48 /s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s119/s105/s116/s104/s111/s117/s116/s32/s83/s107/s68/s79/s83/s47/s49/s48/s48/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110 /s32/s115/s112/s105/s110/s45 /s111/s114/s32/s40/s97/s41 /s40/s98/s41/s32/s115/s112/s105/s110/s45 /s32/s115/s112/s105/s110/s45/s40/s99/s41 /s119/s105/s116/s104/s32/s83/s107 /s40/s100/s41 FIG. 3. The transmission coefficient (a) and density of state s (b) as a function of the electron energy in the absence of an Sk. (c) a nd (d) are the transmission and density of states for cases with an S k at the center. No pumping potential is added in the system ( V= 0). order Runge-Kutta method with a small time step dt. Then the Sk Hamiltonian in Eq. ( 38) can be updated. We repeat the above two-step calculation to simulate the Sk dynamics driv en by quantum parametric pumping, and monitor the pumped current during the time evolution. First, we investigate the static transport properties of th e system without pumping. Fig. 3shows the transmission coef- ficient and density of states (DOS) as a function of the electr on energyEwith and without an Sk locating at the system center. When there is no Sk, Fig. 3(a) and (b) show spin-degenerate transmission coefficients and DOS, which are standard trans - port properties for a metallic square lattice. However, in t he presence of the Sk, spin degeneracy of the system is lifted. In Fig. 3(d), the whole energy range [0,8]can be typically divided into the following three regions, irrespective to t he exchange strength J113,114. (i)0< E <|J|. The conduction electrons are fully spin- polarized. Since J =2 in our calculation, this region corre- sponds to 0< E <2. Only spin-down electrons can trans- mit in this energy region, and the largest spin polarization is reached near E= 2. (ii)|J|< E <8−|J|. Both spin-up and spin-down con- duction electrons exist in the system. (iii)8− |J|< E <8. The conduction electrons are fully polarized with spin up component. Second, we study the parametric pumping effect on the dy- namics of an Sk and the corresponding pumped current. Phys- ically, the pumped current can drive the motion of Sk, while the Sk’s motion can affect the pumped current in turn. The pumped current at time tis defined as89 Ip(t) = Tr/bracketleftbigg ΓRGr fdVp dtGa f/bracketrightbigg , (40) whereΓR= Σr R−Σa Ris the linewidth function of the right metallic lead. Σr,a Rare the retarded and advanced self-/s45/s51/s48/s48/s51/s48 /s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s45/s49/s48/s49/s50/s45/s51/s48/s48/s51/s48/s73 /s112/s32/s119/s105/s116/s104/s32/s83/s107 /s32/s119/s105/s116/s104/s111/s117/s116/s32/s83/s107/s40/s97/s41/s83/s107/s32/s99/s101/s110/s116/s101/s114 /s116/s47/s84/s32/s120 /s32/s121 /s40/s99/s41/s73 /s112/s32/s115/s112/s105/s110/s45 /s32/s115/s112/s105/s110/s45/s40/s98/s41 FIG. 4. (a) The pumped current Ipversus time with or without the Sk. The time is in unit of the pumping period T, withT= 2π/ωp. (b) The pumped spin-dependent current I↑/↓ pwith the Sk. (c) Time evolution of the Sk center position R= (x,y). Parameters: E= 1.9,J= 2,V= 0.8,ωp= 1,φ=π/2. energies. The Sk center R= (x,y)is defined as R=/summationtext i(mz 0−mz i)ri//summationtext i(mz 0−mz i)to characterize its motion, where index isums over sites with mz i<mz 0=−0.1. As shown in Fig. 4(a), in the absence of an Sk, the pumped current is roughly a sine or cosine function in time. When an Sk is introduced at t=t0, the conduction electrons are scattered by the moving Sk. This results in the deviation of the pump current from the smooth curve. Meanwhile, in the presence of the Sk, the pumped current is fully spin-polariz ed at the given energy, where only the spin-down component is nonzero in Fig. 4(b). At the same time, the Sk is driven by the pumped current. Fig. 4(c) displays xandycoordinates of the Sk center as a function of time. The remarkable charac- teristic is the quasi-periodic movement of the Sk along both ˆxandˆydirections. Moreover, the motion of the Sk center has the same period as the pumped current, but is delayed by one quarter cycle in phase. In our system, the pumped current flows in the ±xdirection, and hence the Sk moves faster in this direction. Besides, the Sk acquires a velocity in the ±ˆy direction. This indicates that the Sk Hall effect can also be driven by parametric pumping. We examine the influence of pumping parameters on the Sk dynamics. The pumping amplitude is first evaluated. Fig. 5 shows the time evolution of the Sk center for different pump- ing amplitudes V. We observe that the Sk’s speed along xdi- rection increases with the pumping amplitudes. For V= 0.4, the Sk oscillates around its initial position and does not pr op- agate. As the pumping amplitude increases, the Sk moves faster in+ˆxdirection and then saturates when the amplitude exceedsV= 0.8. The motion along the ˆydirection is always slower than that in the ˆxdirection. The effect of the pumping frequency is also studied. When there is no Sk, the pumped current in adiabatic pumping regime is independent of the pumping frequency.76,82In the presence of an Sk, We expect that the pumped current can8 /s48/s49/s50 /s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s45/s50/s45/s49/s48/s49/s50/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s120 /s32/s48/s46/s52/s32 /s32/s48/s46/s54/s32 /s32/s48/s46/s56/s32 /s32/s49/s46/s48/s32 /s32/s49/s46/s50/s40/s97/s41/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s121 /s116/s47/s84/s40/s98/s41 /s86/s58 FIG. 5. (a) and (b): xandycoordinates of the Sk center Rversus time for different pumping amplitudes V= 0.4,0.6,0.8,1,1.2. The pumping frequency is fixed at ωp= 1. Other parameters: J= 2,E= 1.9,l1= 1,l2= 5,φ=π/2. be simply scaled by the pumping frequency. We examine the pumped current for different pumping frequency when the Sk is fixed at its initial configuration, and numerical results a re presented in Fig. 6(a). Four periods are shown here. It is clear that the pumped currents collapse precisely onto each other when scaled by the corresponding pumping frequencies. For a free Sk, Fig. 6(b) and (c) show the xandycoordinates of the Sk center under the pumping. At small frequencies ωp= 0.2 and0.5, the Sk is driven along +ˆxdirection first, and then re- flected back periodically. Its motion in ydirection is similar. For a larger frequency ωp= 1, there is no such oscillating be- havior even for a time scale of 100 periods (not shown here). Notice that when the phase difference is reversed, both the pumped current and the Sk motion change direction. VI. SUMMARY In conclusion, we have developed a unified microscopic theory of the LLG equation in terms of nonequilibrium Green’s function. Four limiting cases of the microscopic LL G equation are discussed in detail, including the adiabatic, iner- tia, and nonadiabatic limits for the magnetization with fixe d magnitude, as well as the adiabatic limit in spatial domain f or the magnetization with slow varying magnitude and directio n in space. As a demonstration, the microscopic LLG equation is applied to investigate the motion of a Skyrmion state driv en by quantum parametric pumping. Our work not only provides a unified microscopic theory of the LLG equation, but also of- fers a practical formalism to explore magnetization dynami cs with nonequilibrium Green’s functions./s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48/s45/s56/s48/s48/s56/s48/s49/s54/s48 /s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s48 /s49/s48 /s50/s48 /s51/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s73 /s112/s47 /s112 /s116/s47/s84/s32 /s112/s61/s48/s46/s50 /s32 /s112/s61/s48/s46/s53 /s32 /s112/s61/s49/s40/s97/s41 /s102/s105/s120/s101/s100/s32/s83/s107 /s112/s61/s49 /s112/s61/s48/s46/s53/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s120/s40/s98/s41 /s112/s61/s48/s46/s50/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s121 /s116/s47/s84/s40/s99/s41 /s112/s61/s49 /s112/s61/s48/s46/s53/s112/s61/s48/s46/s50 FIG. 6. (a) The pumped current for different pumping frequen cies. The Sk is fixed at its initial configuration during the time evo lution. The pumped current Ipis scaled with ωp. (b) and (c): xandycoordi- nates of the Sk center for ωp= 0.2,0.5,1. The pumping amplitude is fixed at V= 0.8. Other parameters: J= 2,E= 1.9,l1= 1,l2= 5,φ=π/2. ACKNOWLEDGEMENT This work was supported by the National Natural Science Foundation of China (Grants No. 12034014, No. 12174262, No. 12074230, and No. 12074190). L. Zhang thanks the Fund for Shanxi ”1331 Project”. APPENDIX A In this appendix, we express the charge and spin current in terms of the nonequilibirum Green’s functions. In the presence of time-varying magnetization, the nonequi - librium Green’s function Gr(t,t′)depends on two time in- dicestandt′. If the magnetization changes slowly with time, we can treat the time difference (t−t′)in energy space.115 After taking the Fourier transformation, the Green’s funct ion in energy space only depends on one time variable t: Gr(t,E) =/integraldisplay dτeiEτGr(t,t′), whereτ=t−t′. The inverse Fourier transformation gives Gr(t,t′) =/integraldisplaydE 2πe−iEτGr(t,E). With the above definition, it is easy to show that G<(t,t′) =/integraldisplaydE 2πe−iEτGr(t,E)Σ<(E)Ga(E,t′).(41)9 In this representation, the particle current matrix is defin ed116 Iα op(t) =/integraldisplaydE 2π[Gr(t,E)Σ< α(E)+G<(t,E)Σa α(E)+H.c.]. (42) In term of which, the charge current Icα(t)and spin current Isα(t)are expressed as Icα(t) =−qTr[Iα op],Isα(t) =−1 2Tr[σIα op]. (43) As shown in Appendix C, these time-dependent Green’s functions, such as Gr(t,E)(also called Floquet Green’s function115), can be expressed in terms of the instantaneous frozen Green’s function Gr f, which satisfies the following re- cursive relation: Gr(t,E) =Gr f(t,E)−iGr f(t,E)˙Gr(t,E), (44) where˙Gris the time derivative of Gr. The frozen Green’s function contains the effective magnetic field b(t)and is de- fined as Gr f(t,E) = [E−H0−Σr−σ·b(t)/2]−1, (45) whereb(t) =γeB+JM. The self energies (for ferromag- netic leads) are given by Σγ mn(t) =ˆR†Σγ 0,mnˆR, withγ=r,a,< .ˆRis defined in Eq. ( 7).Σγ 0,mnis the self- energy when the magnetization is along z-axis: Σγ 0,mn=/summationdisplay kαt∗ kαmgγ kαtkαn=/parenleftbiggΣγ mn,↑0 0 Σγ mn,↓/parenrightbigg , wheregγ kαis the surface Green’s function of lead α. APPENDIX B In this appendix, we express the microscopic LLG equation Eq. ( 13) in terms of the spin current and spin operators. For the spin transfer torque (STT) to occur in magnetic mul- tilayers, one requires a pair of FM layers in a noncollinear configuration. Then a spin-polarized current can be generat ed from the reference (fixed) layer, and the transverse spin can be transferred to the switchable (free) layer. The current ind uced STT can also be obtained from Eq. ( 13) in such a noncollinear magnetic system. Denoting H1=Htotal−H′and defining the spin operators for lead αand the QD: ˆsα=1 2/summationdisplay kσσ′ˆc† kσασσσ′ˆckσ′α,ˆsD=1 2/summationdisplay nσσ′ˆd† nσσσσ′ˆdnσ′, we have dˆsα dt=−i[ˆsα,H1], dˆsD dt=−i[ˆsD,ˆH1]−i[ˆsD,ˆH′],where −i[ˆsD,ˆH′] =−γeˆsD×B+JˆM׈sD. It can be shown that117[/summationtext αˆsα+ˆsD,H1] = 0 , from which the spin continuity equation of the system is expressed as118 /summationdisplay αdˆsα dt+dˆsD dt=−γeˆsD×B+JˆM׈sD, (46) which indicates that the total spin is not conserved due to sp in precession117. Substituting Eq. ( 46) into Eq. ( 8) and taking quantum average, we have ˙M=−γM×B−/summationdisplay αIsα+A, (47) where the correction term Ais given by A=−dsD dt−γesD×B, (48) and/summationtext αIsα=/summationtext αdsα/dtis the total spin current. Notice that Eq. ( 13) and Eq. ( 47) are equivalent. Form Eq. ( 48), it is found that when sDis time-reversal symmetric, dsD/dtorA is time-reversal antisymmetric. If we decompose/summationtext αIsαand Ainto the time-reversal symmetric (labeled with superscrip t s) and antisymmetric (labeled with superscript a) parts, we have ˙M=−γM×B−/summationdisplay αIs sα+As−JM×ss D. (49) Here we have used the fact that/summationtext αIsα−A=JM×sD. When the external magnetic field is strong enough, the elec- tron spin will approximately align with the direction of the field and we may drop the term −γesa D×B. In the adiabatic limit, the term −dsa D/dt in inAsof Eq. ( 49) corresponds to −ds(0) D/dt. Expanding it to the first order in frequency, we find −ds(0) D dt=−iJM 4/integraldisplaydE 2πTr[G< fσGr fσ+Ga fσG< fσ]dm dt ≡ −ηdm dt, (50) whereηis a second rank tensor. This term can be absorbed by introducing an effective gyromagnetic ratio γ′=γ(1+η)−1. Therefore, the driving force of magnetization precession o rig- inates from the total spin current/summationtext αIs sα, which corresponds to the current induced STT discussed in Ref. [ 119]. APPENDIX C In this appendix, we derive the relation between Floquet Green’s functions and frozen Green’s functions. We start wi th the two-time retarded Green’s function defined as120 [i∂ ∂t1−H(t1)]Gr(t1,t2)+/integraldisplay dt′Σr(t1,t′)Gr(t′,t2) =δ(t1,t2).10 To work in the Floquet Green’s function representation, we make the Fourier transform with respect to the fast time scal e τ=t1−t2, and obtain /bracketleftBig i∂ ∂t+E−H(t)/bracketrightBig Gr(t,E) −/integraldisplayt −∞dt′eiE(t−t′)Σr(t,t′)Gr(t′,E) =I.(51) The last term on the left hand side can be written as /integraldisplayt −∞dt′eiE(t−t′)Σr(t,t′)Gr(t′,E) =/integraldisplaydE′ 2π/integraldisplayt −∞dt′ei(E′−E)t′ei(E−E′)tΣr(t,E′)Gr(t′,E) ≈/integraldisplaydE′ 2π/integraldisplayt −∞dt′ei(E′−E)t′ei(E−E′)tΣr(t,E′)Gr(t,E) =/integraldisplaydE′ 2π2πδ(E′−E)ei(E−E′)tΣr(t,E′)Gr(t,E) = Σr(t,E)Gr(t,E), whereGr(t′,E) =Gr(t−τ,E)≈Gr(t,E)is used and the fast time variable τ=t−t′is neglected. Then we can introduce the frozen Green’s function Gr f(t,E)satisfying [E−H(t)−Σr(t,E)]Gr f(t,E) =I. (52) Substituting Eq. ( 52) into Eq. ( 51), we have Gr(t,E) =Gr f(t,E)−iGr f(t,E)˙Gr(t,E). (53) Similarly, the advanced Green’s function is Ga(E,t) =Ga f(E,t)+i˙Ga(E,t)Ga f(E,t). (54) Up to the linear order in frequency, we find Gr(t,E) =Gr f(t,E)−iGr f(t,E)˙Gr f(t,E), Ga(E,t) =Ga f(E,t)+i˙Ga f(E,t)Ga f(E,t).(55) Notice that the frozen Green’s function is an instantaneous function, since it depends only on the present time. APPENDIX D In this appendix, we compare the Gilbert damping coeffi- cient derived in Ref. [ 28] using the scattering matrix theory with that obtained in this work via nonequilibrium Green’s functions. According to Ref. [ 28], in the absence of exter- nal bias, the Gilbert tensor element at zero temperature is e x- pressed in terms of the scattering matrix at the Fermi energy : Gij(m) =γ2 4πRe{Tr[∂S ∂mi∂S ∂mj]}. (56)To compare with Eq. ( 19), we calculate the dimensionless quantityα′ ij=Gij/γ2. The scattering matrix Sis connected to the Green’s function through the Fisher-Lee relation121,122 Sαβσσ′=−δαβσσ′+i/an}b∇acketle{tWασ|Gr|Wβσ′/an}b∇acket∇i}ht, (S†)βασ′σ=−δαβσσ′−i/an}b∇acketle{tWβσ′|Ga|Wασ/an}b∇acket∇i}ht, whereΓαmnσσ′=|Wαmσ/an}b∇acket∇i}ht/an}b∇acketle{tWαnσ′|.|Wαm/an}b∇acket∇i}htis proportional to the eigenvector of Γα122. From Eq. ( 45), we see that ∂Gr,a f ∂mi=1 2JMGr,a fσiGr,a f. Then we can express Eq. ( 56) with frozen Green’s functions Tr[∂S ∂mi∂S ∂mj] =1 4(JM)2Tr[Ga fΓGr fσiGr fΓGa fσj] = (JM)2Tr[Im(Gr f)σiIm(Gr f)σj]. Hence the dimensionless damping tensor element is α′ ij=1 16π(JM)2Re{Tr[Ga fΓGr fσiGr fΓGa fσj]}.(57) Now we assume that there is no external bias and the temper- ature is zero, which are the same conditions used in deriving Eq. ( 56) in Ref. [ 28]. Note that Eq. ( 56) was derived from the pumped energy current defined as ˙mα′˙mso that the resultant tensorα′ ijis always symmetric. Hence we symmetrize the dimensionless damping tensor in Eq. ( 19): ˜αij=1 2(αij+αji). (58) In the absence of external bias, G< f= (Ga f−Gr f)f(E), and Eq. ( 58) becomes ˜αij=−/integraldisplay dETr[σiAσjGr fGr f−σjAσiGa fGa f +σjAσiGr fGr f−σiAσjGa fGa f] =−/integraldisplay dETr[σiAσj∂E(Gr f−Ga f)+σjAσi∂E(Gr f−Ga f)] =−/integraldisplay dEfTr∂E[σi(Gr f−Ga f)σj(Gr f−Ga f)] = Tr[σi(Gr f−Ga f)σj(Gr f−Ga f)], (59) withA= (Gr f−Ga f)f. Apart from a constant factor (JM)2/16π, this expression is exactly the same as α′ ijshown in Eq. ( 57). Therefore, we confirm the equivalence of the Gilbert damping tensor between our formalism (Eq. ( 19)) and that obtained in Ref. [ 28] in the limiting case. APPENDIX E In this appendix, we derive the nonadiabatic term in Eq. ( 31). Expanding m(r,t)up to the first order in ∂im≡ ∂m/∂xi, we have m(r,t) =m+δm=m+xi∂im,11 where the Einstein summation convention is implied. The nonequilibrium Green’s functions have similar expansions Gr=Gr 0−(JM/4)Gr 0σxi∂imGr 0, G<=G< 0−[(JM/4)Gr 0σxi∂imG< 0−H.c.]. It is straightforward to find the correction on s(0) Ddue to∂im, δs(0) D=JMi 8/integraldisplaydE 2πTrs[σGrσxjG<]xx∂jm+H.c.,(60) where we have focused on the linear response regime52,123and kept only the linear term in ∂jm. HereTrs[...]xxdenotes tracing over spin space and then taking diagonal matrix el- ement in real space and we have dropped the subscript 0in the Green’s function. If we further neglect SOI, the Green’s function is diagonal in spin space in the linear regime, and Trs[σGrσxjG<]xx=1Trs[GrxjG<]xxis also diagonal in spin space. Using the relation xj=mc0[∇j,H−E], which is valid in the adiabatic approximation in spatial do- main (c0is a constant having dimension T−2withTthe di- mension of time), Eq. ( 60) becomes δs(0) D=JMmc 0 8[w1(x)+w2(x)]∂jm,where w1j(x) =i/integraldisplaydE 2π[(G<∇j)Ga(H−E)−(H−E)Gr(∇jG<)], w2j(x) =i/integraldisplaydE 2π[(Gr∇j)(H−E)G<−G<(H−E)(∇jGa)]. 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1808.04385v2.Gilbert_damping_phenomenology_for_two_sublattice_magnets.pdf

Gilbert damping phenomenology for two-sublattice magnets Akashdeep Kamra,1,Roberto E. Troncoso,1Wolfgang Belzig,2and Arne Brataas1,y 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany Abstract We present a systematic phenomenological description of Gilbert damping in two-sublattice mag- nets. Our theory covers the full range of materials from ferro- via ferri- to antiferromagnets. Fol- lowing a Rayleigh dissipation functional approach within a Lagrangian classical eld formulation, the theory captures intra- as well as cross-sublattice terms in the Gilbert damping, parameterized by a 22 matrix. When spin-pumping into an adjacent conductor causes dissipation, we obtain the corresponding Gilbert damping matrix in terms of the interfacial spin-mixing conductances. Our model reproduces the experimentally observed enhancement of the ferromagnetic resonance linewidth in a ferrimagnet close to its compensation temperature without requiring an increased Gilbert parameter. It also predicts new contributions to damping in an antiferromagnet and sug- gests the resonance linewidths as a direct probe of the sublattice asymmetry, which may stem from boundary or bulk. 1arXiv:1808.04385v2 [cond-mat.mtrl-sci] 6 Nov 2018I. INTRODUCTION The fundamental connection1between magnetic moment and spin angular momentum underlies the important role for magnets in nearly all spin-based concepts. An applied mag- netic eld provides the means to manipulate the state of a ferromagnet (FM), and thus the associated spin. Conversely, a spin-polarized current absorbed by the FM aects its mag- netization2{5. Exploiting a related phenomenon, switching the state of an antiferromagnet (AFM) has also been achieved6. Emboldened by this newly gained control, there has been an upsurge of interest in AFMs7{10, which oer several advantages over FMs. These include the absence of stray elds and a larger anisotropy-induced gap in the magnon spectrum. The two-sublattice nature of the AFMs further lends itself to phenomena distinct from FMs11. Concurrently, ferrimagnets (FiMs) have been manifesting their niche in a wide range of phenomena such as ultrafast switching12{14and low-dissipation spin transport15{22. A class of FiMs exhibits the so-called compensation temperature23{28, at which the net magnetization vanishes, similar to the case of AFMs. Despite a vanishing magnetization in the compensated state, most properties remain distinct from that of AFMs29. Thus, these materials can be tuned to mimic FMs and AFMs via the temperature. In conjunction with the possibility of a separate angular-momentum compensation, when the magnetization does not vanish but the total spin does, FiMs provide a remarkably rich platform for physics and applications. An increased complexity in the theoretical description29,30hence accompanies these structurally complicated materials, and may be held responsible for comparatively fewer theoretical studies. Nevertheless, a two-sublattice model with distinct parameters for each sublattice qualitatively captures all the phenomena mentioned above. Dissipation strongly in uences the response of a magnet to a stimulus and is thus cen- tral to the study of magnetic phenomena such as switching, domain wall motion and spin transport. Nevertheless, magnetic damping has conventionally been investigated via the ferromagnetic resonance (FMR) linewidth. It is accounted for phenomenologically in the Landau-Lifshitz description of the magnetization dynamics via the so-called Gilbert damp- ing term31, which produces a good agreement with experiments for a wide range of systems. The Gilbert damping represents the viscous contribution and may be `derived' within a Lagrangian formulation of classical eld theory by including the Rayleigh dissipation func- tional31. While the magnetic damping for FMs has been studied in great detail29,31{35, 2from phenomenological descriptions to microscopic models, a systematic development of an analogous description for ferri- and antiferromagnets has been lacking in literature. Further- more, recent theoretical results on spin pumping in two-sublattice magnets36and damping in AFMs37suggest an important role for the previously disregarded29cross-sublattice terms in Gilbert damping, and thus set the stage for the present study. Yuan and co-workers have recently presented a step in this direction focussing on spin torques in AFMs38. Here, we formulate the magnetization dynamics equations in a general two-sublattice magnet following the classical Lagrangian approach that has previously been employed for FMs31. The Gilbert damping is included phenomenologically via a Rayleigh dissipation functional appropriately generalized to the two-sublattice system, which motivates intra- as well as cross-sublattice terms. The Gilbert damping parameter thus becomes a 2 2 matrix, in contrast with its scalar form for a single-sublattice FM. Solving the system of equations for spatially homogeneous modes in a collinear ground state, we obtain the decay rates of the two eigenmodes nding direct pathways towards probing the dissipation mech- anism and asymmetries in the system. Consistent with recent experiments28,39, we nd an enhancement in the decay rates39close to the magnetization compensation in a FiM with an unaltered damping matrix28. The general description is found to be consistent with the spin pumping mediated damping in the magnet34{36, and allows for relating the Gilbert damping matrix with the interfacial spin-mixing conductances. Focusing on AFMs, we ex- press the magnetization dynamics in terms of the Neel variable thus clarifying the origin of the dierent damping terms in the corresponding dynamical equations38,40. Apart from the usually considered terms, we nd additional contributions for the case when sublattice- symmetry is broken in the AFM36,41{45. Thus, FMR linewidth measurements oer a direct, parameter-free means of probing the sublattice asymmetry in AFMs, complementary to the spin pumping shot noise36. This paper is organized as follows. We derive the Landau-Lifshitz-Gilbert (LLG) equa- tions for the two-sublattice model in Sec. II. The ensuing equations are solved for the resonance frequencies and decay rates of the uniform modes in a collinear magnet in Sec. III. Section IV presents the application of the phenomenology to describe a compensated ferrimagnet and spin pumping mediated Gilbert damping. The case of AFMs is discussed in Sec. V. We comment on the validity and possible generalizations of the theory in Sec. VI. The paper is concluded with a summary in Sec. VII. The discussion of a generalized Rayleigh 3dissipation functional and properties of the damping matrix is deferred to the appendix. II. MAGNETIZATION DYNAMICS AND GILBERT DAMPING We consider a two-sublattice magnet described by classical magnetization elds MMMA MMMA(rrr;t) andMMMBMMMB(rrr;t) corresponding to the sublattices AandB. The system is characterized by a magnetic free energy F[MMMA;MMMB] with the magnetization elds assumed to be of constant magnitudes MA0andMB0. Here, the notation F[ ] is employed to emphasize that the free energy is a functional over the magnetization elds, i.e. an integration of the free energy density over space. The undamped magnetization dynamics is described by equating the time derivative of the spin angular momentum associated with the magnetization to the torque experienced by it. The resulting Landau-Lifshitz equations for the two elds may be written as: d dtMMMA;B j A;Bj =_MMMA;B j A;Bj=MMMA;B0HHHA;B; (1) where A;B(<0) are the gyromagnetic ratios for the two sublattices, and HHHA;Bare the eective magnetic elds experienced by the respective magnetizations. This expression of angular momentum ow may be derived systematically within the Lagrangian classical eld theory31. The same formalism also allows to account for a restricted form of damping via the so-called dissipation functional R[_MMMA;_MMMB] in the generalized equations of motion: d dtL[
] _MMMA;BL[
] MMMA;B=R[_MMMA;_MMMB] _MMMA;B; (2) whereL[
] L [MMMA;MMMB;_MMMA;_MMMB] is the Lagrangian of the magnetic system. Here, L[
]=MMMArepresents the functional derivative of the Lagrangian with respect to the var- ious components of MMMA, and so on. The left hand side of Eq. (2) above represents the conservative dynamics of the magnet and reproduces Eq. (1) with31 0HHHA;B=F[MMMA;MMMB] MMMA;B; (3) while the right hand side accounts for the damping. The Gilbert damping is captured by a viscous Rayleigh dissipation functional parame- terized by a symmetric matrix ijwithfi;jg=fA;Bg: R[_MMMA;_MMMB] =Z Vd3rAA 2_MMMA
_MMMA+BB 2_MMMB
_MMMB+AB_MMMA
_MMMB ; (4) 4whereVis the volume of the magnet. The above form of the functional assumes the damping to be spatially homogeneous, isotropic, and independent of the equilibrium conguration. A more general form with a lower symmetry is discussed in appendix A. Including the dissipation functional via Eq. (2) leads to the following replacements in the equations of motion (1): 0HHHA!0HHHAAA_MMMAAB_MMMB; (5) 0HHHB!0HHHBBB_MMMBAB_MMMA: (6) Hence, the LLG equations for the two-sublattice magnet become: _MMMA=j Aj(MMMA0HHHA) +j AjAA MMMA_MMMA +j AjAB MMMA_MMMB ; (7) _MMMB=j Bj(MMMB0HHHB) +j BjAB MMMB_MMMA +j BjBB MMMB_MMMB :(8) These can further be expressed in terms of the unit vectors ^mmmA;B=MMMA;B=MA0;B0: _^mmmA=j Aj(^mmmA0HHHA) +AA mmmA_^mmmA +AB ^mmmA_^mmmB ; (9) _^mmmB=j Bj(^mmmB0HHHB) +BA ^mmmB_^mmmA +BB ^mmmB_^mmmB ; (10) thereby introducing the Gilbert damping matrix ~ for a two-sublattice system: ~=0 @AAAB BABB1 A=0 @j AjAAMA0j AjABMB0 j BjABMA0j BjBBMB01 A; (11) AB BA=j AjMB0 j BjMA0: (12) As elaborated in appendix B, the positivity of the dissipation functional implies that the eigenvalues and the determinant of ~ must be non-negative, which is equivalent to the following conditions: AA;BB0;  AABB2 AB=)AA;BB0;  AABBABBA: (13) Thus, Eqs. (9) and (10) constitute the main result of this section, and introduce the damping matrix [Eq. (11)] along with the constraints imposed on it [Eq. (12) and (13)] by the underlying formalism. 5III. UNIFORM MODES IN COLLINEAR GROUND STATE In this section, we employ the phenomenology introduced above to evaluate the resonance frequencies and the decay rates of the spatially homogeneous modes that can be probed in a typical FMR experiment. We thus work in the macrospin approximation, i.e. magnetizations are assumed to be spatially invariant. Considering an antiferromagnetic coupling J(>0) between the two sublattices and parameterizing uniaxial easy-axis anisotropies via KA;B(> 0), the free energy assumes the form: F[MMMA;MMMB] =Z Vd3r 0H0(MAz+MBz)KAM2 AzKBM2 Bz+JMMMA
MMMB ;(14) whereH0^zzzis the applied magnetic eld. The magnet is assumed to be in a collinear ground state:MMMA=MA0^zzzandMMMB=MB0^zzzwithMA0>M B0. Employing Eq. (3) to evaluate the eective elds, the magnetization dynamics is expressed via the LLG equations (9) and (10). Considering MMMA=MAx^xxx+MAy^yyy+MA0^zzz,MMMB=MBx^xxx+MBy^yyyMB0^zzzwithjMAx;Ayj MA0,jMBx;Byj MB0, we linearize the resulting dynamical equations. Converting to Fourier space via MAx=MAxexp (i!t) etc. and switching to circular basis via MA(B)= MAx(Bx)iMAy(By), we obtain two sets of coupled equations expressed succinctly as: 0 @! Ai! AA j AjJMA0+i! ABMA0 MB0  j BjJMB0+i! BAMB0 MA0 !+ B+i! BB1 A0 @MA MB1 A=0 @0 01 A; (15) where we dene Aj Aj(JMB0+ 2KAMA0+0H0) and Bj Bj(JMA0+ 2KBMB0 0H0). Substituting !=!r+i!iinto the ensuing secular equation, we obtain the resonance frequencies !rto the zeroth order and the corresponding decay rates !ito the rst order in the damping matrix elements: !r=( A B) +p ( A+ B)24J2j Ajj BjMA0MB0 2; (16) !i !r=!r(AABB) +AA B+BB A2Jj BjMA0AB !r++!r: (17) In the expression above, Eq. (16) and Eq. (17), we have chosen the positive solutions of the secular equations for the resonance frequencies. The negative solutions are equal in magnitude to the positive ones and physically represent the same two modes. The positive- polarized mode in our notation corresponds to the typical ferromagnetic resonance mode, while the negative-polarized solution is sometimes termed `antiferromagnetic resonance'25. 6020406080100120 0 0.2 0.4 0.6 0.8 10.040.050.060.070.080.090.1FIG. 1. Resonance frequencies and normalized decay rates vs. the applied eld for a quasi- ferromagnet ( MA0= 5MB0).j Aj=j Bj= 1;1:5;0:5 correspond to solid, dashed and dash-dotted lines respectively. The curves in blue and red respectively depict the + and modes. The damping parameters employed are AA= 0:06,BB= 0:04 andAB= 0. In order to avoid confusion with the ferromagnetic or antiferromagnetic nature of the un- derlying material, we call the two resonances as positive- and negative-polarized. The decay rates can further be expressed in the following form: !i !r=( A+ B)2Jj BjMA0AB !r++!r
; (18) with (AA+BB)=2 and
(AABB)=2. Eq. (18) constitutes the main result of this section and demonstrates that (i) asymmetric damping in the two sublattices is manifested directly in the normalized decay rates of the two modes (Figs. 1 and 2), and (ii) o-diagonal components of the damping matrix may reduce the decay rates (Fig. 2). Furthermore, it is consistent with and reproduces the mode-dependence of the decay rates observed in the numerical studies of some metallic AFMs37. To gain further insight into the results presented in Eqs. (16) and (18), we plot the 7resonance frequencies and the normalized decay rates vs. the applied magnetic eld for a typical quasi-ferromagnet, such as yttrium iron garnet, in Fig. 1. The parameters employed in the plot arej Bj= 1:81011,MB0= 105,KA=KB= 107, andJ= 105in SI units, and have been chosen to represent the typical order of magnitude without pertaining to a specic material. The plus-polarized mode is lower in energy and is raised with an increasing applied magnetic eld. The reverse is true for the minus-polarized mode whose relatively large frequency makes it inaccessible to typical ferromagnetic resonance experiments. As anticipated from Eq. (18), the normalized decay rates for the two modes dier when AA6= BB. Furthermore, the normalized decay rates are independent of the applied eld for symmetric gyromagnetic ratios for the two sublattices. Alternately, a measurement of the normalized decay rate for the plus-polarized mode is able to probe the sublattice asymmetry in the gyromagnetic ratios. Thus it provides essential information about the sublattices without requiring the measurement of the large frequency minus-polarized mode. IV. SPECIFIC APPLICATIONS We now examine two cases of interest: (i) the mode decay rate in a ferrimagnet close to its compensation temperature, and (ii) the Gilbert damping matrix due to spin pumping into an adjacent conductor. A. Compensated ferrimagnets FMR experiments carried out on gadolinium iron garnet23,39nd an enhancement in the linewidth, and hence the mode decay rate, as the temperature approaches the compensation condition, i.e. when the two eective46sublattices have equal saturation magnetizations. These experiments have conventionally been interpreted in terms of an eective single- sublattice model thereby ascribing the enhancement in the decay rate to an increase in the scalar Gilbert damping constant allowed within the single-sublattice model24. In contrast, experiments probing the Gilbert parameter in a dierent FiM via domain wall velocity nd it to be essentially unchanged around compensation28. Here, we analyze FMR in a compensated FiM using the two-sublattice phenomenology developed above and thus address this apparent inconsistency. 8020406080100120 1 2 3 4 500.050.10.150.20.250.3FIG. 2. Resonance frequencies and normalized decay rates vs. relative saturation magnetizations of the sublattices. The curves which are not labeled as + or represent the common normalized decay rates for both modes. The parameters employed are the same as for Fig. 1 with A= B. The compensation behavior of a FiM may be captured within our model by allowing MA0to vary while keeping MB0xed. The mode frequencies and normalized decay rates are examined with respect to the saturation magnetization variation in Fig. 2. We nd an enhancement in the normalized decay rate, consistent with the FMR experiments23,39, for a xed Gilbert damping matrix. The single-sublattice interpretation ascribes this change to a modication of the eective Gilbert damping parameter24, which is equal to the normalized decay rate within that model. In contrast, the latter is given by Eq. (18) within the two-sublattice model and evolves with the magnetization without requiring a modication in the Gilbert damping matrix. Specically, the enhancement in decay rate observed at the compensation point is analogous to the so-called exchange enhancement of damping in AFMs47. Close to compensation, the FiM mimics an AFM to some extent. We note that while the spherical samples employed in Ref. 23 are captured well by our simple free energy expression [Eq. (14)], the interfacial and shape anisotropies of the thin 9lm sample employed in Ref. 39 may result in additional contributions to decay rates. The similarity of the observed linewidth trends for the two kinds of samples suggests that these additional anisotropy eects may not underlie the observed damping enhancement. Quan- titatively accounting for these thin lm eects requires a numerical analysis, as discussed in Sec VI below, and is beyond the scope of the present work. Furthermore, domain forma- tion may result in additional damping contributions not captured within our single-domain model. B. Spin pumping mediated Gilbert damping Spin pumping34from a FM into an adjacent conductor has been studied in great detail35 and has emerged as a key method for injecting pure spin currents into conductors48. The angular momentum thus lost into the conductor results in a contribution to the magnetic damping on top of the intrinsic dissipation in the bulk of the magnet. A variant of spin pumping has also been found to be the dominant cause of dissipation in metallic magnets37. Thus, we evaluate the Gilbert damping matrix arising due to spin pumping from a two- sublattice magnet36into an adjacent conductor acting as an ideal spin sink. Within the macrospin approximation, the total spin contained by the magnet is given by: SSS=MA0V^mmmA j AjMB0V^mmmB j Bj: (19) The spin pumping current emitted by the two-sublattice magnet has the following general form36: IIIs=~ eX i;j=fA;BgGij ^mmmi_^mmmj ; (20) withGAB=GBA, where the spin-mixing conductances Gijmay be evaluated within dierent microscopic models36,49{51. Equating the spin pumping current to _SSSand employing Eqs. (9) and (10), the spin pumping contribution to the Gilbert damping matrix becomes: 0 ij=~Gijj ij eMi0V; (21) which in turn implies 0 ij=~Gij eMi0Mj0V; (22) 10for the corresponding dissipation functional. The resulting Gilbert damping matrix is found to be consistent with its general form and constraints formulated in Sec. II. Thus, employing the phenomenology developed above, we are able to directly relate the magnetic damping in a two-sublattice magnet to the spin-mixing conductance of its interface with a conductor. V. ANTIFERROMAGNETS Due to their special place with high symmetry in the two-sublattice model as well as the recent upsurge of interest7{10,52{54, we devote the present section to a focused discussion on AFMs in the context of the general results obtained above. It is often convenient to describe the AFM in terms of a dierent set of variables: mmm=^mmmA+^mmmB 2; nnn=^mmmA^mmmB 2: (23) In contrast with ^mmmAand ^mmmB,mmmandnnnare not unit vectors in general. The dynamical equations for mmmandnnnmay be formulated by developing the entire eld theory, starting with the free energy functional, in terms of mmmandnnn. Such a formulation, including damping, has been accomplished by Hals and coworkers40. Here, we circumvent such a repetition and directly express the corresponding dynamical equations by employing Eqs. (9) and (10) into Eq. (23): _mmm=(mmm m0HHHm)(nnn n0HHHn) +X p;q=fm;ngm pq(ppp_qqq); (24) _nnn=(mmm n0HHHn)(nnn m0HHHm) +X p;q=fm;ngn pq(ppp_qqq); (25) with m0HHHmj Aj0HHHA+j Bj0HHHB 2; (26) n0HHHnj Aj0HHHAj Bj0HHHB 2; (27) m mm=n nm=AA+BB+AB+BA 2; (28) m mn=n nn=AABBAB+BA 2; (29) m nn=n mn=AA+BBABBA 2; (30) m nm=n mm=AABB+ABBA 2: (31) 11A general physical signicance, analogous to A;B, may not be associated with m;nwhich merely serve the purpose of notation here. The equations obtained above manifest new damping terms in addition to the ones that are typically considered in the description of AFMs. Accounting for the sublattice symmetry of the antiferromagnetic bulk while allowing for the damping to be asymmetric, we may assume A= BandMA0=MB0, with (AA+BB)=2,
(AABB)=2, andAB=BAod. Thus, the damping parameters simplify to m mm=n nm=+od; (32) m mn=n nn=
; (33) m nn=n mn=od; (34) m nm=n mm=
; (35) thereby eliminating the \new" terms in the damping when AA=BB. However, the sublat- tice symmetry may not be applicable to AFMs, such as FeMn, with non-identical sublattices. Furthermore, the sublattice symmetry of the AFM may be broken at the interface41{43via, for example, spin mixing conductances36,45,55resulting in AA6=BB. The resonance frequencies and normalized decay rates [Eqs. (16) and (18)] take a simpler form for AFMs. Substituting KA=KBK, A= B , andMA0=MB0M0: !r=j j0H0+ 2j jM0p (J+K)K; (36) !i !r=J(od) + 2K 2p (J+K)K
(od) 2r J K+ r K J
; (37) where we have employed JKin the nal simplication. The term /p K=J has typically been disregarded on the grounds KJ. However, recent numerical studies of damping in several AFMs37nd od>0 thus suggesting that this term should be comparable to the one proportional top J=K and hence may not be disregarded. The expression above also suggests measurement of the normalized decay rates as a means of detecting the sublat- tice asymmetry in damping. For AFMs symmetrical in the bulk, such an asymmetry may arise due to the corresponding asymmetry in the interfacial spin-mixing conductance36,45,55. Thus, decay rate measurements oer a method to detect and quantify such interfacial eects complementary to the spin pumping shot noise measurements suggested earlier36. 12VI. DISCUSSION We have presented a phenomenological description of Gilbert damping in two-sublattice magnets and demonstrated how it can be exploited to describe and characterize the system eectively. We now comment on the limitations and possible generalizations of the formal- ism presented herein. To begin with, the two-sublattice model is the simplest description of ferri- and antiferromagnets. It has been successful in capturing a wide range of phenomenon. However, recent measurements of magnetization dynamics in nickel oxide could only be ex- plained using an eight-sublattice model56. The temperature dependence of the spin Seebeck eect in yttrium iron garnet also required accounting for more than two magnon bands57. A generalization of our formalism to a N-sublattice model is straightforward and can be achieved via a Rayleigh dissipation functional with N2terms, counting ijandjias sepa- rate terms. The ensuing Gilbert damping matrix will be N N while obeying the positive determinant constraint analogous to Eq. (13). In our description of the collinear magnet [Eq. (14)], we have disregarded contributions to the free energy which break the uniaxial symmetry of the system about the z-axis. Such terms arise due to spin-nonconserving interactions58, such as dipolar elds and magnetocrys- talline anisotropies, and lead to a mixing between the plus- and minus-polarized modes30. Including these contributions converts the two uncoupled 2 2 matrix equations [(15)] into a single 44 matrix equation rendering the solution analytically intractable. A detailed analysis of these contributions30shows that their eect is most prominent when the two modes are quasi-degenerate, and may be disregarded in a rst approximation. In evaluating the resonance frequencies and the decay rates [Eqs. (16) and (18)], we have assumed the elements of the damping matrix to be small. A precise statement of the assumption employed is !i!r, which simply translates to 1 for a single-sublattice ferromagnet. In contrast, the constraint imposed on the damping matrix within the two- sublattice model by the assumption of small normalized decay rate is more stringent [Eq. (18)]. For example, this assumption for an AFM with AB=
= 0 requires   p K=J1. This stringent constraint may not be satised in most AFMs37, thereby bringing the simple Lorentzian shape description of the FMR into question. It can also be seen from Fig. 2 that the assumption of a small normalized decay rate is not very good for the chosen parameters. 13VII. SUMMARY We have developed a systematic phenomenological description of the Gilbert damping in a two-sublattice magnet via inclusion of a Rayleigh dissipation functional within the La- grangian formulation of the magnetization dynamics. Employing general expressions based on symmetry, we nd cross-sublattice Gilbert damping terms in the LLG equations in con- sistence with other recent ndings36{38. Exploiting the phenomenology, we explain the en- hancement of damping23,39in a compensated ferrimagnet without requiring an increase in the damping parameters28. We also demonstrate approaches to probe the various forms of sublattice asymmetries. Our work provides a unied description of ferro- via ferri- to antiferromagnets and allows for understanding a broad range of materials and experiments that have emerged into focus in the recent years. ACKNOWLEDGMENTS A. K. thanks Hannes Maier-Flaig and Kathrin Ganzhorn for valuable discussions. We acknowledge nancial support from the Alexander von Humboldt Foundation, the Research Council of Norway through its Centers of Excellence funding scheme, project 262633, \QuS- pin", and the DFG through SFB 767 and SPP 1538. Appendix A: Generalized Rayleigh dissipation functional As compared to the considerations in Sec. II, a more general approach to parameterizing the dissipation functional is given by: R[_MMMA;_MMMB] =1 2Z VZ Vd3r0d3rX p;q=fA;BgX i;j=fx;y;zg_Mpi(rrr)ij pq(rrr;rrr0)_Mqj(rrr0): (A1) This form allows to capture the damping in an environment with a reduced symmetry. However, the larger number of parameters also makes it dicult to extract them reliably via typical experiments. The above general form reduces to the case considered in Sec. II whenij pq(rrr;rrr0) =pqij(rrrrrr0) andpq=qp. Furthermore, the coecients ij pqmay depend uponMMMA(rrr) andMMMB(rrr) as has been found in recent numerical studies of Gilbert damping in AFMs37. 14Appendix B: Damping matrix The Rayleigh dissipation functional considered in the main text is given by: R[_MMMA;_MMMB] =Z Vd3rAA 2_MMMA
_MMMA+BB 2_MMMB
_MMMB+AB_MMMA
_MMMB ; (B1) which may be brought into the following concise form with the notation~_MMM[_MMMA_MMMB]|: R[_MMMA;_MMMB] =1 2Z Vd3r~_MMM|~~_MMM; (B2) where ~is the appropriate matrix given by: ~=0 @AAAB ABBB1 A: (B3) Considering an orthogonal transformation~_MMM=~Q~_M, the dissipation functional can be brought to a diagonal form R[_MMMA;_MMMB] =1 2Z Vd3r~_M|~Q|~~Q~_M; (B4) where ~Q|~~Qis assumed to be diagonal. The positivity of the dissipation for arbitrary magnetization dynamics then requires the two diagonal elements to be non-negative which further entails the non-negativity of the determinant of ~ : j~Q|~~Qj0; (B5) j~Q|jj~jj~Qj0; (B6) j~j0 =)AABB2 AB: (B7) akashdeep.kamra@ntnu.no yarne.brataas@ntnu.no 1S. J. Barnett, \Gyromagnetic and electron-inertia eects," Rev. Mod. Phys. 7, 129{166 (1935). 2J.C. Slonczewski, \Current-driven excitation of magnetic multilayers," Journal of Magnetism and Magnetic Materials 159, L1 { L7 (1996). 3L. 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1508.05778v3.Scaling_variables_and_asymptotic_profiles_for_the_semilinear_damped_wave_equation_with_variable_coefficients.pdf

arXiv:1508.05778v3 [math.AP] 10 Oct 2016Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients Yuta Wakasugi Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan Email:yuta.wakasugi@math.nagoya-u.ac.jp Abstract We study the asymptotic behavior of solutions for the semili near damped wave equation with variable coefficients. We prove that if the damping is effectiv e, and the nonlinearity and other lower order terms can be regarded as perturbations, then the solution is approximated by the scaled Gaussian of the corresponding linear parabolic prob lem. The proof is based on the scaling variables and energy estimates. 1 Introduction We consider the Cauchy problem of the semilinear damped wave equat ion with lower order perturba- tions /braceleftbigg utt+b(t)ut= ∆xu+c(t)·∇xu+d(t)u+N(u,∇xu,ut), t>0,x∈Rn, u(0,x) =εu0(x), ut(0,x) =εu1(x), x ∈Rn,(1.1) where the coefficients b,canddare smooth, bsatisfies b(t)∼(1+t)−β,−1≤β <1, (1.2) andc(t)·∇xu,d(t)u,N(u,∇xu,ut) can be regarded as perturbations (the precise assumption will be given in the next section). Also, εdenotes a small parameter. Our purpose is to give the asymptotic profile of global solutions to (1 .1) with small initial data as time tends to infinity. By the assumption (1.2), the damping is effectiv e, and we can expect that the asymptotic profile of solutions is given by the scaled Gaussian (see (2 .7), (2.8) and (2.9)). Theexistenceofglobalsolutionsandtheasymptoticbehaviorofso lutionstodampedwaveequations have been widely investigated for a long time. Matsumura [27] obtain ed decay estimates of solutions to the linear damped wave equation utt−∆u+ut= 0, (1.3) and applied them to nonlinear problems. After that, Yang and Milani [5 2] showed that the solution of (1.3) has the so-called diffusion phenomena , that is, the asymptotic profile of solutions to (1.3) is given by the Gaussian in the L∞-sense. Marcati and Nishihara [26] and Nishihara [31] gave more det ailed informations about the asymptotic behavior of solutions. They fou nd that when n= 1,3, the solution of (1.3) is asymptotically decomposed into the Gaussian and a solution of the wave equation (with an 1exponentially decaying coefficient) in the Lp–Lqsense (see Hosono and Ogawa [12] and Narazaki [30] forn= 2 andn≥4). For the nonlinear problem /braceleftbigg utt−∆u+ut=N(u), (u,ut)(0,x) =ε(u0,u1)(x),(1.4) there are many results about global existence and asymptotic beh avior of solutions (see for example, [13, 14,17, 19, 20, 22, 32]). Inparticular, TodorovaandYordano v[41] and Zhang[53] provedthat when N(u) =|u|p, the critical exponent of (1.4) is given by p= 1+2/n. More precisely, they showed that, for initial data satisfying ( u0,u1)∈H1,0(Rn)×L2(Rn) and having compact support, if p>1+2/n, then the global solution uniquely exists for small ε; ifp≤1+2/nand/integraltext Rn(u0+u1)(x)dx >0, then the local-in-time solution blows up in finite time for any ε >0. The number 1 + 2 /nis the same as the well-known Fujita exponent, which is the critical exponent of the semilinear heat equation vt−∆v=vp(see [7]), though the role of the critical exponent is different in the s emilinear heat equation and the semilinear damped wave equation. In fact, for the subcritical case 1 <p<1+2/n, the solution of the semilinear damped wave equation blows up in finite tim e under the positive mass condition/integraltext Rn(u0+u1)(x)dx>0, while all positive solutions blow up in finite time for the semilinear heat equation. Concerning the asymptotic behavior of the global solution, Hayash i, Kaikina and Naumkin [10] proved that if Nsatisfies|N(u)| ≤C|u|pwithp>1+2/n, then the unique global solution exists for suitably small data and the asymptotic profile of the solution is given b y a constant multiple of the Gaussian. However, they used the explicit formula of the fundamen tal solution of the linear problem in the Fourier space, and hence, it seems to be difficult to apply their m ethod to variable coefficient cases. GallayandRaugel[8]consideredtheone-dimensionaldampedwave equationwithvariableprincipal term and a constant damping utt−(a(x)ux)x+ut=N(u,ux,ut). They used scaling variables s= log(t+t0), y=x√t+t0, (1.5) and showed that if a(x) is positive and has the positive limits lim x→±∞a(x) =a±, then the solution can be asymptotically expanded in terms of the corresponding para bolic equation. Moreover, this expansion can be determined up to the second order. Recently, Ta keda [39, 40] and Kawakami and Takeda [18] obtained the complete expansion for the linear and nonlin ear damped wave equation with constant coefficients. The wave equation with variable coefficient damping utt−∆u+b(t,x)ut= 0 has been also intensively studied. Yamazaki [50, 51] and Wirth [46, 47 , 48, 49] considered time- dependent damping b=b(t). Here we briefly explain their results by restricting the damping bto b(t) =µ(1+t)−βwithµ>0 andβ∈R, although they discussed more general b(t): (i) when β >1 (scattering), the solution scatters to a solution of the free wave equation; (ii) when β= 1 (non-effective weak dissipation), the behavior of solutions depends on the consta ntµ, and the solution scatters with some modification; (iii) when β∈[−1,1) (effective), the asymptotic profile of the solution is given by the scaled Gaussian; (iv) when β <−1 (overdamping), the solution tends to some asymptotic state, which is nontrivial function for nontrivial data. Hence our assumpt ion (1.2) is reasonable because the asymptotic behavior of solutions to the linear problem completely cha nges whenβ <−1 orβ≥1. In the space-dependent damping case b=b(x) = (1 + |x|2)−α/2, Mochizuki [28] (see also [29]) proved that if α >1, then the energy of solution does not decay to zero in general an d solutions 2with data satisfying certain condition scatter to free solutions. On the other hand, Todorova and Yordanov [42] obtained energy decay of solutions when α∈[0,1) and the decay rates agree with those of the corresponding parabolic equation. Moreover, the au thor of this paper [45] proved that the solution actually has the diffusion phenomena when α∈[0,1). In the critical case α= 1, that is,b=µ(1+|x|2)−1/2, Ikehata, Todorova and Yordanov [16] obtained optimal decay es timates of the energy of solutions and found that the decay rate depends on the constantµ. However, the precise asymptotic profile is still open. On the other hand, Radu, Todorova and Yordanov [37, 38] studied the diffusion phenomena for solutions to the abstract damped wave equ ation (L∂2 t+∂t+A)u= 0 by the method of the diffusion approximation, where Ais a nonnegative self-adjoint operator, and Lis a bounded positive self-adjoint operator. Recently, Nishiyama [36 ] studied the abstract damped wave equation having the form ( ∂2 t+M∂t+A)u= 0, where Mis a bounded nonnegative self-adjoint operator. Moreover, as an application, he also determined the asy mptotic profile of solutions to the damped wave equation with variable coefficients under a geometric co ntrol condition. For the semilinear wave equation with space-dependent damping utt−∆u+b(x)ut=N(u), Ikehata, Todorova and Yordanov [15] proved that when b(x)∼(1+|x|)−αwithα∈[0,1) andN(u) = |u|p, the critical exponent is p= 1+2/(n−α) (see also Nishihara [33] for the case N(u) =−|u|p−1u andb(x) = (1+ |x|2)−α/2withα∈[0,1)). Recently, the asymptoticbehaviorofsolutionsto the semilinearwav eequationwith time-dependent damping utt−∆u+b(t)ut=N(u) was also studied. When b(t) = (1 +t)−β(−1< β < 1) andN(u) =|u|p, Lin, Nishihara and Zhai [25] determined the critical exponent as p= 1 + 2/n, provided that the initial data belong to H1,0(Rn)×L2(Rn) with compact support. D’Abbicco, Lucente and Reissig [5] (see also [4]) extended this result to more general b(t) satisfying a monotonicity condition and a polynomial-like behavior. Moreover, they relaxed the assumption on the data to exponentia lly decaying condition. They also dealt with the initial data belong to the class ( L1(Rn)∩H1,0(Rn))×(L1(Rn)∩L2(Rn)) whenn≤4. We also refer the reader to D’Abbicco [3] for the critical case β= 1. On the other hand, Nishihara [34] studied the asymptotic profile of solutions in the case n= 1,b= (1 +t)−β(−1< β <1), (u0,u1)∈H1,0(Rn)×L2(Rn) with compact support and N(u) =−|u|p−1u(see also [35]). He proved thattheasymptoticprofileisgivenbythescaledGaussian. However ,the asymptoticprofileofsolutions in higher dimensional cases n≥2 remains open. Furthermore, even for the small data global exist ence, there are no results for non exponentially decaying initial data when n≥5. Here we also refer the reader to [21, 23, 24, 43, 44] for space and time dependent dampin g cases. In this paper, we shall prove the existence of the global-in-time solu tion to the Cauchy problem (1.1) with suitably small εand determine the asymptotic profile. Our result extends that of [ 34] to higher dimensional cases n≥2, more general damping b=b(t), non exponentially decaying initial data and with lower order perturbations. Moreover, in the one-dim ensional case, we can treat more general nonlinear terms N=N(u,ux,ut) including first order derivatives. For the proof, we basically follow the method of Gallay and Raugel [8]. To extend their argument t o variable damping cases, we introduce new scaling variables s= log(B(t)+1), y= (B(t)+1)−1/2x, B(t) =/integraldisplayt 0dτ b(τ) instead of (1.5). Then, we decompose the solution to the asymptot ic profile and the remainder term, and prove that remainder term decays to zero as time tends to infin ity by using the energy method. To estimate the energy of the remainder term, in [8], they used the p rimitive of the remainder term 3F(s,y) =/integraltexty −∞f(s,z)dz. However, this does not work in higher dimensional cases n≥2. To overcome this difficulty, we employ the idea from Coulaud [2] in which asymptotic pr ofiles for the second grade fluids equationwerestudied in the three dimensionalspace. Namely, weshall usethe fractionalintegral of the form ˆF(ξ) =|ξ|−n/2−δˆf(ξ) with 0< δ <1, and apply the energy method to ˆFin the Fourier side. Since the remainder term fsatisfies ˆf(0) = 0, ˆFmakes sense and enables us to control the term /ba∇dblˆf/ba∇dblL2in energy estimates. This paper is organized as follows. In the next section, we state the precise assumptions and our main result. Section 3 is devoted to a proofof the main result. The pr oof of energy estimates is divided into the one-dimensional case and the higher dimensional cases. Af ter that, we will unify both cases and complete the proof of our result except for the estimates of t he error terms. These error estimates will be given in Section 4. We end up this section with some notations used in this paper. For a co mplex number ζ, we denote by Reζits real part. The letter Cindicates a generic positive constant, which may change from line to line. In particular, we denote by C(∗,...,∗) constants depending on the quantities appearing in parenthesis. We use the symbol f∼g, which stands for C−1g≤f≤Cgwith some C≥1. For a function u=u(t,x) : [0,∞)×Rn→R, we write ut=∂u ∂t(t),∂xiu=∂u ∂xi(i= 1,...,n),∇xu= t(∂x1u,...,∂ xnu) and ∆u(t,x) =/summationtextn i=1∂2 xiu(t,x). Furthermore, we sometimes use /an}b∇acketle{tx/an}b∇acket∇i}ht:=/radicalbig 1+|x|2. For a function f=f(x) :Rn→R, we denote the Fourier transform of fbyˆf=ˆf(ξ), that is, ˆf(ξ) = (2π)−n/2/integraldisplay Rnf(x)e−ixξdx. LetLp(Rn) andHk,m(Rn) be usual Lebesgue and weighted Sobolev spaces, respectively, e quipped with the norms defined by /ba∇dblf/ba∇dblLp=/parenleftbigg/integraldisplay Rn|f(x)|pdx/parenrightbigg1/p (1≤p<∞),/ba∇dblf/ba∇dblL∞= esssupx∈Rn|f(x)|, /ba∇dblf/ba∇dblHk,m=/summationdisplay |α|≤k/ba∇dbl(1+|x|)m∂α xf/ba∇dblL2(k∈Z≥0,m≥0). For an interval Iand a Banach space X, we define Cr(I;X) as the space of r-times continuously differentiable mapping from ItoXwith respect to the topology in X. 2 Main result Let us introduce our main result. First, we put the following assumpt ions: Assumptions (i) The initial data ( u0,u1) belong to H1,m(Rn)×H0,m(Rn), wherem= 1 (n= 1) andm > n/2+1 (n≥2). (ii) The coefficient of the damping term b(t) satisfies C−1(1+t)−β≤b(t)≤C(1+t)−β,/vextendsingle/vextendsingle/vextendsingle/vextendsingledb dt(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(1+t)−1b(t) (2.1) with someβ∈[−1,1). (iii) The functions c(t) andd(t) satisfy |c(t)| ≤C(1+t)−γ,|d(t)| ≤C(1+t)−ν(2.2) with someγ >(1+β)/2 andν >1+β. 4(iv)-(1) When n= 1, the nonlinearity Nis of the form N=k/summationdisplay i=1Ni(u,ux,ut) for somek≥0 and each Ni=Ni(z) =Ni(z1,z2,z3) satisfies |Ni(z)| ≤C|z1|pi1|z2|pi2|z3|pi3, pij≥1 or = 0, pi1>1, pi2+pi3≤1, pi1+2pi2+/parenleftbigg 3−2β 1+β/parenrightbigg pi3>3,(2.3) where we note that when β=−1, the number −2β/(1+β) is interpreted as an arbitrary large number. Moreover, to ensure the existence of local-in-time solutio ns, we assume that, for any R>0, there exists a constant C(R)>0 such that |Ni(z)−Ni(w)| ≤C(R)[|z1−w1|(1+|z2|+|w2|+|z3|+|w3|)+|z2−w2|+|z3−w3|] (2.4) forzi,wi∈R(i= 1,2,3) satisfying |z1|,|w1| ≤R. (iv)-(2) When n≥2, the nonlinearity Nis of classC1and independent of ∇xu,ut, that is,N=N(u). Moreover,Nsatisfies /braceleftBigg|N(u)| ≤C|u|p, 2<p<+∞(n= 2),1+2 n<p≤n n−2(n≥3).(2.5) Also, to ensure the existence of local-in-time solutions, we assume t hat |N(u)−N(v)| ≤C|u−v|(|u|+|v|)p−1. (2.6) Remark 2.1. (i) By the above assumptions, as we will see later, we can rega rd the terms c(t)·∇xu, d(t)uandN(u,∇xu,ut)as perturbations. (ii) We can treat the case where the coefficients b(t),c(t)andd(t)depend on both tandx. More precisely, our result is also valid for b=b(t,x),c=c(t,x)andd=d(t,x)such thatb(t,x) = b0(t)+b1(t,x)withb0satisfying Assumption (ii), b1(t,x)fulfilling |b1(t,x)| ≤C(1+t)−µ(µ>β) andc(t,x),d(t,x)satisfying |c(t,x)| ≤C(1+t)−γ(γ >(1+β)/2),|d(t,x)| ≤C(1+t)−ν(ν > 1+β). (iii) A typical example satisfying the assumptions (2.3)and(2.4)is N=|u|pu+|u|qux+|u|rut withp>2,q>1andr>1. (iv) The assumption 1+2/n<pin(2.5)is sharp in the sense that, if N(u) =|u|p,1<p≤1+2/n and the initial data satisfies/integraltext Rn(u0+b∗u1)(x)dx>0withb∗=/integraltext∞ 0exp(−/integraltextt 0b(τ)dτ)dt, then the local-in-time solution blows up in finite time (see [15, 22, 2 4, 41, 53]). (v) Whenn= 1, we can also treat the principal term with variable coefficien t(a(x)ux)xsatisfying inf x∈Ra(x)>0,lim x→±∞a(x) =a±>0 instead ofuxx. However, the argument is the same as in Gallay and Raugel [8] and hence, we do not pursue here for simplicity. 5(vi) There are no mutual implication relations between the a ssumptions on the damping bin ours and Wirth [48], D’Abbicco, Lucente and Reissig [5]. To state our result, we put B(t) =/integraldisplayt 0dτ b(τ)(2.7) and G(t,x) = (4πt)−n/2exp/parenleftbigg −|x|2 4t/parenrightbigg . (2.8) We note that the assumption (2.1) implies that B(t) is strictly increasing, and lim t→∞B(t) = +∞. The main result of this paper is the following: Theorem 2.1. Under the Assumptions (i)–(iv), there exists some ε0>0such that, for any ε∈(0,ε0], there exists a unique solution u∈C([0,∞);H1,m(Rn))∩C1([0,∞);H0,m(Rn)) for the Cauchy problem (1.1). Moreover, there exists the limit α∗= lim t→∞/integraldisplay Rnu(t,x)dx such that the solution usatisfies /ba∇dblu(t,·)−α∗G(B(t),·)/ba∇dblL2≤Cε(B(t)+1)−n/4−λ/ba∇dbl(u0,u1)/ba∇dblH1,m×H0,m (2.9) fort≥1. Hereλis defined by λ= min/braceleftbigg1 2,m 2−n 4,λ0,λ1/bracerightbigg −η with arbitrary small number η>0, andλ0andλ1are defined by λ0= min/braceleftbigg1−β 1+β,γ 1+β−1 2,ν 1+β−1/bracerightbigg , where we interpret 1/(1+β)as an arbitrary large number when β=−1, and λ1= 1 2min i=1,...,k/braceleftbigg pi1+2pi2+/parenleftbigg 3−2β 1+β/parenrightbigg pi3−3/bracerightbigg , n= 1, n 2/parenleftbigg p−1−2 n/parenrightbigg , n ≥2. Here we interpret −2βpi3/(1+β)as an arbitrary large number when pi3/ne}ationslash= 0andβ=−1. Remark 2.2. IfN=c=d= 0, namely there are no perturbation terms, and if βis close to 1so that min{1/2,m/2−n/4,(1−β)/(1+β)}= (1−β)/(1+β), thenλ= (1−β)/(1+β)−ηwith arbitrary smallη>0, and we expect that the gain of the decay rate (1−β)/(1+β)is optimal, in other words, the second order approximation of udecays as (B(t) +1)−n/4−(1−β)/(1+β). The higher order asymptotic expansion will be discussed in a forthcoming paper. 63 Proof of the main theorem 3.1 Scaling variables We introduce the following scaling variables: s= log(B(t)+1), y= (B(t)+1)−1/2x (3.1) and v(s,y) =ens/2u(t(s),es/2y), w(s,y) =b(t(s))e(n+2)s/2ut(t(s),es/2y), or equivalently, u(t,x) = (B(t)+1)−n/2v(log(B(t)+1),(B(t)+1)−1/2x), ut(t,x) =b(t)−1(B(t)+1)−n/2−1w(log(B(t)+1),(B(t)+1)−1/2x),(3.2) where we have used the notation t(s) =B−1(es−1). Then, the problem (1.1) is transformed as vs−y 2·∇yv−n 2v=w, s>0,y∈Rn, e−s b(t(s))2/parenleftBig ws−y 2·∇yw−/parenleftBign 2+1/parenrightBig w/parenrightBig +w= ∆yv+r(s,y), s>0,y∈Rn, v(0,y) =εv0(y) =εu0(y), w(0,y) =εw0(y) =εb(0)u1(y), y∈Rn,(3.3) where r(s,y) =1 b(t(s))2db dt(t(s))w+es/2c(t(s))·∇yv+esd(t(s))v +e(n+2)s/2N/parenleftBig e−ns/2v,e−(n+1)s/2∇yv,b(t(s))−1e−(n+2)s/2w/parenrightBig . (3.4) 3.2 Preliminary lemmas First, we collect frequently used relations and estimates. Lemma 3.1. We have d dsb(t(s)) =db dt(t(s))b(t(s))es,d ds1 b(t(s))2=−2 b(t(s))2db dt(t(s))es. (3.5) Proof.First, we note that the function σ=B(t) is strictly increasing, and hence, the inverse t= B−1(σ) exists and d dσB−1(σ) =/parenleftbiggdB dt(t)/parenrightbigg−1 =b(t). Combining this with s= log(B(t)+1), we obtain d dsb(t(s)) =d dsb/parenleftbig B−1(es−1)/parenrightbig =db dt(t(s))d dsB−1(es−1) =db dt(t(s))/parenleftbiggdB dt(t(s))/parenrightbigg−1d ds(es−1) =db dt(t(s))b(t(s))es. 7This shows the first assertion of (3.5). Moreover, we have d ds1 b(t(s))2=−2 b(t(s))3d dsb(t(s)) =−2 b(t(s))2db dt(t(s))es, which shows the second assertion of (3.5). Next, the assumption (2.1) implies the following: Lemma 3.2. Under the assumption (2.1), we have the following estimates. (i) Whenβ∈(−1,1), we have b(t(s))∼e−βs/(1+β),e−s b(t(s))2∼e−(1−β)s/(1+β),1 b(t(s))2/vextendsingle/vextendsingle/vextendsingle/vextendsingledb dt(t(s))/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−(1−β)s/(1+β). (ii) Whenβ=−1, we have b(t(s))∼exp(es),e−s b(t(s))2∼exp(−2es−s),1 b(t(s))2/vextendsingle/vextendsingle/vextendsingle/vextendsingledb dt(t(s))/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cexp(−2es). Proof.(i) Whenβ∈(−1,1), from (2.7) and (3.1) we compute as es=B(t(s))+1 =/integraldisplayt(s) 0dτ b(τ)+1∼/integraldisplayt(s) 0(1+τ)βdτ+1∼(1+t(s))1+β. Therefore, one has 1+ t(s)∼es/(1+β), and hence, b(t(s))∼(1+t(s))−β∼e−βs/(1+β). By the assumption (2.1), the other estimates can be obtained in a sim ilar way. (ii) Whenβ=−1, we have es=B(t(s))+1∼/integraldisplayt(s) 0(1+τ)−1dτ+1 = log(1+ t(s))+1, and hence, b(t(s))∼1+t(s)∼exp(es) holds. We can prove the other estimates in the same way, and the proof is omitted. We sometimes employ the Gagliardo-Nirenberg inequality: Lemma 3.3 (Gagliardo-Nirenberg inequality) .Let1<p<∞(n= 1,2)and1<p≤n/(n−2) (n≥ 3). Then for any f∈H1,0(Rn), we have /ba∇dblf/ba∇dblL2p≤C/ba∇dbl∇f/ba∇dblσ L2/ba∇dblf/ba∇dbl1−σ L2, whereσ=n(p−1)/(2p). For the proof, see for example [6, 9]. 83.3 Local existence of solutions We prove the local existence of solutions for the equation (1.1) and the system (3.3), respectively. To this end, putting U(t,x) =/an}b∇acketle{tx/an}b∇acket∇i}htmu,U0(x) =/an}b∇acketle{tx/an}b∇acket∇i}htmu0andU1=/an}b∇acketle{tx/an}b∇acket∇i}htmu1, we change the problem (1.1) to /braceleftbigg Utt+b(t)Ut= ∆xU+˜c(t,x)·∇xU+˜d(t,x)U+˜N(U,∇xU,Ut), t>0,x∈Rn, U(0,x) =εU0(x), Ut(0,x) =εU1(x), x ∈Rn,(3.6) where ˜c=c−2m/an}b∇acketle{tx/an}b∇acket∇i}ht−2x,˜d=d−c·(m/an}b∇acketle{tx/an}b∇acket∇i}ht−2x)−m/an}b∇acketle{tx/an}b∇acket∇i}ht−4(n/an}b∇acketle{tx/an}b∇acket∇i}ht2−(m+2)|x|2) and ˜N(U,∇xU,Ut) =/an}b∇acketle{tx/an}b∇acket∇i}htmN/parenleftbig /an}b∇acketle{tx/an}b∇acket∇i}ht−mU,/an}b∇acketle{tx/an}b∇acket∇i}ht−m∇xU−m/an}b∇acketle{tx/an}b∇acket∇i}ht−m−2xU,/an}b∇acketle{tx/an}b∇acket∇i}ht−mUt/parenrightbig . We further put U=t(U,Ut) andU0=t(U0,U1). Then, the equation (3.6) is written as /braceleftbiggUt=AU+N(U), U(0) =εU0,(3.7) where A=/parenleftbigg 0 1 ∆ 0/parenrightbigg ,N(U) =/parenleftbigg0 −bUt+˜c·∇xU+˜dU+˜N(U,∇xU,Ut)/parenrightbigg . The operator AonH1,0(Rn)×L2(Rn) with the domain D(A) =H2,0(Rn)×H1,0(Rn) ism-dissipative (see [1, Proposition 2.6.9]) with dense domain, and hence, Agenerates a contraction semigroup etAon H1,0(Rn)×L2(Rn) (see [1, Theorem 3.4.4]). Thus, we consider the integral form U(t) =εetAU0+/integraldisplayt 0e(t−τ)AN(U(τ))dτ (3.8) of the equation (3.7) in C([0,T);H1,0(Rn)×L2(Rn)). First, we define the mild and strong solutions and the lifespan of solut ions. Definition 3.4. We say that uis a mild solution of the Cauchy problem (1.1)on the interval [0,T) ifuhas the regularity u∈C([0,T);H1,m(Rn))∩C1([0,T);H0,m(Rn)). (3.9) and satisfies the integral equation (3.8)inC([0,T);H1,0(Rn)×L2(Rn)). We also call ua strong solution of the Cauchy problem (1.1)on the interval [0,T)ifuhas the regularity u∈C([0,T);H2,m(Rn))∩C1([0,T);H1,m(Rn))∩C2([0,T);H0,m(Rn)) (3.10) and satisfies the equation (1.1)inC([0,T);H0,m(Rn)). Moreover, we say that (v,w)defined by (3.2) is a mild (resp. strong) solution of the Cauchy problem (3.3)on the interval [0,S)ifuis a mild (resp. strong) solution of (1.1)on the interval [0,t(S)). We note that if (v,w)is a mild solution of (3.3)on [0,S), then(v,w)has the regularity (v,w)∈C([0,S);H1,m(Rn)×H0,m(Rn)), and if(v,w)is a strong solution of (3.3)on[0,S), then(v,w)has the regularity (v,w)∈C([0,S);H2,m(Rn)×H1,m(Rn))∩C1([0,S);H1,m(Rn)×H0,m(Rn)) (3.11) and satisfies the system (3.3)inC([0,S);H1,m(Rn)×H0,m(Rn)). We also define the lifespan of the mild solutions uand(v,w)by T(ε) = sup{T∈(0,∞);there exists a unique mild solution uto(1.1)} and S(ε) = sup{S∈(0,∞);there exists a unique mild solution (v,w)to(3.3)}, respectively. 9Proposition 3.5. Under the assumptions (i)–(iv) in the previous section, the re existsT >0depending only onε/ba∇dbl(u0,u1)/ba∇dblH1,m×H0,m(the size of the initial data) such that the Cauchy problem (1.1)admits a unique mild solution u. Also, if (u0,u1)∈H2,m(Rn)×H1,m(Rn)in addition to Assumption (i), then the corresponding mild solution ubecomes a strong solution of (1.1). Moreover, if the lifespan T(ε)is finite, then usatisfies limt→T(ε)/ba∇dbl(u,ut)(t)/ba∇dblH1,m×H0,m=∞. Furthermore, for arbitrary fixed timeT0>0, we can extend the solution to the interval [0,T0)by takingεsufficiently small. From this proposition, we easily have the following. Proposition 3.6. Under the assumptions (i)–(iv) in the previous section, the re existsS >0depending only onε/ba∇dbl(v0,w0)/ba∇dblH1,m×H0,m(the size of the initial data) such that the Cauchy problem (3.3)admits a unique mild solution (v,w). Also, if (u0,u1)∈H2,m(Rn)×H1,m(Rn)in addition to Assumption (i), then the corresponding mild solution (v,w)becomes a strong solution of (3.3). Moreover, if the lifespan S(ε)is finite, then (v,w)satisfies lims→S(ε)/ba∇dbl(v,w)(s)/ba∇dblH1,m×H0,m=∞. Furthermore, for arbitrary fixed timeS0>0, we can extend the solution to the interval [0,S0]by takingεsufficiently small. Proof of Proposition 3.5. By using the assumption (iv), and the Sobolev inequality for n= 1, or the Gagliardo-Nirenberginequality for n≥2 (see Lemma 3.3), we can see that N(U) is a locally Lipschitz mapping on H1,0(Rn)×L2(Rn). Therefore, by [1, Proposition 4.3.3], there exists a unique solution U ∈C([0,T);H1,0(Rn)×L2(Rn)) to the integral equation (3.8). This shows the existence of a uniq ue mild solution uto the Cauchy problem (1.1). If (u0,u1)∈H2,m(Rn)×H1,m(Rn), then we have U0∈D(A), and hence, [1, Proposition 4.3.9] implies that U ∈C([0,T);D(A))∩C1([0,T);H1,0(Rn)×L2(Rn)) andUbecomes the strong solution of the equation (3.7), namely, Usatisfies the equation (3.7) in C([0,T);H1,0(Rn)×L2(Rn)). Then, by the definition of U, we conclude that uhas the regularity in (3.10) and satisfies the equation (1.1) in C([0,T);H0,m(Rn)). Moreover, employing [1, Theorem 4.3.4], we see that if the lifespan T(ε) is finite, thenusatisfies lim t→T(ε)/ba∇dbl(u,ut)(t)/ba∇dblH1,m×H0,m=∞. Next, we prove that for any fixed T0>0, the solution ucan be extended over the interval [0 ,T0] by takingεsufficiently small. To verify this, we reconsider the Cauchy problem (3 .6) and its inhomo- geneous linear version /braceleftbigg Utt+b(t)Ut= ∆xU+˜c(t,x)·∇xU+˜d(t,x)U+˜N(t,x), t>0,x∈Rn, U(0,x) =εU0(x), Ut(0,x) =εU1(x), x ∈Rn.(3.12) For˜N∈L1(0,T0;L2(Rn)), the existence of a unique solution in the distribution sense is prov ed by [11, Theorem 23.2.2]. We also recall the standard energy estimate (s ee [11, Lemma 23.2.1]) sup 0<t<T 0/ba∇dbl(U,Ut)(t)/ba∇dblH1,0×L2≤C(T0)/parenleftBigg ε/ba∇dbl(U0,U1)/ba∇dblH1,0×L2+/integraldisplayT0 0/ba∇dbl˜N(t)/ba∇dblL2dt/parenrightBigg .(3.13) We again construct the solution Uto (3.6) in K:=/braceleftbigg U∈C([0,T0];H1,0(Rn))∩C1([0,T0];L2(Rn)); sup 0<t<T 0/ba∇dbl(U,Ut)(t)/ba∇dblH1,0×L2≤2C(T0)I0ε/bracerightbigg , whereI0:=/ba∇dbl(U0,U1)/ba∇dblH1,0×L2. For each V∈K, we define the mapping by U=M(V), where Uis the solution to (3.12) with ˜N=˜N(V,∇xV,Vt). Then, by using the Sobolev inequality or the Gagliardo-Nirenberg inequality again with the estimate (3.13), we can see that sup 0<t<T 0/ba∇dbl(U,Ut)(t)/ba∇dblH1,0×L2≤C(T0)I0ε+C(T0)(2C(T0)I0ε)pT0. (3.14) Thus, noting p>1 and taking ε>0 sufficiently small, we deduce that MmapsKto itself. Further- more, in the same manner, we easily obtain sup 0<t<T 0/ba∇dbl(U1,U1 t)−(U2,U2 t)/ba∇dblH1,0×L1≤C(T0)(4C(T0)I0ε)p−1T0sup 0<t<T 0/ba∇dbl(V1,V1 t)−(V2,V2 t)/ba∇dblH1,0×L2, 10whereUj=M(Vj) (j= 1,2). Thus, noting p>1 again and taking εsufficiently small, we see that M is acontractionmappingon K. Therefore, by the contractionmappingprinciple, wefind aunique fi xed point˜Uof the mapping Min the setK, and˜Usatisfies the equation (3.6) in the distribution sense. Also, the uniqueness of the solution in the distribution sense to (3.6) in the class C([0,T0];H1,0(Rn))∩ C1([0,T0];L2(Rn)) follows from (3.13). Since the mild solution Uconstructed before also satisfies the equation (3.6) in the distribution sense, we have U(t) =˜U(t) fort∈[0,min{T(ε),T0}). However, noting that the estimate (3.14) implies sup0<t<T 0/ba∇dbl(˜U,˜Ut)(t)/ba∇dblH1,0×L2is finite, we have T0<T(ε) and this completes the proof. 3.4 A priori estimate implies the global existence In what follows, to justify the energy method, we tacitly assume th at (u0,u1)∈H2,m(Rn)×H1,m(Rn), and the solution ( v,w) is in the class (3.11). Therefore, the following calculations make sen se. Once we obtain the desired asymptotic estimate (2.9) for such a data, we can easily have the same estimate for general ( u0,u1)∈H1,m(Rn)×H0,m(Rn) by applying the usual approximation argument. Let(v,w) bethelocal-in-timesolutionto(3.3)ontheinterval[0 ,S). Bythelocalexistencetheorem, it suffices to show an a priori estimate of solutions. The first goal of this section is the following a priori estimate: Proposition 3.7. Under the assumptions (i)–(iv) in the previous section, the re exist constants s0>0, ε1>0andC∗>0such that the following holds: if ε∈(0,ε1]and(v,w)is a mild solution of (1.1)on some interval [0,S]withS >s0, then(v,w)satisfies /ba∇dblv(s)/ba∇dbl2 H1,m+e−s b(t(s))2/ba∇dblw(s)/ba∇dbl2 H0,m≤C∗ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m. (3.15) Before proving the above proposition, we show that Propositions 3 .6 and 3.7 imply the global existence of solutions for small ε. Proof of global existence part of Theorem 2.1. First,wenotethatProposition3.6guaranteesthatthere exitsε2>0 such that the mild solution ( v,w) uniquely exists on the interval [0 ,s0] forε∈(0,ε2], wheres0is the constant described in Proposition 3.7. In particular, we have S(ε)>s0forε∈(0,ε2]. Letε0:= min{ε1,ε2}, whereε1is the constant described in Proposition 3.7. Then, we have S(ε) =∞ forε∈(0,ε0]. Indeed, supposethat S(ε∗)<∞forsomeε∗∈(0,ε0] and let (v,w) be the corresponding mild solution of (3.3). Applying Proposition 3.7, we have the a priori est imate (3.15) with ε=ε∗. On the other hand, Proposition 3.6 also implies lim s→S(ε∗)/ba∇dbl(v,w)(s)/ba∇dblH1,m×H0,m=∞. However, it contradicts the a priori estimate (3.15). Thus, we hav eS(ε) =∞forε∈(0,ε0]. 3.5 Spectral decomposition In the following, we prove the a priori estimate (3.15) in Proposition 3 .7. At first, we decompose v andwinto the leading terms and the remainder terms, respectively. Letα(s) be α(s) =/integraldisplay Rnv(s,y)dy. (3.16) Sincev(s)∈H1,m(Rn) for eachs∈[0,S) andm>n/2,α(s) is well-defined. We also put ϕ0(y) = (4π)−n/2exp/parenleftbigg −|y|2 4/parenrightbigg . 11Then, it is easily verified that /integraldisplay Rnϕ0(y)dy= 1 (3.17) and ∆ϕ0=−y 2·∇yϕ0−n 2ϕ0. (3.18) We also put ψ0(y) = ∆ϕ0(y). We decompose v,was v(s,y) =α(s)ϕ0(y)+f(s,y), w(s,y) =dα ds(s)ϕ0(y)+α(s)ψ0(y)+g(s,y).(3.19) We shall prove that f,gcan be regarded as remainder terms. First, we note the following lemma. Lemma 3.8. We have dα ds(s) =/integraldisplay Rnw(s,y)dy, (3.20) e−s b(t(s))2d2α ds2(s) =e−s b(t(s))2dα ds(s)−dα ds(s)+/integraldisplay Rnr(s,y)dy, (3.21) whereris defined by (3.4). Proof.Notingv∈C1([0,S);H1,m(Rn)),w∈C([0,S);H0,m(Rn)) andm > n/ 2, we immediately obtain (3.20) from dα ds(s) =/integraldisplay Rnvs(s,y)dy=/integraldisplay Rn/parenleftBigy 2·∇yv+n 2v+w/parenrightBig dy=/integraldisplay Rnw(s,y)dy. Next, by the regularity (3.11), we see thatdα ds(s)∈C1([0,S);R). Differentiatingdα ds(s) again and using the second equation of (3.3), we have e−s b(t(s))2d2α ds2(s) =e−s b(t(s))2/integraldisplay Rnws(s,y)dy =e−s b(t(s))2/integraldisplay Rn/parenleftBigy 2·∇yw+/parenleftBign 2+1/parenrightBig w/parenrightBig dy−/integraldisplay Rnwdy+/integraldisplay Rn∆yvdy+/integraldisplay Rnrdy =e−s b(t(s))2/integraldisplay Rnwdy−/integraldisplay Rnwdy+/integraldisplay Rnrdy. Thus, we finish the proof. Next, we consider the remainder term ( f,g). Sincefandgare defined by (3.19), and we assumed that (v,w) has the regularity in (3.11), so is ( f,g): (f,g)∈C([0,S);H2,m(Rn)×H1,m(Rn))∩C1([0,S);H1,m(Rn)×H0,m(Rn)).(3.22) 12Therefore, from the system (3.3) and the equation (3.18), we see thatfandgsatisfy the following system: fs−y 2·∇yf−n 2f=g, s> 0,y∈Rn, e−s b(t(s))2/parenleftBig gs−y 2·∇yg−/parenleftBign 2+1/parenrightBig g/parenrightBig +g= ∆yf+h, s>0,y∈Rn, f(0,y) =v(0,y)−α(0)ϕ0(y), y ∈Rn, g(0,y) =w(0,y)−˙α(0)ϕ0(y)−α(0)ψ0(y), y ∈Rn,(3.23) wherehis given by h(s,y) =e−s b(t(s))2/parenleftbigg −2dα ds(s)ψ0(y)+α(s)/parenleftBigy 2·∇yψ0(y)+/parenleftBign 2+1/parenrightBig ψ0(y)/parenrightBig/parenrightbigg +r(s,y)−/parenleftbigg/integraldisplay Rnr(s,y)dy/parenrightbigg ϕ0(y). (3.24) Moreover, from (3.16), (3.17) and (3.20), it follows that /integraldisplay Rnf(s,y)dy=/integraldisplay Rng(s,y)dy= 0. (3.25) We also notice that the condition (3.25) implies /integraldisplay Rnh(s,y)dy= 0. (3.26) We note that it suffices to show a priori estimates of f,g,αanddα dsfor the proof of global existence of solutions to the system (3.3). Therefore, hereafter, we cons ider the system (3.23) instead of (3.3). 3.6 Energy estimates for n= 1 To obtain the decay estimates for f,g, we introduce F(s,y) =/integraldisplayy −∞f(s,z)dz, G(s,y) =/integraldisplayy −∞g(s,z)dz. (3.27) From the following lemma and the condition (3.25), we see that F,G∈C([0,S);L2(R)). Lemma 3.9 (Hardy-type inequality) .Letf=f(y)belong toH0,1(R)and satisfy/integraltext Rf(y)dy= 0, and letF(y) =/integraltexty −∞f(z)dz. Then it holds that /integraldisplay RF(y)2dy≤4/integraldisplay Ry2f(y)2dy. (3.28) Proof.First, we prove (3.28) when f∈C∞ 0(R). In this case/integraltext Rf(y)dy= 0 leads to F∈C∞ 0(R). Therefore, we apply the integration by parts and have /integraldisplay RF(y)2dy=−2/integraldisplay RyF(y)f(y)dy≤2/integraldisplay Ry2f(y)2dy+1 2/integraldisplay RF(y)2dy. Thus, we obtain (3.28). For general f∈H0,1(R) satisfying/integraltext Rf(y)dy= 0, we can easily prove (3.28) by appropriately approximations. 13Moreover, by the regularity assumption (3.22) on ( f,g), we see that (F,G)∈C([0,S);H3,0(R)×H2,0(R))∩C1([0,S);H2,0(R)×H1,0(R)). (3.29) Sincefandgsatisfy the equation (3.23), we can show that FandGsatisfy the following system: Fs−y 2Fy=G, s> 0,y∈R, e−s b(t(s))2/parenleftBig Gs−y 2Gy−G/parenrightBig +G=Fyy+H, s> 0,y∈R, F(0,y) =/integraldisplayy −∞f(0,z)dz, G(0,y) =/integraldisplayy −∞g(0,z)dz, y∈R,(3.30) where H(s,y) =/integraldisplayy −∞h(s,z)dz. (3.31) We define the following energy. E0(s) =/integraldisplay R/parenleftbigg1 2/parenleftbigg F2 y+e−s b(t(s))2G2/parenrightbigg +1 2F2+e−s b(t(s))2FG/parenrightbigg dy, E1(s) =/integraldisplay R/parenleftbigg1 2/parenleftbigg f2 y+e−s b(t(s))2g2/parenrightbigg +f2+2e−s b(t(s))2fg/parenrightbigg dy, E2(s) =/integraldisplay Ry2/bracketleftbigg1 2/parenleftbigg f2 y+e−s b(t(s))2g2/parenrightbigg +1 2f2+e−s b(t(s))2fg/bracketrightbigg dy. By using Lemma 3.2, the following equivalents are valid for s≥s1with sufficiently large s1>0. E0(s)∼/integraldisplay R/parenleftbigg F2 y+e−s b(t(s))2G2+F2/parenrightbigg dy, E1(s)∼/integraldisplay R/parenleftbigg f2 y+e−s b(t(s))2g2+f2/parenrightbigg dy, (3.32) E2(s)∼/integraldisplay Ry2/bracketleftbigg f2 y+e−s b(t(s))2g2+f2/bracketrightbigg dy. Next, we prove the following energy identity. Lemma 3.10. We have d dsE0(s)+1 2E0(s)+L0(s) =R0(s), (3.33) where L0(s) =/integraldisplay R/parenleftbigg1 2F2 y+G2/parenrightbigg dy, R0(s) =3 2e−s b(t(s))2/integraldisplay RG2dy−1 b(t(s))2db dt(t(s))/integraldisplay R/parenleftbig G2+2FG/parenrightbig dy+/integraldisplay R(F+G)Hdy. Moreover, we have d dsE1(s)+1 2E1(s)+L1(s) =R1(s), (3.34) 14where L1(s) =/integraldisplay R/parenleftbig f2 y+g2/parenrightbig dy−/integraldisplay Rf2dy, R1(s) = 3e−s b(t(s))2/integraldisplay Rg2dy+2e−s b(t(s))2/integraldisplay Rfgdy−1 b(t(s))2db dt(t(s))/integraldisplay R(g2+4fg)dy+/integraldisplay R(2f+g)hdy. Furthermore, we have d dsE2(s)+1 2E2(s)+L2(s) =R2(s), (3.35) where L2(s) =/integraldisplay Ry2/parenleftbigg1 2f2 y+g2/parenrightbigg dy+2/integraldisplay Ryfy(f+g)dy, R2(s) =3 2e−s b(t(s))2/integraldisplay Ry2g2dy−1 b(t(s))2db dt(t(s))/integraldisplay Ry2(2f+g)gdy+/integraldisplay Ry2(f+g)hdy. Proof.The proofs of (3.34) and (3.35) are the almost same as that of (3.33 ), and we only prove (3.33). We calculate the derivatives of each term of E0(s). First, we have d ds/bracketleftbigg1 2/integraldisplay RF2dy/bracketrightbigg =/integraldisplay RFFsdy =/integraldisplay RF/parenleftBigy 2Fy+G/parenrightBig dy =/integraldisplay R/parenleftbigg/parenleftBigy 4F2/parenrightBig y−1 4F2+FG/parenrightbigg dy =−1 4/integraldisplay RF2dy+/integraldisplay RFGdy. Here we have used thaty 2FFy∈L1(R), which enables us to justify the integration by parts. By Lemma 3.1, we also have d ds/bracketleftbigge−s b(t(s))2/integraldisplay RFGdy/bracketrightbigg =−2 b(t(s))2db dt(t(s))/integraldisplay RFGdy−e−s b(t(s))2/integraldisplay RFGdy+e−s b(t(s))2/integraldisplay R(FsG+FGs)dy =−2 b(t(s))2db dt(t(s))/integraldisplay RFGdy−e−s b(t(s))2/integraldisplay RFGdy+e−s b(t(s))2/integraldisplay R/parenleftBigy 2Fy+G/parenrightBig Gdy +e−s b(t(s))2/integraldisplay RF/parenleftBigy 2Gy+G/parenrightBig dy−/integraldisplay RFGdy+/integraldisplay RFFyydy+/integraldisplay RFHdy =−1 2e−s b(t(s))2/integraldisplay RFGdy−2 b(t(s))2db dt(t(s))/integraldisplay RFGdy +e−s b(t(s))2/integraldisplay RG2dy−/integraldisplay RFGdy−/integraldisplay RF2 ydy+/integraldisplay RFHdy. Adding up the above identities, we conclude that d ds/bracketleftbigg/integraldisplay R/parenleftbigg1 2F2+e−s b(t(s))2FG/parenrightbigg dy/bracketrightbigg =−1 4/integraldisplay RF2dy−1 2e−s b(t(s))2/integraldisplay RFGdy−2 b(t(s))2db dt(t(s))/integraldisplay RFGdy +e−s b(t(s))2/integraldisplay RG2dy−/integraldisplay RF2 ydy+/integraldisplay RFHdy. (3.36) 15We also have d ds/bracketleftbigg1 2/integraldisplay RF2 ydy/bracketrightbigg =/integraldisplay RFyFysdy =/integraldisplay RFy/parenleftbiggy 2Fyy+1 2Fy+Gy/parenrightbigg dy =1 4/integraldisplay RF2 ydy+/integraldisplay RFyGydy and d ds/bracketleftbigg1 2e−s b(t(s))2/integraldisplay RG2dy/bracketrightbigg =−1 b(t(s))2db dt(t(s))/integraldisplay RG2dy−1 2e−s b(t(s))2/integraldisplay RG2dy+e−s b(t(s))2/integraldisplay RGGsdy =−1 b(t(s))2db dt(t(s))/integraldisplay RG2dy−1 2e−s b(t(s))2/integraldisplay RG2dy +e−s b(t(s))2/integraldisplay RG/parenleftBigy 2Gy+G/parenrightBig dy−/integraldisplay RG2dy+/integraldisplay RGFyydy+/integraldisplay RGHdy =−1 b(t(s))2db dt(t(s))/integraldisplay RG2dy+1 4e−s b(t(s))2/integraldisplay RG2dy −/integraldisplay RG2dy−/integraldisplay RFyGydy+/integraldisplay RGHdy. Adding up the above two identities, one has d ds/bracketleftbigg1 2/integraldisplay R/parenleftbigg F2 y+e−s b(t(s))2G2/parenrightbigg dy/bracketrightbigg =1 4/integraldisplay RF2 ydy+1 4e−s b(t(s))2/integraldisplay RG2dy−1 b(t(s))2db dt(t(s))/integraldisplay RG2dy−/integraldisplay RG2dy+/integraldisplay RGHdy. (3.37) From (3.36) and (3.37), we conclude that d dsE0(s)+1 2E0(s)+/integraldisplay R/parenleftbigg1 2F2 y+G2/parenrightbigg dy=R0(s). This completes the proof. 3.7 Energy estimates for n≥2 Next, we consider higher dimensional cases n≥2. In this case, we cannot use the primitives (3.27). Therefore, instead of (3.27), we define ˆF(s,ξ) =|ξ|−n/2−δˆf(s,ξ),ˆG(s,ξ) =|ξ|−n/2−δˆg(s,ξ),ˆH(s,ξ) =|ξ|−n/2−δˆh(s,ξ), where 0< δ <1, andˆf(s,ξ) denotes the Fourier transform of f(s,y) with respect to the space variable. First, to ensure that ˆF,ˆGandˆHmake sense as L2-functions, instead of Lemma 3.9, we prove the following lemma. Lemma3.11. Letm>n/2+1andf(y)∈H0,m(Rn)be afunction satisfying ˆf(0) = (2π)−n/2/integraltext Rnf(y)dy= 0. LetˆF(ξ) =|ξ|−n/2−δˆf(ξ)with some 0<δ <1. Then, there exists a constant C(n,m,δ)>0such that /ba∇dblF/ba∇dblL2≤C(n,m,δ)/ba∇dblf/ba∇dblH0,m (3.38) holds. 16Proof.By the Plancherel theorem, it suffices to show that /ba∇dblˆF/ba∇dblL2≤C/ba∇dblf/ba∇dblH0,m. Using the definition of ˆFand the condition ˆf(0) = 0 , we compute /integraldisplay Rn|ˆF(ξ)|2dξ=/integraldisplay Rn|ξ|−n−2δ|ˆf(ξ)|2dξ =/integraldisplay |ξ|≤1|ξ|−n−2δ|ˆf(ξ)|2dξ+/integraldisplay |ξ|>1|ξ|−n−2δ|ˆf(ξ)|2dξ =/integraldisplay |ξ|≤1|ξ|−n−2δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1 0d dθˆf(θξ)dθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dξ+/integraldisplay |ξ|>1|ξ|−n−2δ|ˆf(ξ)|2dξ ≤ /ba∇dbl∇ξˆf/ba∇dbl2 L∞/integraldisplay |ξ|≤1|ξ|2−n−2δdξ+/ba∇dblˆf/ba∇dbl2 L2 ≤C(n,δ)/parenleftBig /ba∇dbl∇ξˆf/ba∇dbl2 L∞+/ba∇dblˆf/ba∇dbl2 L2/parenrightBig . Sincem>n/2+1, we have /ba∇dbl∇ξˆf/ba∇dblL∞=/ba∇dbl/hatwideryf/ba∇dblL∞≤C/ba∇dblyf/ba∇dblL1≤C(n,m)/ba∇dbl(1+|y|)mf/ba∇dblL2≤C(n,m)/ba∇dblf/ba∇dblH0,m. Consequently, we obtain /ba∇dblˆF/ba∇dblL2≤C(n,δ)/parenleftBig /ba∇dbl∇ξˆf/ba∇dblL∞+/ba∇dblˆf/ba∇dblL2/parenrightBig ≤C(n,m,δ)/ba∇dblf/ba∇dblH0,m, which completes the proof. We also notice that, for any small η>0, the inequality /integraldisplay Rn|ˆf|2dξ=/integraldisplay |ξ|≥√η−1|ˆf|2dξ+/integraldisplay |ξ|<√η−1|ˆf|2dξ ≤η/integraldisplay |ξ|≥√η−1|ξ|2|ˆf|2dξ+η(2−n−2δ)/2/integraldisplay |ξ|<√η−1|ξ|2−n−2δ|ˆf|2dξ ≤η/integraldisplay Rn|ξ|2|ˆf|2dξ+η(2−n−2δ)/2/integraldisplay Rn|ξ|2|ˆF|2dξ (3.39) holds. This is provedby noting that 2 −n−2δ<0 (here we assumed that n≥2). The above inequality enables us to control /ba∇dblˆf/ba∇dblL2by/ba∇dbl|ξ|ˆf/ba∇dblL2and/ba∇dbl|ξ|ˆF/ba∇dblL2. Moreover, the coefficient in front of /ba∇dbl|ξ|ˆf/ba∇dblL2 can be taken arbitrarily small. By applying the Fourier transform to (3.23), we obtain ˆfs+1 2∇ξ·/parenleftBig ξˆf/parenrightBig −n 2ˆf= ˆg, s> 0,ξ∈Rn, e−s b(t(s))2/parenleftbigg ˆgs+1 2∇ξ·(ξˆg)−/parenleftBign 2+1/parenrightBig ˆg/parenrightbigg + ˆg=−|ξ|2ˆf+ˆh, s>0,ξ∈Rn.(3.40) By noting that 1 2∇ξ·/parenleftBig ξˆf/parenrightBig =ξ 2·∇ξˆf+n 2ˆf, we rewrite (3.40) as ˆfs+ξ 2·∇ξˆf= ˆg, s> 0,ξ∈Rn, e−s b(t(s))2/parenleftbigg ˆgs+ξ 2·∇ξˆg−ˆg/parenrightbigg + ˆg=−|ξ|2ˆf+ˆh, s>0,ξ∈Rn. 17Making use of this, we calculate ˆFs=|ξ|−n/2−δˆfs =|ξ|−n/2−δ/parenleftbigg −ξ 2·∇ξˆf+ ˆg/parenrightbigg =|ξ|−n/2−δ/parenleftbigg −ξ 2·∇ξ/parenleftBig |ξ|n/2+δˆF/parenrightBig +|ξ|n/2+δˆG/parenrightbigg =−ξ 2·∇ξˆF−1 2/parenleftBign 2+δ/parenrightBig ˆF+ˆG and e−s b(t(s))2ˆGs=e−s b(t(s))2|ξ|−n/2−δˆgs =|ξ|−n/2−δ/bracketleftbigge−s b(t(s))2/parenleftbigg −ξ 2·∇ξˆg+ ˆg/parenrightbigg −ˆg−|ξ|2ˆf+ˆh/bracketrightbigg =|ξ|−n/2−δ/bracketleftbigge−s b(t(s))2/parenleftbigg −ξ 2·∇ξ/parenleftBig |ξ|n/2+δˆG/parenrightBig +|ξ|n/2+δˆG/parenrightbigg −|ξ|n/2+δˆG−|ξ|2+n/2+δˆF+|ξ|n/2+δˆH/bracketrightBig =e−s b(t(s))2/parenleftbigg −ξ 2·∇ξˆG−1 2/parenleftBign 2+δ−2/parenrightBig ˆG/parenrightbigg −ˆG−|ξ|2ˆF+ˆH. Hence,ˆFandˆGsatisfy the following system. ˆFs+ξ 2·∇ξˆF+1 2/parenleftBign 2+δ/parenrightBig ˆF=ˆG, s> 0,ξ∈Rn, e−s b(t(s))2/parenleftbigg ˆGs+ξ 2·∇ξˆG+1 2/parenleftBign 2+δ−2/parenrightBig ˆG/parenrightbigg +ˆG=−|ξ|2ˆF+ˆH, s> 0,ξ∈Rn. We consider the following energy. E0(s) = Re/integraldisplay Rn/parenleftbigg1 2/parenleftbigg |ξ|2|ˆF|2+e−s b(t(s))2|ˆG|2/parenrightbigg +1 2|ˆF|2+e−s b(t(s))2ˆF¯ˆG/parenrightbigg dξ, E1(s) =/integraldisplay Rn/parenleftbigg1 2/parenleftbigg |∇yf|2+e−s b(t(s))2g2/parenrightbigg +/parenleftBign 4+1/parenrightBig/parenleftbigg1 2f2+e−s b(t(s))2fg/parenrightbigg/parenrightbigg dy, E2(s) =/integraldisplay Rn|y|2m/bracketleftbigg1 2/parenleftbigg |∇yf|2+e−s b(t(s))2g2/parenrightbigg +1 2f2+e−s b(t(s))2fg/bracketrightbigg dy. By using Lemma 3.2 again, the following equivalents are valid for s≥s1with sufficiently large s1. E0(s)∼/integraldisplay Rn/parenleftbigg |ξ|2|ˆF|2+e−s b(t(s))2|ˆG|2+|ˆF|2/parenrightbigg dξ, E1(s)∼/integraldisplay Rn/parenleftbigg |∇yf|2+e−s b(t(s))2g2+f2/parenrightbigg dy, (3.41) E2(s)∼/integraldisplay Rn|y|2m/bracketleftbigg |∇yf|2+e−s b(t(s))2g2+f2/bracketrightbigg dy. Then, in a similar way to the case n= 1, we obtain the following energy identities. Lemma 3.12. We have d dsE0(s)+δE0(s)+L0(s) =R0(s), (3.42) 18where L0(s) =1 2/integraldisplay Rn|ξ|2|ˆF|2dξ+/integraldisplay Rn|ˆG|2dξ, R0(s) =3 2e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ−1 b(t(s))2db dt(t(s))Re/integraldisplay Rn/parenleftBig 2ˆF+ˆG/parenrightBig¯ˆGdξ+Re/integraldisplay Rn/parenleftBig ˆF+ˆG/parenrightBig¯ˆHdξ. Moreover, we have d dsE1(s)+δE1(s)+L1(s) =R1(s), (3.43) where L1(s) =1 2(1−δ)/integraldisplay Rn|∇yf|2dy+/integraldisplay Rng2dy−/parenleftbiggn 4+δ 2/parenrightbigg/parenleftBign 4+1/parenrightBig/integraldisplay Rnf2dy, R1(s) =/parenleftBign 2+δ/parenrightBig/parenleftBign 4+1/parenrightBige−s b(t(s))2/integraldisplay Rnfgdy+1 2(n+3+δ)e−s b(t(s))2/integraldisplay Rng2dy −1 b(t(s))2db dt(t(s))/integraldisplay Rn/parenleftBig 2/parenleftBign 4+1/parenrightBig f+g/parenrightBig gdy+/integraldisplay Rn/parenleftBig/parenleftBign 4+1/parenrightBig f+g/parenrightBig hdy. Furthermore, we have d dsE2(s)+(˜δ−η)E2(s)+L2(s) =R2(s), (3.44) where˜δ=m−n/2,η∈(0,˜δ)is an arbitrary number, L2(s) =η 2/integraldisplay Rn|y|2mf2dy+1 2(η+1)/integraldisplay Rn|y|2m|∇yf|2dy+/integraldisplay Rn|y|2mg2dy +2m/integraldisplay Rn|y|2m−2(y·∇yf)(f+g)dy, R2(s) =−ηe−s b(t(s))2/integraldisplay Rn|y|2mfgdy−1 2(η−3)e−s b(t(s))2/integraldisplay Rn|y|2mg2dy −1 b(t(s))2db dt(t(s))/integraldisplay Rn|y|2m(2f+g)gdy+/integraldisplay Rn|y|2m(f+g)hdy. Proof.The proofs of (3.43) and (3.44) are the almost same as that of (3.42 ), and we only prove (3.42). First, we calculate d ds/bracketleftbigg1 2/integraldisplay Rn|ˆF|2dξ/bracketrightbigg = Re/integraldisplay Rn/parenleftbigg −ξ 2·∇ξˆF−1 2/parenleftBign 2+δ/parenrightBig ˆF+ˆG/parenrightbigg ¯ˆFdξ =−δ 2/integraldisplay Rn|ˆF|2dξ+Re/integraldisplay Rn¯ˆFˆGdξ and d ds/bracketleftbigge−s b(t(s))2Re/integraldisplay RnˆF¯ˆGdξ/bracketrightbigg =−2 b(t(s))2db dt(t(s))Re/integraldisplay RnˆF¯ˆGdξ−e−s b(t(s))2Re/integraldisplay RnˆF¯ˆGdξ +e−s b(t(s))2Re/integraldisplay Rn/parenleftBig ˆFs¯ˆG+¯ˆFˆGs/parenrightBig dξ =−2 b(t(s))2db dt(t(s))Re/integraldisplay RnˆF¯ˆGdξ−δe−s b(t(s))2Re/integraldisplay RnˆF¯ˆGdξ +e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ−Re/integraldisplay RnˆF¯ˆGdξ −/integraldisplay Rn|ξ|2|ˆF|2dξ+Re/integraldisplay RnˆF¯ˆHdξ. 19Adding up these identities, we see that d ds/bracketleftbigg Re/integraldisplay Rn/parenleftbigg |ˆF|2+e−s b(t(s))2ˆF¯ˆG/parenrightbigg dξ/bracketrightbigg =−δ 2/integraldisplay Rn|ˆF|2dξ−2 b(t(s))2db dt(t(s))Re/integraldisplay RnˆF¯ˆGdξ−δe−s b(t(s))2Re/integraldisplay RnˆF¯ˆGdξ +e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ−/integraldisplay Rn|ξ|2|ˆF|2dξ+Re/integraldisplay RnˆF¯ˆHdξ (3.45) We also have d ds/bracketleftbigg1 2/integraldisplay Rn|ξ|2|ˆF|2dξ/bracketrightbigg = Re/integraldisplay Rn|ξ|2/parenleftbigg −ξ 2·∇ξˆF−1 2/parenleftBign 2+δ/parenrightBig ˆF+ˆG/parenrightbigg ¯ˆFdξ =1 2(1−δ)/integraldisplay Rn|ξ|2|ˆF|2dξ+Re/integraldisplay Rn|ξ|2¯ˆFˆGdξ and d ds/bracketleftbigg1 2e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ/bracketrightbigg =−1 b(t(s))2db dt(t(s))/integraldisplay Rn|ˆG|2dξ−1 2e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ +e−s b(t(s))2Re/integraldisplay RnˆGs¯ˆGdξ =−1 b(t(s))2db dt(t(s))/integraldisplay Rn|ˆG|2dξ+1 2(1−δ)e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ −/integraldisplay Rn|ˆG|2dξ−Re/integraldisplay Rn|ξ|2ˆF¯ˆGdξ+Re/integraldisplay RnˆG¯ˆHdξ. Summing up the above identities, we have d ds/bracketleftbigg1 2e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ/bracketrightbigg =1 2(1−δ)/integraldisplay Rn|ξ|2|ˆF|2dξ−1 b(t(s))2db dt(t(s))/integraldisplay Rn|ˆG|2dξ +1 2(1−δ)e−s b(t(s))2/integraldisplay Rn|ˆG|2dξ−/integraldisplay Rn|ˆG|2dξ+Re/integraldisplay RnˆG¯ˆHdξ. (3.46) From (3.45) and (3.46), we conclude (3.42). 3.8 Proof of Proposition 3.7 In either case when n= 1 orn≥2, we have proved energy identities of Ej(s) with remainder terms Rj(j= 0,1,2). Hereafter, we unify the both cases and complete the proof of Proposition 3.7. We define E3(s) =1 2e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 +e−2λsα(s)2 and E4(s) =C0E0(s)+C1E1(s)+E2(s)+E3(s), whereλ >0 is determined later, and C0,C1are positive constants such that 1 ≪C1≪C0. By recalling the equivalences (3.32) and (3.41), the following equivalence is valid for s≥s1: E4(s)∼ /ba∇dblf(s)/ba∇dbl2 H1,m+e−s b(t(s))2/ba∇dblg(s)/ba∇dbl2 H0,m+e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 +e−2λsα(s)2.(3.47) To obtain the energy estimate of E4(s), we first notice the following lemma. 20Lemma 3.13. We have d dsE3(s)+2λE3(s)+/parenleftbiggdα ds(s)/parenrightbigg2 =R3(s), where R3(s) =1 2(2λ+1)e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 −1 b(t(s))2db dt(t(s))/parenleftbiggdα ds(s)/parenrightbigg2 +dα ds(s)/parenleftbigg/integraldisplay Rnr(s,y)dy/parenrightbigg +2e−2λsα(s)dα ds(s). (3.48) Then, we can also see the following energy estimate. Lemma 3.14. We have d dsE4(s)+2λE4(s)+L4(s) =R4(s), (3.49) where L4(s) =/parenleftbigg1 2−2λ/parenrightbigg (C0E0(s)+C1E1(s)+E2(s))+C0L0(s)+C1L1(s)+L2(s)+/parenleftbiggdα ds(s)/parenrightbigg2 , forn= 1, L4(s) =C0(δ−2λ)E0(s)+C1(δ−2λ)E1(s)+(˜δ−η−2λ)E2(s) +C0L0(s)+C1L1(s)+L2(s)+/parenleftbiggdα ds(s)/parenrightbigg2 forn≥2, and R4(s) =C0R0(s)+C1R1(s)+R2(s)+R3(s). HereR0,R1,R2andL0,L1,L2are defined in Lemmas 3.10 (n= 1)and 3.12 (n≥2), andR3is defined by (3.48). Then, by the Schwarz inequality and the inequality (3.39), we obtain t he following lower estimate ofL4. Here we recall that δ∈(0,1) is an arbitrary number, ˜δ=m−n/2 andη >0 is an arbitrary small number. Lemma 3.15. If0<λ≤1/4 (n= 1),0<λ<min{1 2,m 2−n 4}(n≥2), then L4(s)≥C/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg holds fors≥s1. Proof.Letλsatisfy 0<λ≤1/4 (n= 1) and 0 <λ<min{1 2,m 2−n 4}(n≥2). We take the parameter δso that 2λ<δ < 1. Then, recalling that ˜δ=m−n/2, we have 2 λ<min{δ,˜δ}. We also note that the equivalences (3.32) and (3.41) of E0(s),E1(s),E2(s) yield the positivity of the first three terms of L4(s) fors≥s1. Therefore, it suffices to consider the terms L0(s),L1(s) andL2(s). Whenn= 1, notingFy=fand applying the Schwarz inequality, we easily have /integraldisplay Rf2dy=/integraldisplay RF2 ydy 21and 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Ryfy(f+g)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1 4/integraldisplay Ry2f2 ydy+8/integraldisplay R(f2+g2)dy. Hence, taking C1>8 andC0>2C1, we obtain the desired estimate. Next, when n≥2, we note that, for any small µ>0, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Rn|y|2m−2(y·∇yf)(f+g)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤µ/integraldisplay Rn|y|2m|∇yf|2dy+8µ−1/integraldisplay Rn|y|2m−2(f2+g2)dy and µ−1/integraldisplay Rn|y|2m−2(f2+g2)dy=µ−1/integraldisplay |y|>µ−1|y|2m−2(f2+g2)dy+µ−1/integraldisplay |y|≤µ−1|y|2m−2(f2+g2)dy ≤µ/integraldisplay |y|>µ−1|y|2m(f2+g2)dy+µ−2m+1/integraldisplay |y|≤µ−1(f2+g2)dy ≤µ/integraldisplay Rn|y|2m(f2+g2)dy+µ−2m+1/integraldisplay Rn(f2+g2)dy. We takeµsufficiently small so that µ≪ηand thenC0,C1sufficiently large so that µ−2m+1≪C1≪ C0. Then, applying (3.39) to estimate the last term, we have the desire d estimate. Finally, we put E5(s) =E4(s)+1 2α(s)2+e−s b(t(s))2α(s)dα ds(s). Then, we easily obtain Lemma 3.16. There exists s2≥s1such that we have E5(s)∼ /ba∇dblf(s)/ba∇dbl2 H1,m+e−s b(t(s))2/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 , E5(s)+/ba∇dblg(s)/ba∇dbl2 H0,m+/parenleftbiggdα ds(s)/parenrightbigg2 ∼ /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2 fors≥s2. Proof.By the Schwarz inequality, we have /vextendsingle/vextendsingle/vextendsingle/vextendsinglee−s b(t(s))2α(s)dα ds(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(˜η)e−s b(t(s))2α(s)2+ ˜ηe−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 , where ˜η >0 is a small number determined later. By the equivalence (3.47) of E4(s) and taking ˜ η sufficiently small, we control the second term of the right-hand side and have ˜ηe−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 ≤1 2E4(s) fors≥s1. On the other hand, by Lemma 3.2 and taking s2≥s1sufficiently large, we estimate the first term as C(˜η)e−s b(t(s))2α(s)2≤1 4α(s)2 22fors≥s2. Combining them, we conclude that E5(s)≥1 2E4(s)+1 4α(s)2 holds fors≥s2. Then, using the lower bound (3.47) again, we have the lower bound /ba∇dblf(s)/ba∇dbl2 H1,m+e−s b(t(s))2/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 ≤CE5(s) fors≥s2. The upper bound of E5(s) immediately follows from the equivalence (3.47) of E4(s) and we have the first assertion. The second assertion is also directly pr oved from the first one. By using (3.21), we also have d ds/bracketleftbigg1 2α(s)2+e−s b(t(s))2α(s)dα ds(s)/bracketrightbigg =e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 −2 b(t(s))2α(s)db dt(t(s))dα ds(s) +α(s)/parenleftbigg/integraldisplay Rnr(s,y)dy/parenrightbigg =:˜R5(s). LettingR5(s) =R4(s)+˜R5(s), we obtain d dsE5(s)+2λE4(s)+L4(s) =R5(s). (3.50) We give an estimate for the remainder term R5(s): Lemma 3.17 (Estimate for the remainder terms) .Letλ0,λ1be λ0= min/braceleftbigg1−β 1+β,γ 1+β−1 2,ν 1+β−1/bracerightbigg (3.51) (where we interpret 1/(1+β)as an arbitrary large number when β=−1) and λ1= 1 2min i=1,...,k/braceleftbigg pi1+2pi2+/parenleftbigg 3−2β 1+β/parenrightbigg pi3−3/bracerightbigg , n= 1, n 2/parenleftbigg p−1−2 n/parenrightbigg , n ≥2(3.52) (where we interpret −2βpi3/(1+β)as an arbitrary large number when pi3/ne}ationslash= 0andβ=−1). Then, there exists s0≥s2such that we have the following estimates: (i)Whenn= 1,R4(s)andR5(s)satisfy |R4(s)| ≤˜ηL4(s)+C(˜η)e−2λ0sE5(s)+C(˜η)e−2λ1sk/summationdisplay i=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3), |R5(s)| ≤˜ηL4(s)+C(˜η)e−λ0sE5(s)+C(˜η)e−2λ1sk/summationdisplay i=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3) +C(˜η)e−λ1sk/summationdisplay i=1E5(s)(pi1+pi2+pi3+1)/2 fors≥s0, where˜η>0is an arbitrary small number. 23(ii)Whenn≥2,R4(s)andR5(s)satisfy |R4(s)| ≤˜ηL4(s)+C(˜η)e−2λ0sE5(s)+C(˜η)e−2λ1sE5(s)p, |R5(s)| ≤˜ηL4(s)+C(˜η)e−λ0sE5(s)+C(˜η)e−2λ1sE5(s)p+C(˜η)e−λ1sE5(s)(p+1)/2 fors≥s0, where˜η>0is an arbitrary small number. We postpone the proof of this lemma until the next section, and now we completes the proofs of Proposition 3.7 and Theorem 2.1. We first consider the case n≥2. Taking ˜η= 1/2 in Lemma 3.17 and using (3.50) and Lemmas 3.15, 3.16, we have d dsE5(s)≤Ce−λ0sE5(s)+Ce−2λ1sE5(s)p+Ce−λ1sE5(s)(p+1)/2(3.53) fors≥s0. Let Λ(s) := exp/parenleftbigg −C/integraldisplays s0e−λ0τdτ/parenrightbigg . We note that e−Ce−λ0s0/λ0≤Λ(s)≤1 fors≥s0and Λ(s0) = 1. Multiplying (3.53) by Λ( s) and integrating it over [ s0,s], we see that Λ(s)E5(s)≤E5(s0)+C/integraldisplays s0/bracketleftBig Λ(τ)e−2λ1τE5(τ)p+Λ(τ)e−λ1τE5(τ)(p+1)/2/bracketrightBig dτ holds fors≥s0. Putting M(s) := sup s0≤τ≤sE5(τ), we further obtain M(s)≤CM(s0)+C(s0,λ0,λ1)/parenleftBig M(s)p+M(s)(p+1)/2/parenrightBig (3.54) fors≥s0. On the other hand, we easily estimate M(s0) as M(s0)≤C(s0)/parenleftbig /ba∇dbl(f(s0),g(s0))/ba∇dbl2 H1,m×H0,m+α(s0)2+ ˙α(s0)2/parenrightbig ≤C(s0)/ba∇dbl(v(s0),w(s0))/ba∇dbl2 H1,m×H0,m ≤C(s0)ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m (3.55) by using the local existence result (see the proof of Proposition 3.5 ). Combining (3.54) with (3.55), we have M(s)≤C2ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m+C2/parenleftBig M(s)p+M(s)(p+1)/2/parenrightBig fors≥s0with some constant C2>0. Letε1be ε1:=/parenleftBig/radicalbig C22(p+1)/4/parenrightBig−1 . Then, a direct calculation implies 2C2ε2I0>C2ε2I0+C2/bracketleftBig (2C2ε2I0)p+(2C2ε2I0)(p+1)/2/bracketrightBig holdsforε∈(0,ε1], whereI0=/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m. Combiningthiswith M(s0)≤C2ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m and the continuity of M(s) with respect to s, we conclude that M(s)≤2C2ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m (3.56) 24holds fors≥s0andε∈(0,ε1]. Therefore, from Lemma 3.16, we obtain /ba∇dblf(s)/ba∇dbl2 H1,m+e−s b(t(s))2/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 ≤Cε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m fors≥s0andε∈(0,ε1]. This implies /ba∇dblv(s)/ba∇dbl2 H1,m+e−s b(t(s))2/ba∇dblw(s)/ba∇dbl2 H0,m≤C∗ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m (3.57) with some constant C∗>0. This completes the proof of Proposition 3.7. Whenn= 1, to control the additional term E5(s)pi1L4(s) appearing in the estimate of R5(s), we use (3.50) as d dsE5(s)+L4(s)≤Ce−λ0sE5(s)+C(˜η)e−2λ1sk/summationdisplay i=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3) +C(˜η)e−λ1sk/summationdisplay i=1E5(s)(pi1+pi2+pi3+1)/2 instead of (3.53). In the same way as before, we multiply the both sid es by Λ(s) and integrate it over [s0,s] to obtain Λ(s)E5(s)+/integraldisplays s0Λ(τ)L4(τ)dτ ≤E5(s0)+C/summationdisplay i=1,...,k pi3=1/integraldisplays s0Λ(τ)e−2λ1τE5(τ)pi1+pi2L4(τ)pi3dτ +Ck/summationdisplay i=1/integraldisplays s0/bracketleftBig Λ(τ)e−2λ1τE5(τ)pi1+pi2+pi3+Λ(τ)e−λ1τE5(τ)(pi1+pi2+pi3+1)/2/bracketrightBig dτ. As before, putting M(s) := sups0≤τ≤sE5(τ) and noting that Λ( s) is bounded by both above and below, we see that M(s)+/integraldisplays s0L4(τ)dτ≤C2ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m+C2/summationdisplay i=1,...,k pi3=1M(s)pi1+pi2/integraldisplays s0L4(τ)pi3dτ +C2k/summationdisplay i=1/parenleftBig M(s)pi1+pi2+pi3+M(s)(pi1+pi2+pi3+1)/2/parenrightBig fors≥s0with some constant C2>0. Takingε1sufficiently small so that 2C2ε2I0+/integraldisplays s0L4(τ)dτ >C 2ε2I0+C2/summationdisplay i=1,...,k pi3=1(2C2ε2I0)pi1+pi2/integraldisplays s0L4(τ)pi3dτ +C2k/summationdisplay i=1/parenleftBig (2C2ε2I0)pi1+pi2+pi3+(2C2ε2I0)(pi1+pi2+pi3+1)/2/parenrightBig holds forε∈(0,ε1], whereI0=/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m. Combining this with M(s0)≤C2ε2I0and the continuity of M(s) with respect to s, we conclude that M(s)≤2C2ε2/ba∇dbl(v0,w0)/ba∇dbl2 H1,m×H0,m, which leads to (3.57) and completes the proof of Proposition 3.7 for n= 1. 253.9 Proof of Theorem 2.1: asymptotic behavior Next, we prove the asymptotic behavior (2.9). For simplicity, we only consider the case n≥2, since the proof of the one-dimensional case is similar. Putting λ= min/braceleftbigg1 2,m 2−n 4,λ0,λ1/bracerightbigg −η, (3.58) whereη >0 is an arbitrary small number, and λ0,λ1are defined by (3.51), (3.52), and turning back to (3.49) and using Lemma 3.17 with ˜ η=1 2toR4(s), we have d dsE4(s)+2λE4(s)+1 2L4(s)≤Ce−2λ0sE5(s)+Ce−2λ1sE5(s)p ≤Ce−2λ2sε2/ba∇dbl(u0,u1)/ba∇dbl2 H1,m×H0,m, whereλ2= min{λ0,λ1}. Multiplying the above inequality by e2λs, we obtain d ds/bracketleftbig e2λsE4(s)/bracketrightbig +e2λs 2L4(s)≤Ce−2ηsε2/ba∇dbl(u0,u1)/ba∇dbl2 H1,m×H0,m. Integrating it over [ s0,s] and using Lemma 3.15, we have E4(s)+/integraldisplays s0e−2λ(s−τ)/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+/parenleftbiggdα dτ(τ)/parenrightbigg2/parenrightBigg dτ≤Ce−2λsε2/ba∇dbl(u0,u1)/ba∇dbl2 H1,m×H0,m. In particular, for s0≤˜s≤s, one has |α(s)−α(˜s)|2=/parenleftBigg/integraldisplays ˜s/parenleftbiggdα dτ(τ)/parenrightbigg2 dτ/parenrightBigg2 ≤/parenleftbigg/integraldisplays ˜se−2λτdτ/parenrightbigg/parenleftBigg/integraldisplays ˜se2λτ/parenleftbiggdα dτ(τ)/parenrightbigg2 dτ/parenrightBigg ≤Ce−2λ˜sε2/ba∇dbl(u0,u1)/ba∇dbl2 H1,m×H0,m, and hence, the limit α∗= lims→+∞α(s) exists and it follows that |α(s)−α∗|2≤Ce−2λsε2/ba∇dbl(u0,u1)/ba∇dbl2 H1,m×H0,m. Finally, we have /ba∇dblv(s)−α∗ϕ0/ba∇dbl2 H1,m≤ /ba∇dblf(s)/ba∇dbl2 H1,m+|α(s)−α∗|2/ba∇dblϕ0/ba∇dbl2 H1,m≤Ce−2λsε2/ba∇dbl(u0,u1)/ba∇dbl2 H1,m×H0,m. Recalling the relation (3.2) and ( B(t) + 1)−n/2ϕ0((B(t) +1)−1/2x) =G(B(t) + 1,x), where Gis the Gaussian defined by (2.8), we obtain /ba∇dblu(t,·)−α∗G(B(t)+1,·)/ba∇dbl2 L2≤Cε2(B(t)+1)−n/2−2λ/ba∇dbl(u0,u1)/ba∇dbl2 H1,m×H0,m, which completes the proof of Theorem 2.1. 4 Estimates of the remainder terms In this section, we give a proof to Lemma 3.17. 26Lemma 4.1. Under the assumptions (2.1),(2.3)and(2.5), we have /vextenddouble/vextenddouble/vextenddoublee3s/2N/parenleftBig e−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig/vextenddouble/vextenddouble/vextenddouble2 H0,1 ≤Ce−2λ1sk/summationdisplay i=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg /ba∇dblg(s)/ba∇dblH0,1+α(s)+dα ds(s)/parenrightbigg2pi3 (4.1) forn= 1,s≥0and /vextenddouble/vextenddouble/vextenddoublee(n 2+1)sN/parenleftbig e−n 2sv/parenrightbig/vextenddouble/vextenddouble/vextenddouble2 H0,m≤Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p(4.2) forn≥2,s≥0, whereλ1is given by (3.52). Proof.Whenn= 1,β∈(−1,1), by the assumption (2.3) and Lemma 3.2, we compute (1+y2)e3sNi/parenleftBig e−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig2 ≤C(1+y2)e−2λ1s|v|2pi1|vy|2pi2|w|2pi3, whereλ1is defined by (3.52). By the Sobolev inequality /ba∇dblv(s)/ba∇dblL∞≤C/ba∇dblv(s)/ba∇dblH1,0, we calculate (1+y2)e−2λ1s|v|2pi1|vy|2pi2|w|2pi3 ≤Ce−2λ1s|v2|pi1+pi2+pi3−1((1+y2)v2)1−pi2−pi3((1+y2)v2 y)pi2((1+y2)w2)pi3 ≤Ce−2λ1s/ba∇dblv(s)/ba∇dbl2(pi1+pi2+pi3−1) H1,0 ((1+y2)v2)1−pi2−pi3((1+y2)v2 y)pi2((1+y2)w2)pi3. Therefore, by the H¨ older inequality, we conclude /vextenddouble/vextenddouble/vextenddoublee3s/2Ni/parenleftBig e−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig/vextenddouble/vextenddouble/vextenddouble2 H0,1 ≤Ce−2λ1s/ba∇dblv(s)/ba∇dbl2(pi1+pi2−1) H1,0/ba∇dblv(s)/ba∇dbl2(1−pi2) H1,1/ba∇dblv(s)/ba∇dbl2pi2 H1,1/ba∇dblw(s)/ba∇dbl2pi3 H0,1 ≤Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg /ba∇dblg(s)/ba∇dblH0,1+α(s)+dα ds(s)/parenrightbigg2pi3 Whenn= 1,pi3/ne}ationslash= 0,β=−1, we obtain (1+y2)e3sNi/parenleftBig e−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig2 ≤C(1+y2)e(3−pi1−2pi2−3pi3)sb(t(s))−pi3|v|2pi1|vy|2pi2|w|2pi3 ≤C(1+y2)e−λi1s|v|2pi1|vy|2pi2|w|2pi3, where we can take λi1as an arbitrary large number, since Lemma 3.2 shows b(t(s))−pi3∼exp(−pi3es). Therefore, by the same way, we obtain the desired estimate. Next, we consider the case n≥2. By the assumption (2.5) and Lemma 3.3, we have /vextenddouble/vextenddouble/vextenddoublee(n 2+1)sN/parenleftbig e−n 2sv/parenrightbig/vextenddouble/vextenddouble/vextenddouble2 H0,m≤C/integraldisplay Rne2(n 2+1)s/an}b∇acketle{ty/an}b∇acket∇i}ht2m/vextendsingle/vextendsinglee−n 2sv(s,y)/vextendsingle/vextendsingle2pdy ≤Ce−2λ1s/integraldisplay Rn/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{ty/an}b∇acket∇i}htm/pv(s,y)/vextendsingle/vextendsingle/vextendsingle2p dy ≤Ce−2λ1s/vextenddouble/vextenddouble/vextenddouble∇/parenleftBig /an}b∇acketle{ty/an}b∇acket∇i}htm/pv(s)/parenrightBig/vextenddouble/vextenddouble/vextenddouble2pσ L2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{ty/an}b∇acket∇i}htm/pv(s)/vextenddouble/vextenddouble/vextenddouble2p(1−σ) L2 ≤Ce−2λ1s/ba∇dblv(s)/ba∇dbl2p H1,m ≤Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p, which completes the proof. 27From Lemmas 3.2, 4.1 and the assumption (2.2), we immediately obtain t he following estimate: Lemma 4.2. Letrbe defined by (3.4). Under the assumptions (2.1)–(2.5), we have /ba∇dblr(s)/ba∇dbl2 H0,m≤Ce−2λ0s/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg +Ce−2λ1sk/summationdisplay i=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg /ba∇dblg(s)/ba∇dblH0,1+α(s)+dα ds(s)/parenrightbigg2pi3 forn= 1,s≥0and /ba∇dblr(s)/ba∇dbl2 H0,m≤Ce−2λ0s/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg +Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p forn≥2,s≥0, whereλ0,λ1are defined by (3.51),(3.52), respectively. Proof.By Lemma 4.1, it suffices to estimate 1 b(t(s))2db dt(t(s))w+es/2c(t(s))·∇yv+esd(t(s))v. Applying Lemma 3.2, we have /vextenddouble/vextenddouble/vextenddouble/vextenddouble1 b(t(s))2db dt(t(s))w(s)/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 H0,m≤C/parenleftBigg /ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg ×/braceleftBigg e−2(1−β)s/(1+β)β∈(−1,1), exp(−4es)β=−1. Also, the assumption (2.2) implies /vextenddouble/vextenddouble/vextenddoublees/2c(t(s))·∇yv(s)/vextenddouble/vextenddouble/vextenddouble2 H0,m≤C/parenleftbig /ba∇dblf(s)/ba∇dbl2 H1,1+α(s)2/parenrightbig ×/braceleftBigg e−((2γ)/(1+β)−1)sβ∈(−1,1), exp(−2γes+s)β=−1 and /ba∇dblesd(t(s))v(s)/ba∇dbl2 H0,m≤C/parenleftbig /ba∇dblf(s)/ba∇dbl2 H1,1+α(s)2/parenrightbig ×/braceleftBigg e−(2ν/(1+β)−2)β∈(−1,1), exp(−2νes+2s)β=−1. Summing up the above estimates and (4.1), (4.2), we obtain the desir ed estimate. Next, we estimate the term hgiven by (3.24). By Lemmas 3.2 and 4.2, we can easily have the following estimate: Lemma 4.3. Lethbe defined by (3.24). Under the assumption (2.1)–(2.5), we have /ba∇dblh(s)/ba∇dbl2 H0,m≤Ce−2λ0s/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg +Ce−2λ1sk/summationdisplay i=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg /ba∇dblg(s)/ba∇dblH0,1+α(s)+dα ds(s)/parenrightbigg2pi3 forn= 1,s≥0and /ba∇dblh(s)/ba∇dbl2 H0,m≤Ce−2λ0s/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg +Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p forn≥2,s≥0, whereλ0,λ1are defined by (3.51),(3.52), respectively. 28Proof.We easily estimate /vextenddouble/vextenddouble/vextenddouble/vextenddoublee−s b(t(s))2/parenleftbigg −2dα ds(s)ψ0(y)+α(s)/parenleftBigy 2·∇yψ0(y)+/parenleftBign 2+1/parenrightBig ψ0(y)/parenrightBig/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 H0,m ≤Ce−2λ0s/parenleftBigg α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg . For the term r(s), we apply Lemma 4.2. Finally, for the term (/integraltext Rnr(s,y)dy)ϕ0(y), we note that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Rnr(s,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dblr(s)/ba∇dblH0,m, (4.3) holds due to m>n/2. Thus, we apply Lemma 4.2 again to obtain the conclusion. Moreover, combining (3.26) and the Hardy-type inequalities (3.28), (3.38), we also have Lemma 4.4. LetHbe defined by (3.31). Under the assumption (2.1)–(2.5), we have /ba∇dblH(s)/ba∇dbl2 H0,m≤Ce−2λ0s/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg +Ce−2λ1sk/summationdisplay i=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg /ba∇dblg(s)/ba∇dblH0,1+α(s)+dα ds(s)/parenrightbigg2pi3 forn= 1,s≥0and /ba∇dblH(s)/ba∇dbl2 H0,m≤Ce−2λ0s/parenleftBigg /ba∇dblf(s)/ba∇dbl2 H1,m+/ba∇dblg(s)/ba∇dbl2 H0,m+α(s)2+/parenleftbiggdα ds(s)/parenrightbigg2/parenrightBigg +Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p forn≥2,s≥0, whereλ0,λ1are defined by (3.51),(3.52), respectively. Now we are at the position to prove Lemma 3.17. Proof of Lemma 3.17. We first prove the estimate for R4(s). Let ˜η>0 be an arbitrary small number. Then, by the Schwarz inequality and Lemmas 3.15 and 3.16, there exis tss3≥s2such that the terms not including the nonlinearity are easily bounded by ˜ ηL4(s)+C(˜η)e−2λ0sE5(s) fors≥s3. The terms including the nonlinearity consist of the following three terms: /integraldisplay Rn(F+G)Hdy,/integraldisplay Rn(1+|y|2m)(f+g)hdy,/parenleftbigg/integraldisplay Rnr(s,y)dy/parenrightbiggdα ds(s). By the Schwarz inequality and Lemmas 3.9, 3.11, 3.15, 3.16, 4.3, 4.4, th ere existss4≥s2such that the first two terms are easily bounded by ˜ηL4(s)+C(˜η)e−2λ0sE5(s)+ Ce−2λ1sk/summationdisplay i=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3) (n= 1), Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p(n≥2) fors≥s4. For the third term, we apply the Schwarz inequality to obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg/integraldisplay Rnr(s,y)dy/parenrightbiggdα ds(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤˜η/parenleftbiggdα ds(s)/parenrightbigg2 +C(˜η)/parenleftbigg/integraldisplay Rnr(s,y)dy/parenrightbigg2 . 29Noting (4.3) and applying Lemma 4.2, and then Lemmas 3.15 and 3.16, we have the desired estimate. Finally, we prove the estimate for R5(s). Let ˜η >0 be an arbitrary small number. Recall that R5(s) =R4(s)+˜R5(s) with ˜R5(s) =e−s b(t(s))2/parenleftbiggdα ds(s)/parenrightbigg2 −2 b(t(s))2α(s)db dt(t(s))dα ds(s)+α(s)/parenleftbigg/integraldisplay Rnr(s,y)dy/parenrightbigg . We have already estimated R4(s) and hence, it suffices to estimate ˜R5(s). By Lemmas 3.15 and 3.16, there exists s5≥s2such that the first two terms are easily estimated by ˜ ηL4(s) +C(˜η)e−2λ0sE5(s) fors≥s5. Moreover, by (4.3), α(s)≤CE5(s)1/2and Lemma 4.2, there exists s6≥s2such that the third term is estimated as α(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Rnr(s,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤α(s)/ba∇dblr(s)/ba∇dblH0,m ≤˜ηL4(s)+Ce−λ0sE5(s) + Ce−λ1sk/summationdisplay i=1E5(s)(pi1+pi2+1)/2/parenleftBig E5(s)pi3/2+L4(s)pi3/2/parenrightBig (n= 1), Ce−λ1sE5(s)(p+1)/2(n≥2) fors≥s6. 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1811.00020v2.Anisotropic_and_controllable_Gilbert_Bloch_dissipation_in_spin_valves.pdf

Anisotropic and controllable Gilbert-Bloch dissipation in spin valves Akashdeep Kamra,1,Dmytro M. Polishchuk,2Vladislav Korenivski,2and Arne Brataas1 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway 2Nanostructure Physics, Royal Institute of Technology, Stockholm, Sweden Spin valves form a key building block in a wide range of spintronic concepts and devices from magnetoresistive read heads to spin-transfer-torque oscillators. We elucidate the dependence of the magnetic damping in the free layer on the angle its equilibrium magnetization makes with that in the xed layer. The spin pumping-mediated damping is anisotropic and tensorial, with Gilbert- and Bloch-like terms. Our investigation reveals a mechanism for tuning the free layer damping in-situ from negligible to a large value via the orientation of xed layer magnetization, especially when the magnets are electrically insulating. Furthermore, we expect the Bloch contribution that emerges from the longitudinal spin accumulation in the non-magnetic spacer to play an important role in a wide range of other phenomena in spin valves. Introduction. { The phenomenon of magnetoresistance is at the heart of contemporary data storage technolo- gies [1, 2]. The dependence of the resistance of a multi- layered heterostructure comprising two or more magnets on the angles between their respective magnetizations has been exploited to read magnetic bits with a high spatial resolution [3]. Furthermore, spin valves comprised of two magnetic layers separated by a non-magnetic conductor have been exploited in magnetoresistive random access memories [2, 4, 5]. Typically, in such structures, one `free layer' is much thinner than the other `xed layer' allowing for magnetization dynamics and switching in the former. The latter serves to spin-polarize the charge currents owing across the device and thus exert spin- torques on the former [6{9]. Such structures exhibit a wide range of phenomena from magnetic switching [5] to oscillations [10, 11] driven by applied electrical currents. With the rapid progress in taming pure spin cur- rents [12{20], magnetoresistive phenomena have found a new platform in hybrids involving magnetic insulators (MIs). The electrical resistance of a non-magnetic metal (N) was found to depend upon the magnetic congura- tion of an adjacent insulating magnet [21{24]. This phe- nomenon, dubbed spin Hall magnetoresistance (SMR), relies on the pure spin current generated via spin Hall eect (SHE) in N [25, 26]. The SHE spin current accu- mulates spin at the MI/N interface, which is absorbed by the MI depending on the angle between its magne- tization and the accumulated spin polarization. The net spin current absorbed by the MI manifests as ad- ditional magnetization-dependent contribution to resis- tance in N via the inverse SHE. The same principle of magnetization-dependent spin absorption by MI has also been exploited in demonstrating spin Nernst eect [27], i.e. thermally generated pure spin current, in platinum. While the ideas presented above have largely been ex- ploited in sensing magnetic elds and magnetizations, tunability of the system dissipation is a valuable, un- derexploited consequence of magnetoresistance. Such an electrically controllable resistance of a magnetic wire FIG. 1. Schematic depiction of the device under investigation. The blue arrows denote the magnetizations. The xed layer F2magnetization remains static. The free layer F 1magneti- zation precesses about the z-axis with an average cone angle 1. The two layers interact dynamically via spin pumping and back ow currents. hosting a domain wall [28] has been suggested as a ba- sic circuit element [29] in a neuromorphic computing [30] architecture. In addition to the electrical resistance or dissipation, the spin valves should allow for controlling the magnetic damping in the constituent magnets [31]. Such an in-situ control can be valuable in, for example, architectures where a magnet is desired to have a large damping to attain low switching times and a low dissipa- tion for spin dynamics and transport [13, 16]. Further- more, a detailed understanding of magnetic damping in spin valves is crucial for their operation as spin-transfer- torque oscillators [10] and memory cells [5]. Inspired by these new discoveries [21, 27] and previous related ideas [31{34], we suggest new ways of tuning the magnetic damping of the free layer F 1in a spin valve (Fig. 1) via controllable absorption by the xed layer F2of the spin accumulated in the spacer N due to spin pumping [31, 35]. The principle for this control over spin absorption is akin to the SMR eect discussed above and capitalizes on altering the F 2magnetization direction. When spin relaxation in N is negligible, the spin lost by F1is equal to the spin absorbed by F 2. This lost spin appears as tensorial Gilbert [36] and Bloch [37] damp-arXiv:1811.00020v2 [cond-mat.mes-hall] 10 Apr 20192 ing in F 1magnetization dynamics. In its isotropic form, the Gilbert contribution arises due to spin pumping and is well established [31{33, 35, 38{40]. We reveal that the Bloch term results from back ow due to a nite dc longitudinal spin accumulation in N. Our results for the angular and tensorial dependence of the Gilbert damping are also, to best of our knowledge, new. We show that the dissipation in F 1, expressed in terms of ferromagnetic resonance (FMR) linewidth, varies with the anglebetween the two magnetizations (Fig. 3). The maximum dissipation is achieved in collinear or or- thogonal congurations depending on the relative size of the spin-mixing g0 rand longitudinal spin glconduc- tances of the NjF2subsystem. For very low gl, which can be achieved employing insulating magnets, the spin pumping mediated contribution to the linewidth vanishes for collinear congurations and attains a -independent value for a small non-collinearity. This can be used to strongly modulate the magnetic dissipation in F 1electri- cally via, for example, an F 2comprised by a magneto- electric material [41]. FMR linewidth. { Disregarding intrinsic damping for convenience, the magnetization dynamics of F 1including a dissipative spin transfer torque arising from the spin current lost IIIs1may be expressed as: _^mmm=j j(^mmm0HHHe) +j j MsVIIIs1: (1) Here, ^mmmis the unit vector along the F 1magnetization MMMtreated within the macrospin approximation, (<0) is the gyromagnetic ratio, Msis the saturation magneti- zation,Vis the volume of F 1, andHHHeis the eective magnetic eld. Under certain assumptions of linearity as will be detailed later, Eq, (1) reduces to the Landau- Lifshitz equation with Gilbert-Bloch damping [36, 37]: _^mmm=j j(^mmm0HHHe) + ( ^mmmGGG)BBB: (2) Considering the equilibrium orientation ^mmmeq=^zzz, Eq. (2) is restricted to the small transverse dynamics described bymx;y1, while the z-component is fully determined by the constraint ^mmm
^mmm= 1. Parameterized by a diagonal dimensionless tensor  , the Gilbert damping has been in- corporated via GGG=xx_mx^xxx+yy_my^yyyin Eq. (2). The Bloch damping is parametrized via a diagonal frequency tensor  asBBB= xxmx^xxx+ yymy^yyy. A more familiar, although insucient for the present considerations, form of Bloch damping can be obtained by assuming isotropy in the transverse plane: BBB= 0(^mmm^mmmeq). This form, restricted to transverse dynamics, makes its eect as a relaxation mechanism with characteristic time 1 = 0ev- ident. The Bloch damping, in general, captures the so- called inhomogeneous broadening and other, frequency independent contributions to the magnetic damping. Considering uniaxial easy-axis and easy-plane anisotropies, parametrized respectively by Kzand 0 30 60 9000.10.20.30.40.5FIG. 2. Normalized damping parameters for F 1magneti- zation dynamics vs. spin valve conguration angle (Fig. 1). ~xx6= ~yysignies the tensorial nature of the Gilbert damping. The Bloch parameters ~ xx~ yyare largest for the collinear conguration. The curves are mirror symmetric about= 90. ~g0 r= 1, ~gl= 0:01,  = 0:1,!0= 102 GHz, and!ax= 12GHz. Kx[42], the magnetic free energy density Fmis ex- pressed as: Fm=0MMM
HHHextKzM2 z+KxM2 x;with HHHext=H0^zzz+hhhrfas the applied static plus microwave eld. Employing the eective eld 0HHHe=@Fm=@MMM in Eq. (2) and switching to Fourier space [ exp(i!t)], we obtain the resonance frequency !r=p !0(!0+!ax). Here,!0j j(0H0+ 2KzMs) and!axj j2KxMs. The FMR linewidth is evaluated as: j j0
H=(xx+yy) 2!+t( xx+ yy) 2 +t!ax 4(yyxx); (3) where!is the frequency of the applied microwave eld hhhrfand is approximately !rclose to resonance, and t !=p !2+!2ax=41 for a weak easy-plane anisotropy. Thus, in addition to the anisotropic Gilbert contribu- tions, the Bloch damping provides a nearly frequency- independent oset in the linewidth. Spin ow. { We now examine spin transport in the device with the aim of obtaining the damping parame- ters that determine the linewidth [Eq. (3)]. The N layer is considered thick enough to eliminate static exchange interaction between the two magnetic layers [31, 40]. Fur- thermore, we neglect the imaginary part of the spin- mixing conductance, which is small in metallic systems and does not aect dissipation in any case. Disregarding longitudinal spin transport and relaxation in the thin free layer, the net spin current IIIs1lost by F 1is the dierence between the spin pumping and back ow currents [31]: IIIs1=gr 4 ~^mmm_^mmm^mmms^mmm ; (4) wheregris the real part of the F 1jN interfacial spin- mixing conductance, and sis the spatially homogeneous3 spin accumulation in the thin N layer. The spin current absorbed by F 2may be expressed as [31]: IIIs2=g0 r 4^mmm2s^mmm2+gl 4(^mmm2
s)^mmm2; X i;j=fx;y;zggij 4sj^iii; (5) whereglandg0 rare respectively the longitudinal spin conductance and the real part of the interfacial spin- mixing conductance of the N jF2subsystem, ^mmm2denotes the unit vector along F 2magnetization, and gij=gji are the components of the resulting total spin conduc- tance tensor. glquanties the absorption of the spin current along the direction of ^mmm2, the so-called longi- tudinal spin current. For metallic magnets, it is domi- nated by the diusive spin current carried by the itin- erant electrons, which is dissipated over the spin re- laxation length [31]. On the other hand, for insulat- ing magnets, the longitudinal spin absorption is domi- nated by magnons [43, 44] and is typically much smaller than for the metallic case, especially at low tempera- tures. Considering ^mmm2= sin^yyy+ cos^zzz(Fig. 1), Eq. (5) yields gxx=g0 r,gyy=g0 rcos2+glsin2, gzz=g0 rsin2+glcos2,gxy=gyx=gxz=gzx= 0, andgyz=gzy= (glg0 r) sincos. Relegating the consideration of a small but nite spin relaxation in the thin N layer to the supplemental ma- terial [45], we assume here that the spin current lost by F1is absorbed by F 2, i.e.,IIIs1=IIIs2. Imposing this spin current conservation condition, the spin accumulation in N along with the currents themselves can be determined. We are primarily interested in the transverse (x and y) components of the spin current since these fully deter- mine the magnetization dynamics ( ^mmm
^mmm= 1): Is1x=1 4grgxx gr+gxx(~_my+mxsz); Is1y=1 4grgyy gr+gyy(~_mx+mysz) +gyzsz(1ly) ; sz=~gr(lxmx_mylymy_mxp_mx) gzzpgyz+gr lxm2x+lym2y+ 2pmy
; (6) wherelx;ygxx;yy=(gr+gxx;yy) andpgyz=(gr+gyy). The spin lost by F 1appears as damping in the magneti- zation dynamics [Eqs. (1) and (2)] [31, 35]. We pause to comment on the behavior of szthus ob- tained [Eq. (6)]. Typically, szis considered to be rst or second order in the cone angle, and thus negligibly small. However, as discussed below, an essential new nding is that it becomes independent of the cone an- gle and large under certain conditions. For a collinear conguration and vanishing gl,gzz=gyz= 0 results in ~szsz=~!!1 [38]. Its nite dc value con- tributes to the Bloch damping [Eq. (6)] [38]. For a non-collinear conguration, sz~grp_mx=(gzzpgyz) 0 45 90 135 18000.10.20.30.40.50.6FIG. 3. Normalized ferromagnetic resonance (FMR) linewidth of F 1for dierent values of the longitudinal spin conductance ~ glgl=grof NjF2bilayer. The various parame- ters employed are ~ g0 rg0 r=gr= 1,  = 0:1 rad,!0= 102 GHz, and!ax= 12GHz.grandg0 rare the spin-mixing conductances of F 1jN and NjF2interfaces respectively. Only the spin pumping-mediated contribution to the linewidth has been considered and is normalized to its value for the case of spin pumping into a perfect spin sink [31]. and contributes to Gilbert damping via Is1y[Eq. (6)]. Thus, in general, we may express the spin accumulation assz=sz0+sz1[46], where sz0is the dc value andsz1/_mxis the linear oscillating component. sz0 andsz1contribute, respectively, to Bloch and Gilbert damping. Gilbert-Bloch dissipation. { Equations (1) and (6) com- pletely determine the magnetic damping in F 1. However, these equations are non-linear and cannot be captured within our linearized framework [Eqs. (2) and (3)]. The leading order eects, however, are linear in all but a nar- row range of parameters. Evaluating these leading or- der terms within reasonable approximations detailed in the supplemental material [45], we are able to obtain the Gilbert and Bloch damping tensors  and . Obtaining the general result numerically [45], we present the ana- lytic expressions for two cases covering a large range of the parameter space below. First, we consider the collinear congurations in the limit of ~glgl=gr!0. As discussed above, we obtain ~sz0sz0=~!!1 and ~sz1sz1=~!!0 [Eq. (6)]. Thus the components of the damping tensors can be di- rectly read from Eq. (6) as ~ xx;yyxx;yy=ss=ly;x= g0 r=(gr+g0 r) = ~g0 r=(1+~g0 r);and~ xx;yy xx;yy=(ss!) = lx;ysz0=(~!) =g0 r=(gr+g0 r) =~g0 r=(1 + ~g0 r). Here, we dened ~ g0 rg0 r=grandss~grj j=(4MsV) is the Gilbert constant for the case of spin-pumping into an ideal spin sink [31, 35]. Substituting these values in Eq. (3), we nd that the linewidth, or equivalently damping, vanishes. This is understandable since the system we have considered is not able to relax the z component of the spin at all. There can, thus, be no net contribution to4 FIG. 4. Normalized FMR linewidth of F 1for very small ~ gl. The squares and circles denote the evaluated points while the lines are guides to the eye. The linewidth increases from being negligible to its saturation value as becomes comparable to the average cone angle . ~ g0 r= 1,!0= 102GHz, and !ax= 12GHz. magnetic damping. sz0accumulated in N opposes the Gilbert relaxation via a negative Bloch contribution [38]. The latter may also be understood as an anti-damping spin transfer torque due to the accumulated spin [6]. Next, we assume the system to be in a non-collinear conguration such that ~ sz0!0 and may be disre- garded, while ~ sz1simplies to: ~sz1=_mx !(~gl~g0 r) sincos ~g0r~gl+ ~glcos2+ ~g0rsin2; (7) where ~glgl=grand ~g0 rg0 r=gras above. This in turn yields the following Gilbert parameters via Eq. (6), with the Bloch tensor vanishing on account of ~ sz0!0: ~xx=~g0 r~gl ~g0r~gl+ ~glcos2+ ~g0rsin2;~yy=~g0 r 1 + ~g0r;(8) where ~xx;yyxx;yy=ssas above. Thus, ~ yyis- independent since ^mmm2lies in the y-z plane and the x- component of spin, the absorption of which is captured by ~yy, is always orthogonal to ^mmm2. ~xx, on the other hand, strongly varies with and is generally not equal to ~yyhighlighting the tensorial nature of the Gilbert damping. Figure 2 depicts the congurational dependence of nor- malized damping parameters. The Bloch parameters are appreciable only close to the collinear congurations on account of their proportionality to sz0. Therange over which they decrease to zero is proportional to the cone angle  [Eq. (6)]. The Gilbert parameters are described suciently accurately by Eq. (8). The linewidth [Eq. (3)] normalized to its value for the case of spin pump- ing into a perfect spin sink has been plotted in Fig. 3. For low ~gl, the Bloch contribution partially cancels the Gilbert dissipation, which results in a smaller linewidthclose to the collinear congurations [38]. As ~ glincreases, the relevance of Bloch contribution and sz0diminishes, and the results approach the limiting condition described analytically by Eq. (8). In this regime, the linewidth dependence exhibits a maximum for either collinear or orthogonal conguration depending on whether ~ gl=~g0 ris smaller or larger than unity. Physically, this change in the angle with maximum linewidth is understood to re- ect whether transverse or longitudinal spin absorption is stronger. We focus now on the case of very low ~ glwhich can be realized in structures with electrically-insulating mag- nets. Figure 4 depicts the linewidth dependence close to the collinear congurations. The evaluated points are marked with stars and squares while the lines smoothly connect the calculated points. The gap in data for very small angles re ects the limited validity of our linear theory, as discussed in the supplemental material [45]. As per the limiting case ~ gl!0 discussed above, the linewidth should vanish in perfectly collinear states. A more precise statement for the validity of this limit is re ected in Fig. 4 and Eq. (6) as ~ gl=2!0. For su- ciently low ~ gl, the linewidth changes sharply from a neg- ligible value to a large value over a range approximately equal to the cone angle . This shows that systems com- prised of magnetic insulators bearing a very low ~ glare highly tunable as regards magnetic/spin damping by rel- atively small deviation from the collinear conguration. The latter may be accomplished electrically by employ- ing magnetoelectric material [41] for F 2or via current driven spin transfer torques [6, 9, 47]. Discussion. { Our identication of damping contribu- tions as Gilbert-like and Bloch-like [Eq. (6)] treats sz as an independent variable that may result from SHE, for example. When it is caused by spin pumping cur- rent andsz/!, this Gilbert-Bloch distinction is less clear and becomes a matter of preference. Our results demonstrate the possibility of tuning the magnetic damp- ing in an active magnet via the magnetization of a passive magnetic layer, especially for insulating magnets. In ad- dition to controlling the dynamics of the uniform mode, this magnetic `gate' concept [48] can further be employed for modulating the magnon-mediated spin transport in a magnetic insulator [43, 44]. The anisotropy in the result- ing Gilbert damping may also oer a pathway towards dissipative squeezing [49] of magnetic modes, comple- mentary to the internal anisotropy-mediated `reactive' squeezing [50, 51]. We also found the longitudinal accu- mulated spin, which is often disregarded, to signicantly aect the dynamics. This contribution is expected to play an important role in a wide range of other phenom- ena such as spin valve oscillators. Summary. { We have investigated the angular modu- lation of the magnetic damping in a free layer via control of the static magnetization in the xed layer of a spin valve device. The damping can be engineered to become5 larger for either collinear or orthogonal conguration by choosing the longitudinal spin conductance of the xed layer smaller or larger than its spin-mixing conductance, respectively. The control over damping is predicted to be sharp for spin valves made from insulating magnets. Our results pave the way for exploiting magneto-damping eects in spin valves. Acknowledgments. { We acknowledge nancial support from the Research Council of Norway through its Centers of Excellence funding scheme, project 262633, \QuSpin", and from the Swedish Research Council, project 2018- 03526, and Stiftelse Olle Engkvist Byggm astare. akashdeep.kamra@ntnu.no [1] Albert Fert, \Nobel lecture: Origin, development, and future of spintronics," Rev. Mod. Phys. 80, 1517{1530 (2008). [2] S. Parkin, Xin Jiang, C. Kaiser, A. Panchula, K. Roche, and M. Samant, \Magnetically engineered spintronic sen- sors and memory," Proceedings of the IEEE 91, 661{680 (2003). [3] Keiichi Nagasaka, \Cpp-gmr technology for magnetic read heads of future high-density recording systems," Journal of Magnetism and Magnetic Materials 321, 508 { 511 (2009), current Perspectives: Perpendicular Record- ing. [4] Johan Akerman, \Toward a universal memory," Science 308, 508{510 (2005), http://science.sciencemag.org/content/308/5721/508.full.pdf. 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[45] See supplemental material for a discussion on the case of collinear congurations with small ~ gl, details on the numerical analysis with related approximations, depen- dence of the FMR linewidth on the easy-plane anisotropy and spin-mixing conductance g0 r, and the eect of nite spin relaxation in the spacer layer. [46] As detailed in the supplemental material [45], we have disregarded the term in szwhich oscillates with a fre- quency 2!. Strictly speaking, this term needs to be in- cluded even in our linear analysis, since it produces terms oscillating with !when multiplied with another term at !. However, such a contribution is only relevant in a nar- row parameter range which may be hard to resolve in an experiment. Furthermore, it requires a non-linear so- lution to the equations and is beyond the scope of the present work. [47] Jairo Sinova, Sergio O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, \Spin hall eects," Rev. Mod. Phys. 87, 1213{1260 (2015). [48] L. J. Cornelissen, J. Liu, B. J. van Wees, and R. A. Duine, \Spin-current-controlled modulation of the magnon spin conductance in a three-terminal magnon transistor," Phys. Rev. Lett. 120, 097702 (2018). [49] Andreas Kronwald, Florian Marquardt, and Aashish A Clerk, \Dissipative optomechanical squeezing of light," New Journal of Physics 16, 063058 (2014). [50] Akashdeep Kamra and Wolfgang Belzig, \Super- poissonian shot noise of squeezed-magnon mediated spin transport," Phys. Rev. Lett. 116, 146601 (2016). [51] Akashdeep Kamra, Utkarsh Agrawal, and Wolfgang Belzig, \Noninteger-spin magnonic excitations in untex- tured magnets," Phys. Rev. B 96, 020411 (2017). [52] A.I. Akhiezer, V.G. Bar'iakhtar, and S.V. Peletminski, Spin waves (North-Holland Publishing Company, Ams- terdam, 1968).1 Supplemental material with the manuscript Anisotropic and controllable Gilbert-Bloch dissipation in spin valves by Akashdeep Kamra, Dmytro M. Polishchuk, Vladislav Korenivski and Arne Brataas COLLINEAR CONFIGURATION WITHOUT LONGITUDINAL SPIN RELAXATION In order to appreciate some of the subtleties, we rst examine the collinear conguration in the limit of vanishing longitudinal spin conductance. = 0;andgl= 0 imply the following values for the various parameters: gxx=gyy=g0 r; g zz=gyz=p= 0; lx;y=g0 r gr+g0rl; (S1) whence we obtain: sz ~=(mx_mymy_mx) m2x+m2y; (S2) =!0+!ax 1 +!ax 2!0[1cos(2!t)]; (S3) where we have assumed magnetization dynamics as given by the Landau-Lifshitz equation without damping, and the phase of mxis treated as the reference and set to zero. In order to obtain analytic expressions, we make the assumption !ax=!01 such that we have: sz=sz0+sz2; with (S4) sz0=~ !0+!ax 2 ; (S5) sz2=~!ax 4 ei2!t+ei2!t
: (S6) In contrast with our assumptions in the main text, a term oscillating with 2 !appears. Furthermore, it yields contributions to the Bloch damping via products such as mysz, which now have contributions oscillating at !due to thesz0as well assz2. We obtain: ~xx= ~yy=l; (S7) ~ xx=l!0+3!ax 4 !0+!ax 2and ~ yy=l!0+!ax 4 !0+!ax 2; (S8) substituting which into Eq. (3) from the main text yields a vanishing linewidth and damping. This is expected from the general spin conservation argument that there can be no damping in the system if it is not able to dissipate the z-component of the spin. In fact, in the above considerations, sz2contributed with the opposite sign to ~ xxand ~ yy, and thus dropped out of the linewidth altogether. This also justies our ignoring this contribution in the main text. Figure 1 depicts the dependence of the accumulated z-polarized spin and the normalized linewidth for small but niteglin the collinear conguration. The accumulated longitudinal (z-polarized) spin increases with the cone angle and the linewidth accordingly decreases to zero [38]. NUMERICAL EVALUATION Despite the additional complexity in the previous section, we could treat the dynamics within our linearized frame- work. However, in the general case, szhas contributions at all multiples of !and cannot be evaluated in a simple manner. A general non-linear analysis must be employed which entails treating the magnetization dynamics numer- ically altogether. Such an approach prevents us from any analytic description of the system, buries the underlying physics, and is thus undesirable. Fortunately, the eects of non-linear terms are small for all, but a narrow, range of parameters. Hence, we make some simplifying assumptions here and continue treating our system within the linearized theory. We only show2 10-310-210-100.10.20.30.40.5 FIG. 1. Ferromagnetic resonance linewidth and the dc spin accumulation created in the spacer as a function of the average cone angle in the collinear conguration. Depending on ~ gl, there is a complementary transition of the two quantities between small and large values as the cone angle increases. ~ g0 r= 1,!0= 102GHz, and!ax= 12GHz. results in the parameter range where our linear analysis is adequate. Below, we describe the numerical routine for evaluating the various quantities. To be begin with the average cone angle  is dened as: 2= m2 x+m2 y ; (S9) whereh
idenotes averaging over time. The spin accumulation is expressed as sz=sz0+sz1with: sz0=* ~gr(lxmx_mylymy_mxp_mx) gzzpgyz+gr lxm2x+lym2y+ 2pmy
+ ; (S10) sz1=* grp gzzpgyz+gr lxm2x+lym2y+ 2pmy
+ ~_mx: (S11) The above expressions combined with the equations for the spin current ow (Eqs. (6) in the main text) directly yield the Gilbert and Bloch damping tensors. VARIATION WITH ADDITIONAL PARAMETERS Here, we discuss the dependence of the FMR linewidth on the easy-plane anisotropy and the spin-mixing conduc- tanceg0 rof the NjF2interface. The results are plotted in Fig. 2. A high easy-plane anisotropy is seen to diminish the conguration dependence of the linewidth and is thus detrimental to the dissipation tunability. The easy-axis anisotropy, on the other hand, is absorbed in !0and does not need to be examined separately. We also see an increase in the conguration dependence of the damping with an increasing g0 r. This is understood simply as an increased damping when the spin is absorbed more eciently due to a larger g0 r. The damping is expected to reach the case of spin pumping into a perfect spin sink in the limit of ~ g0 r!1 and= 0;. EFFECT OF SPIN RELAXATION IN THE SPACER LAYER We now address the role of the small but nite spin relaxation in the non-magnetic spacer layer. To this end, we consider that a part of the spin current injected into N by F 1is lost as the \spin-leakage current" IIIsl, as depicted in Fig. 3, such that IIIs1=IIIs2+IIIsl. In order to evaluate the leakage, we consider the spin diusion equation in N which reads [31]: D@2 xs=s sf; (S12)3 0 45 90 135 18000.10.20.30.40.50.6 (a) 0 45 90 135 18000.20.40.60.81 (b) FIG. 2. Normalized ferromagnetic resonance (FMR) linewidth of F 1. (a) Same as Fig. 3 in the main text with additional plots for a large easy-plane anisotropy. (b) Linewidth dependence for dierent spin-mixing conductances of N jF2interface. The parameters employed are the same as Fig. 2 in the main text. FIG. 3. Schematic depiction of the spin currents owing through the device, including the spin-leakage current IIIslthat is lost on account of a nite spin relaxation in the spacer layer N. whereDandsfare diusion constant and spin- ip time, respectively. We now integrate the equation over the thickness of N: Z d(D@xs) =Zd 0s sfdx: (S13) Since the N-layer thickness dis typically much smaller than the spin diusion length in N (e.g., a few nm versus a few hundred nm for Cu), we treat son the right hand side as a constant. Furthermore, in simplifying the left hand side, we invoke the expression for the spin current [31]: IIIs= (~NSD=2)@xs, withNthe one-spin density of states per unit volume and Sthe interfacial area. Thus, we obtain 2 ~NS(IIIs1IIIs2) =d sfs; (S14) which simplies to the desired relation IIIs1=IIIs2+IIIslwith IIIsl=~NVN 2sfsgsl 4s; (S15) whereVNis the volume of the spacer layer N. It is easy to see that accounting for spin leakage, as derived in Eq. (S15), results in the following replacements to Eqs. (6) of the main text: gxx!gxx+gsl; g yy!gyy+gsl; g zz!gzz+gsl: (S16)4 Since all our specic results are based on Eqs. (6) of the main text, this completes our assessment of the role played by spin relaxation in N. Physically, this new result means that the condition for no spin relaxation in the system, which was previously treated as gl!0, is now amended to gl+gsl!0. This, however, does not aect the generality and signicance of the key results presented in the main text.

2102.10394v2.Fast_magnetization_reversal_of_a_magnetic_nanoparticle_induced_by_cosine_chirp_microwave_field_pulse.pdf

Fast magnetization reversal of a magnetic nanoparticle induced by cosine chirp microwave field pulse M. T. Islam,1,a)M. A. S. Akanda,1M. A. J. Pikul,1and X. S. Wang2,b) 1)Physics Discipline, Khulna University, Khulna 9208, Bangladesh 2)School of Physics and Electronics, Hunan University, Changsha 410082, China We investigate the magnetization reversal of single-domain magnetic nanoparticle driven by the circularly polarized cosine chirp microwave pulse (CCMP). The numerical findings, based on the Landau-Lifshitz-Gilbert equation, reveal that the CCMP is by itself capable of driving fast and energy-efficient magnetization reversal. The microwave field am- plitude and initial frequency required by a CCMP are much smaller than that of the linear down-chirp microwave pulse. This is achieved as the frequency change of the CCMP closely matches the frequency change of the magnetization pre- cession which leads to an efficient stimulated microwave energy absorption (emission) by (from) the magnetic particle before (after) it crosses over the energy barrier. We further find that the enhancement of easy-plane shape anisotropy significantly reduces the required microwave amplitude and the initial frequency of CCMP. We also find that there is an optimal Gilbert damping for fast magnetization reversal. These findings may provide a pathway to realize the fast and low-cost memory device. I. INTRODUCTION Achieving fast and energy-efficient magnetization rever- sal of high anisotropy materials has drawn much attention since it has potential application in non-volatile data stor- age devices1–3and fast data processing4. For high thermal stability and low error rate, high anisotropy materials are required5in device application. But one of the challenging issues is to find out the way which can induce the fastest mag- netization reversal with minimal energy consumption. Over the last two decades, many magnetization reversal methods has been investigated, such as by constant magnetic fields6,7, by the microwave field of constant frequency, either with or without a polarized electric current8–10and by spin-transfer torque (STT) or spin-orbit torque (SOT)11–28. However, all the means are suffering from their own limitations. For in- stance, in the case of external magnetic field, reversal time is longer and has scalability and field localization issues6. In case of the constant microwave field driven magnetiza- tion reversal, the large field amplitude and the long rever- sal time are emerged as limitations29–31. In the case of the STT-MRAM, the threshold current density is a large and thus, Joule heat which may lead the device malfunction durability and reliability issues32–38. Moreover, there are several stud- ies showing magnetization reversal induced by microwaves of time-dependent frequency39–45. In the study39, the magnetiza- tion reversal, with the assistance of external field, is obtained by a radio-frequency microwave field pulse. Here, a dc ex- ternal field acts as the main reversal force. In the study40, to obtain magnetization reversal, the applied microwave fre- quency needs to be the same as the resonance frequency, and in the studies42,43, optimal microwave forms are constructed. These microwave forms are difficult to be realized in practice. The study44reports magnetization reversal induced by the mi- crowave pulse, but the pulse is applied such that the magne- tization just crosses over the energy barrier, i.e., only positive a)Electronic mail: torikul@phy.ku.ac.bd b)Electronic mail: justicewxs@hnu.edu.cnfrequency range ( +f0to 0) is employed. A recent study45has demonstrated that the circularly polar- ized linear down-chirp microwave pulse (DCMP) (whose fre- quency linearly decreases with time from the initial frequency +f0tof0) can drive fast magnetization reversal of uniax- ial nanoparticles. The working principle of the above model is that the DCMP triggers stimulated microwave energy ab- sorption (emission) by (from) the magnetization before (after) crossing the energy barrier. However, the efficiency of trig- gered microwave energy absorption or emission depends on how closely the frequency of chirp microwave pulse matches the magnetization precession frequency. In DCMP-driven case, the frequency linearly decreases from f0tof0with time but, in fact, the decrement of magnetization precession frequency is not linear46,47during magnetization reversal . So the frequency of DCMP only roughly matches the magneti- zation precession frequency. Thus, the DCMP triggers in- efficient energy absorption or emission and the required mi- crowave amplitude is still large. Therefore, to achieve more efficient magnetization reversal, we need to find a microwave pulse of proper time-dependent frequency that matches the intrinsic magnetization precession frequency better. In this study, we demonstrate that a cosine chirp microwave pulse (CCMP), defined as a microwave pulse whose frequency sweeps in a cosine function with time from +f0tof0in first half-period of the microwave pulse, is ca- pable of driving the fast and energy efficient magnetization reversal. This is because the frequency change of the CCMP matches the nonlinear frequency change of magnetization pre- cession better than the DCMP. In addition, this study empha- sizes how the shape anisotropy influences the required param- eters of CCMP and how the Gilbert damping affects the mag- netization reversal. We find that the increase of easy-plane shape anisotropy makes the magnetization reversal easier. The materials with larger damping are better for fast magnetization reversal. These investigations might be useful in device appli- cations.arXiv:2102.10394v2 [cond-mat.mes-hall] 15 Sep 20212 mf0 0-f(b) (a) τ t FIG. 1. (a) Schematic diagram of the system in which mrepresents a unit vector of the magnetization. A circularly polarized cosine (nonlinear) chirp microwave pulse is applied onto the single domain nanoparticle. (b) The frequency sweeping (from +f0tof0) of a cosine chirp microwave pulse. II. ANALYTICAL MODEL AND METHOD We consider a square magnetic nanoparticle of area Sand thickness dwhose uniaxial easy-axis anisotropy directed in thez-axis as shown in FIG. 1(a). The size of the nanopar- ticle is chosen so that the magnetization is considered as a macrospin represented by the unit vector mwith the mag- netic moment SdM s, where Msis the saturation magnetization of the material. The demagnetization field can be approxi- mated by a easy-plane shape anisotropy. The shape anisotropy field coefficient is hshape=m0(NzNx)Ms, and the shape anisotropy field is hshape=hshapemzˆz, where NzandNxare demagnetization factors48,49andm0=4p107N=A2is the vacuum magnetic permeability. The strong uniaxial mag- netocrystalline anisotropy hani=hanimzˆzdominates the total anisotropy so that the magnetization of the nanoparticle has two stable states, i.e., mparallel to ˆzandˆz. The magnetization dynamics min the presence of circularly polarized CCMP is governed by the Landau-Lifshitz-Gilbert (LLG) equation50 dm dt=gmheff+amdm dt; (1) where aandgare the dimensionless Gilbert damping con- stant and the gyromagnetic ratio, respectively, and heffis the total effective field which includes the microwave magnetic fieldhmw, and the effective anisotropy field hkalong zdirec- tion (note that although we consider small nanoparticles that can be treated as macrospins approximately, we still perform full micromagnetic simulations with small meshes and full de- magnetization field to be more accurate). The effective anisotropy field can be expressed in terms of uniaxial anisotropy haniand shape anisotropy hshape ashk= hani+hshape= [hanim0(NzNx)Ms]mzˆz. Thus, the resonant frequency of the nanoparticle is obtained from the well-known Kittel formula f0=g 2p[hanim0(NzNx)Ms]: (2). For microwave field-driven magnetization reversal from the LLG equation, the rate of energy change is expressed as dE dt=agjmheffj2m
dhmw dt: (3) The first term is always negative since damping ais posi- tive. The second term can be either positive or negative for time-dependent external microwave field. Therefore, the mi- crowave field pulse can trigger the stimulated energy absorp- tion or emission, depending on the angle between the instan- taneous magnetization manddhmw dt. Initially, because of easy-axis anisotropy, the magnetiza- tion prefers to stay in one of the two stable states, ˆz, cor- responding to two energy minima. The objective of magne- tization reversal is to get the magnetization from one stable state to the other. Along the reversal process, the magnetiza- tion requires to overcome an energy barrier at mz=0 which separates two stable states. For fast magnetization reversal, the external field is required to supply the necessary energy to the magnetization until crossing the energy barrier and, af- ter crossing the energy barrier, the magnetization releases en- ergy through damping and =or the external field is required to extract (by negative work done) energy from the magne- tization. It is mentioned that there is as intrinsic anisotropy fieldhanidue to the anisotropy which induces a magnetization natural =resonant frequency proportional to mz. When mag- netization goes from one stable state to another, the magneti- zation resonant frequency decreases while the magnetization climbs up and becomes zero momentarily while crossing the energy barrier and then increases with the opposite preces- sion direction while it goes down from the barrier. In princi- ple, for fast and energy-efficient reversal, one requires a chirp microwave pulse whose frequency always matches the mag- netization precession frequency to ensure the term m
˙hmw to be maximal (minimal) before (after) crossing the energy barrier. The study45, employs the DCMP (whose frequency linearly decreases with time) to match the magnetization pre- cession frequency roughly. In fact, during the reversal from mz= +1 to mz=1 , the decreasing of the resonant fre- quency ( while the spin climbs up the energy barrier) and in- creasing of the resonant frequency (while it goes down from the barrier) are not linear46,47. This leads us to consider a cosine chirp microwave pulse (CCMP) (a microwave pulse whose frequency decreases non-linearly with time) in order to match the change of magnetization precession frequency closely. Thus the CCMP might trigger more efficient stim- ulated microwave absorptions (emissions) by (from) magne- tization before (after) the spin crosses the energy barrier to induce fast and energy-efficient magnetization reversal. In order to substantiate the above mentioned prediction, we apply a circularly polarized cosine down-chirp microwave pulse in the xyplane of the nanoparticle and solve the LLG equation numerically using the MUMAX3 package51. The cosine chirp microwave pulse (CCMP) takes the form hmw= hmw[cosf(t)ˆx+sinf(t)ˆy], where hmwis the amplitude of the microwave field and f(t)is the phase. Since the phase f(t)is 2pf0cos(2pRt)t, where R(in units of GHz) is the controlling3 /s45/s49 /s48 /s49/s45/s49/s48/s49 /s45/s49/s48/s49 /s109 /s122 /s109/s121 /s109 /s120 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s49/s48/s49 /s46 /s116/s32/s40/s110/s115/s41/s109 /s122 /s45/s49/s48/s45/s53/s48/s53/s49/s48/s101 /s32/s40/s49/s48/s49/s53 /s32/s74/s47/s115/s32/s109/s51 /s41(a) (b) (c) FIG. 2. Model parameters of nanoparticle of Ms= 106A/m, Hk= 0:75 T, g= 1:761011rad/(T
s), and a=0:01. (a) Temporal evo- lutions of mzofV= (888)nm3driven by the CCMP (with the minimal hmw=0:035 T, f0=18:8 GHz and R= 0.32 GHz) (red line) and DCMP (with hmw=0:035 T, f0=18:8 GHz and R= 1.53 GHz ) (blue line). For CCMP case, (b) the corresponding magnetization reversal trajectories and (c) temporal evolutions of mz(red lines) and the energy changing rate ˙eof the magnetization against time (blue lines). parameter, the instantaneous frequency f(t)of CCMP is ob- tained as f(t) =1 2pdf dt=f0[cos(2pRt)(2pRt)sin(2pRt)] which decreases with time from f0to finalf0at a time dependent chirp rate h(t)(in units of ns2) as shown in FIG. 1(b). The chirp rate takes the form h(t) = f0h (4pR)sin(2pRt)+(2pR)2tcos(2pRt)i . According to the applied CCMP, the second term of right hand side of Eq. (3), i.e., the energy changing rate can be expressed as ˙e=Hmwsinq(t)sinF(t)f(t) td dtf(t) t t (4) whereF(t)is the angle between mt(the in-plane component ofmandhmw. Therefore, the microwave field pulse can trig- ger the stimulated energy absorption (before crossing the en- ergy barrier) withF(t)and emission (after crossing the en- ergy barrier) with F(t). The material parameters of this study are chosen from typ- ical experiments on microwave-driven magnetization reversal asMs=106A=m,hani=0:75 T, g=1:761011rad=(T
s), exchange constant A=131012J=m and a=0:01. Al- though, the strategy and other findings of this study would work for other materials also. The cell size (222)nm3 is used in this study. We consider the switching time win- dow 1 ns at which the magnetization switches/reverses to mz=0:9. III. NUMERICAL RESULTS We first investigate the possibility of reversing the magne- tization of cubic sample (888)nm3by the cosine chirpmicrowave pulse (CCMP). Accordingly, we apply the CCMP with the microwave amplitude hmw=0:045 T, initial fre- quency f0=21 GHz and R=1:6 ns1which are same as estimated in the study45), to the sample and found that CCMP can drive the fast magnetization reversal. Then, we search the minimal hmw,f0, and Rof the CCMP such that the fast re- versal is still valid. Interestingly, the CCMP with significantly smaller parameters i.e., hmw=0:035 T, f0=18:8 GHz and optimal R=0:32 ns1, is capable of reversing the magnetiza- tion efficiently shown by red line in FIG. 2(a). Then we intend to show how efficient the CCMP driven magnetization reversal compare to DCMP driven case. For fair comparison, we choose the same pulse duration t(as shown in Fig. 1, tis the time at which the frequency changes from +f0tof0). For the CCMP, we solve cos (2pRt) (2pRt)sin(2pRt) =1, which gives the relation t=1:307 2pR. But, in case of DCMP, we know that t=2f0 hand for f0= 18:8 GHz, the chirp rate becomes h=57:86 ns2and hence find the parameter R(=1=t)= 1.53 ns1. Then we apply the DCMP with the hmw=0:035 T, f0=18:8 GHz (which are same as CCMP-driven case), and R=1.53 ns1and found that the magnetization only precesses around the initial state, i.e., the DCMP is not able to reverse the magnetization as shown by blue line in FIG. 2(a). So, it is mentioned that the CCMP can reverse the magnetization with lower energy consumption which is the desired in device application. To be more explicit, the trajectories of magnetization reversal induced by CCMP is shown FIG. 2(b) which shows the magnetization reverses swiftly.For further justification of CCMP driven reversal, we calculate the energy changing rate dE=dtrefers to (4) by de- termining the angles F(t)andq(t)and plotted with time in FIG. 2(c). The stimulated energy absorption (emission) peaks are obtained before (after) crossing the energy barrier as ex- pected for faster magnetization reversal. This is happened be- cause the frequency of the CCMP closely matches the fre- quency of magnetization precession frequency, i.e., before crossing the energy barrier, the F(t)remains around90and after crossing the energy barrier F(t)around 90to maximize the energy absorption and emission respectively. Then, this study emphasizes how shape-anisotropy field hshape affects the magnetization switching time, microwave amplitude hmwand initial frequency f0of CCMP. Since de- magnetization field or shape-anisotropy field hshape should have significant effect on magnetization reversal process as it opposes the magnetocrystalline anisotropy field haniwhich stabilizes the magnetization along two stable states. Accord- ingly, to induce the hshape in the sample, we choose square cuboid shape samples and, to increase the strength of hshape, the cross-sectional area S=xyis enlarged gradually for the fixed thickness d(=z) =8 nm. Specifically, we focus on the samples of S1=1010,S2=1212,S3=1414, S4=1616,S5=1818,S6=2020 and S7=2222 nm2 with d=8 nm. For the samples of different S, by determining the analytic demagnetization factors NzandNx48,52, the shape anisotropy field hshape=m0(NzNx)Msmzˆzare determined. Thehshape actually opposes the anisotropy field haniand hence resonance frequencies f0=g 2p[hanim0(NzNx)Ms]4 TABLE I. Shape anisotropy coefficient, resonant frequency f0, simulated frequency, f0and frequency-band Crosssectional area Shape anisotropy coefficient Resonant frequency, Simulated minimal frequency, Simulated frequency-band S(nm2) hshape (T) f0(GHz) f0(GHz) (GHz) S1 0.09606 18.3 17.8 17.8 19.9 S2 0.17718 16 14.9 14.9 18.3 S3 0.2465 14.1 13.7 13.7 15.4 S4 0.3064 12.4 12.2 12.2 13.8 S5 0.3588 11 10.7 10.7 12.3 S6 0.4049 9.6 8.8 8.8 11.5 S7 0.4459 8.5 7.7 7.4 8.7 decreases as shown in the Table I. Then, for the samples of different S, with the fixed hmw=0:035 T, the corresponding minimal f0and optimal Rof CCMP are determined through the study of the magnetization reversal. The temporal evolu- tions of mzfor different Sare shown in FIG. 3(a) and found that for S72222 nm2, the magnetization smoothly reveres with the shortest time. The switching time tsas a function of the coefficient hshape (corresponding to S) is plotted in FIG. 3(c). It is observed that, with the increase of hshape orS, the tsshows slightly increasing trend but for S7(2222)nm2or hshape=0:4459 T , tsdrops to 0.43 ns. For further increment ofSorhshape,tsremains constant around 0.43 ns which is close to the theoretical limit (0.4 ns) refers to the study37. This is because the hshape reduces the effective anisotropy and thus reduces the height of energy barrier (energy differ- ence between the initial state and saddle point) which is shown in the FIG. 3(b). Therefore, after decreasing certain height of energy barrier, the magnetization reversal becomes fastest even with the same filed amplitude hmw=0:035 T. Due to the reduction of height of the energy barrier with the hshape, one can expect that the hmwand f0should also decrease with the increase of the anisotropy coefficient hshape and theses find- ings are presented subsequently. Here we present the effect of the shape anisotropy hshape on the microwave amplitude hmwand initial frequency f0of CCMP. Purposely, for each sample Sorhshape, we numerically determine (by tuning hmw,f0and optimal R) the minimally re- quired hmwandf0of CCMP for the the magnetization reversal time window 1 ns. Interestingly, we find that the fast and effi- cient reversal is valid for a wide range of initial frequency f0. FIG. 4(a) shows the estimated frequency bands of f0by verti- cal dashed lines for different S( for example, f0=19:917:8 forhshape=0:096 T) for time window 1 ns. So there is a great flexibility in choosing the initial frequency f0which is useful in device application. FIG. 3(d) shows how the minimal f0 (red circles) and hmw(blue square) decrease with the increase ofhshape. The decay of f0is expected as hshape reduces ef- fective anisotropy (refers to Eq.(2)). For more justification, in same FIG. 3(d), theoretically resonant frequency (black solid line) and simulated minimal frequency (red circles) as func- tion of hshape indicate the agreement. The minimal frequency f0always smaller than the theoretical =resonant frequency. Moreover, the decreasing trend of hmwwith hshape can be attributed by the same reason as the height of the energy bar- 1 2 3 4 5 6 72 4 6 70 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s53/s49/s48/s49/s53/s50/s48/s32/s102 /s111 /s32/s102 /s111/s32/s40/s84/s104/s101/s111/s114/s101/s116/s105/s99/s97/s108/s41 /s32/s72 /s109/s119 /s104 /s115/s104/s97/s112/s101/s32/s40/s84 /s41/s102 /s111/s32/s40/s71/s72/s122/s41 /s48/s46/s48/s50/s48/s48/s46/s48/s50/s53/s48/s46/s48/s51/s48/s104 /s109/s119/s32/s40/s84/s41(a) (b) (c) (d) /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s52/s53/s54/s55/s56/s57/s116 /s115/s32/s40/s110/s115/s41 /s104 /s115/s104/s97/s112/s101/s32/s40/s84 /s41FIG. 3. (a) Temporal evolution of mzinduced by CCMP (with hmw= 0:035 T fixed) for different cross-sectional area, S. (b) The energy landscape Ealong the line f=0. The symbols i and s represent the initial state and saddle point. (c) tsas a function of hshape. (d) The minimal f0(red dotted) and hmw(blue square) as a function of hshape while switching time window 1 ns. 0 1 32 4 5 760 1 2 3(a) (b) FIG. 4. (a) Minimal switching time tsas a function of estimated fo of CCMP with fixed hmwandRcorresponding to different S. (b) Minimal tsas a function of Gilbert damping afor different S. rier decreases with hshape (refers to FIG. 3(b)). For the larger hshape or lower height of energy barrier, the smaller microwave field hmwcan induce the magnetization reversal swiftly. The Gilbert damping parameter, ahas also a crucial effect on the magnetization magnetization dynamics and hence re- versal process and reversal time53–55. In case of CCMP-driven5 /s45/s49 /s48 /s49/s45/s49/s48/s49 /s45/s49/s48/s49 /s109 /s122 /s109/s121 /s109 /s120/s45/s49 /s48 /s49/s45/s49/s48/s49 /s45/s49/s48/s49 /s109 /s122 /s109/s121 /s109 /s120(a) (b) FIG. 5. Magnetization reversal trajectories of biaxial shape (10 108)nm3driven by CCMP for (a) a=0:010. (b) a=0:045. magnetization reversal, smaller (larger) ais preferred while the magnetization climbing the energy barrier (the magnetiza- tion goes down to stable states). Therefore, it is meaningful to find the optimal afor samples of different Sat which the re- versal is fastest. For fixed hmw=0:035 T, using the optimal f0 andRcorresponding to S0,S1,S2andS3, we study the CCMP- driven magnetization reversal as a function of afor different S. FIG. 4(b) shows the dependence of switching time on the Gilbert damping for different S. For each S, there is certain value or range of afor which the switching time is minimal. For instance, the switching time is lowest at a=0:045 for the sample of S1=1010 nm2. To be more clear, one may look at FIG. 5(a) and FIG. 5(b) which show the trajectories of magnetization reversal for a=0:01 and a=0:045 respec- tively and observed that for a=0:045, the reversal path is shorter. This is because, after crossing over the energy bar- rier, the larger damping dissipates the magnetization energy promptly and thus it leads to faster magnetization reversal. This finding suggests that larger ashows faster magnetiza- tion reversal. IV. DISCUSSIONS AND CONCLUSIONS This study investigates the CCMP-driven magnetization re- versal of a cubic sample at zero temperature limit and found that the CCMP with significantly smaller hmw=0:035 T, f0=18:8 GHz and R=0:32 ns1than that of DCMP (i.e., hmw=0:045 T, f0=21 GHz and R= 1.6 ns1) can drive fast magnetization reversal. Since the frequency change of CCMP closely matches the magnetization precession frequency and thus it leads fast magnetization reversal with lower energy- cost. Then we study the influence of demagnetization field = shape anisotropy field hshape, on magnetization reversal pro- cess and the optimal parameters of CCMP. Interestingly we find that, with the increase of hshape, the parameters hmwand f0of CCMP decreases with increasing hshape. So, we search a set of minimal parameters of CCMP for the fast ( ts1 ns) magnetization reversal of the sample 22  228 nm3, and estimated as hmw=0:03 T, f0=7:7 GHz andR=0:24 ns1which are (significantly smaller than that of DCMP) useful for device application. This is happened because with increase of hshape, the effective anisotropy field decreases and thus the energy barrier (which separates two stable states) decreases. In addition, it is observed that thematerials with the larger damping are better for fast magneti- zation reversal. There is a recent study56reported that ther- mal effect assists the magnetization reversal i.e., thermal ef- fect reduces the controlling parameters of chirp microwave field pulse. Thus, it is expected that the parameters of CCMP also might be reduced further at room temperature. To gener- ate such a cosine down-chirp microwave pulse, several recent technologies46,47are available. Therefore, the strategy of the cosine chirp microwave chirp driven magnetization reversal and other findings may lead to realize the fast and low-cost memory device. ACKNOWLEDGMENTS This work was supported by the Ministry of Education (BANBEIS, Grant No. SD2019972). X. S. W. acknowledges the support from the Natural Science Foundation of China (NSFC) (Grant No. 11804045) and the Fundamental Research Funds for the Central Universities. Appendix A: Calculation of ˙e In this Appendix, we show the details of the derivation of ˙e in Eq. 4. The rate of change of hmwis ˙hmw=dhmw dt =d dt(hmw[cosf(t)ˆx+sinf(t)ˆy]) =hmw[sinf(t)ˆx+cosf(t)ˆy]df dt =hmw[sinf(t)ˆx+cosf(t)ˆy]f(t) td dtf(t) t t The magnetization is given by m=mxˆx+myˆy =sinq(t)cosfm(t)ˆx+sinq(t)sinfm(t)ˆy where q(t)is the polar angle and fm(t)is the azimuthal angle of the magnetization m. Substituting mxand˙hmwin Eq. 3, we get, ˙e=m
˙hmw =hmwsinq(t)[sinf(t)cosfm(t)+cosf(t)sinfm(t)]
f(t) td dtf(t) t t =hmwsinq(t)sin(f(t)fm(t))f(t) td dtf(t) t t Defining F(t) =fm(t)f(t), we have finally ˙e=hmwsinq(t)sinF(t)f(t) td dtf(t) t t whereh f(t) td dt f(t) t ti represents w(t).6 REFERENCES 1S. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, “Monodisperse FePt nanoparticles and ferromagnetic FePt nanocrystal superlattices,” sci- ence287, 1989–1992 (2000). 2S. Woods, J. Kirtley, S. Sun, and R. Koch, “Direct investigation of su- perparamagnetism in Co nanoparticle films,” Physical review letters 87, 137205 (2001). 3D. Zitoun, M. Respaud, M.-C. Fromen, M. J. Casanove, P. Lecante, C. Amiens, and B. Chaudret, “Magnetic enhancement in nanoscale CoRh particles,” Physical review letters 89, 037203 (2002). 4B. Hillebrands and K. Ounadjela, Spin dynamics in confined magnetic structures I & II , V ol. 83 (Springer Science & Business Media, 2003). 5S. Mangin, D. Ravelosona, J. Katine, M. Carey, B. Terris, and E. E. 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1710.04766v1.Mode_Dependent_Damping_in_Metallic_Antiferromagnets_Due_to_Inter_Sublattice_Spin_Pumping.pdf

Mode-Dependent Damping in Metallic Antiferromagnets Due to Inter-Sublattice Spin Pumping Qian Liu,1,H. Y. Yuan,2,Ke Xia,1, 3and Zhe Yuan1,y 1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 Beijing, China 2Department of Physics, South University of Science and Technology of China, Shenzhen, Guangdong, 518055, China 3Synergetic Innovation Center for Quantum Eects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China (Dated: October 16, 2017) Damping in magnetization dynamics characterizes the dissipation of magnetic energy and is essen- tial for improving the performance of spintronics-based devices. While the damping of ferromagnets has been well studied and can be articially controlled in practice, the damping parameters of an- tiferromagnetic materials are nevertheless little known for their physical mechanisms or numerical values. Here we calculate the damping parameters in antiferromagnetic dynamics using the gen- eralized scattering theory of magnetization dissipation combined with the rst-principles transport computation. For the PtMn, IrMn, PdMn and FeMn metallic antiferromagnets, the damping coef- cient associated with the motion of magnetization ( m) is one to three orders of magnitude larger than the other damping coecient associated with the variation of the N eel order ( n), in sharp contrast to the assumptions made in the literature. Damping describes the process of energy dissipation in dynamics and determines the time scale for a nonequi- librium system relaxing back to its equilibrium state. For magnetization dynamics of ferromagnets (FMs), the damping is characterized by a phenomenological dissipa- tive torque exerted on the precessing magnetization [1]. The magnitude of this torque that depends on material, temperature and magnetic congurations, has been well studied in experiment [2{10] and theory [11{16]. Recently, magnetization dynamics of antiferromagnets (AFMs) [17{20], especially that controlled by an electric or spin current [21{32], has attracted lots of attention in the process of searching the high-performance spintronic devices. However, the understanding of AFM dynamics, in particular the damping mechanism and magnitude in real materials, is quite limited. Magnetization dynam- ics of a collinear AFM can be described by two coupled Landau-Lifshitz-Gilbert (LLG) equations corresponding to the precessional motion of the two sublattices, respec- tively [33], i.e. ( i= 1;2) _mi= mihi+imi_mi; (1) where is the gyromagnetic ratio, miis the magnetiza- tion direction on the i-th sublattice and _mi=@tmi.hi is the eective magnetic eld on mi, which contains the anisotropy eld, the external eld and the exchange eld arising from the magnetization on the both sublattices. The last contribution to himakes the dynamic equation of one sublattice coupled to the equation of the other one. Specically, if the free energy of the AFM is given by the following formF[m1;m2]0MsVE[m1;m2] with the permeability of vacuum 0, the magnetization on each sublatticeMsand the volume of the AFM V, one has hi=E=mi.iin Eq. (1) is the damping param- eter representing the dissipation rate of the magnetiza-tionmi. Due to the sublattice permutation symmetry, the damping magnitudes of the two sublattices should be equal. This approach has been used to investigate the AFM resonance [33, 34], temperature gradient induced domain wall (DW) motion [35] and spin-transfer torques in an AFMjFM bilayer [36]. An alternative way to deal with the AFM dynamics is introducing the net magnetization mm1+m2and the N eel order nm1m2so that the precessional motion ofmandncan be derived from the Lagrangian equa- tion [26]. The damping eect is then included articially with two parameters mandnthat characterize the dissipation rate of mandn, respectively. This approach is widely used to investigate spin super uid in an AFM insulator [37, 38], AFM nano-oscillator [39], and DW mo- tion induced by an electrical current [26, 40], spin waves [41] and spin-orbit torques [42, 43]. Using the above def- initions of mandn, one can reformulate Eq. (1) and derive the following dynamic equations _n= ( hmm_m)n+ ( hnn_n)m;(2) _m= ( hmm_m)m+ ( hnn_n)n;(3) where hnandhmare the eective magnetic elds exerted onnandm, respectively. They can also be written as the functional derivative of the free energy [26, 41], i.e. hn=E=nandhm=E=m. The damping pa- rameters in Eqs. (1{3) have the relation n=m= 1=2 =2=2 [36]. Indeed, the assumption m=nis commonly adopted in the theoretical study of AFM dy- namics with only a few exceptions, where mis ignored in the current-induced skyrmion motion in AFM materials [44] and the magnon-driven DW motion [45]. However, the underlying damping mechanism of an AFM and the relation between mandnhave not been fully justied yet [46, 47].arXiv:1710.04766v1 [cond-mat.mtrl-sci] 13 Oct 20172 In this paper, we generalize the scattering theory of magnetization dissipation in FMs [48] to AFMs and cal- culate the damping parameters from rst-principles for metallic AFMs PtMn, IrMn, PdMn and FeMn. The damping coecients in an AFM are found to be strongly mode-dependent with mup to three orders of magni- tude larger than n. By analyzing the dependence of damping on the disorder and spin-orbit coupling (SOC), we demonstrate that narises from SOC in analog to the Gilbert damping in FMs, while mis dominated by the spin pumping eect between sublattices. Theory.| In analogue to the scattering theory of mag- netization dissipation in FMs [48], the damping parame- ters in AFMs, nandm, can be expressed in terms of the scattering matrix. Following the previous denition of the free energy, the energy dissipation rate of an AFM reads _E=0MsV_E=0MsV E m
_mE n
_n =0MsV(hm
_m+hn
_n): (4) By replacing the eective elds hmandhnby the time derivative of magnetization order and N eel order using Eq. (2) and (3), one arrives at [49] _E=0MsV n_n2+m_m2
: (5) If we place an AFM between two semi-innite nonmag- netic metals, the propagating electronic states coming from the metallic leads are partly re ected and trans- mitted. The probability amplitudes of the re ection and transmission form the so-called scattering matrix S[50]. For such a scattering structure with only the order pa- rameter nof the AFM varying in time (see the insets of Fig. 1), the energy loss that is pumped into the reservoir is given by _E=~ 4Tr _S_Sy =~ 4Tr@S @n@Sy @n _n2Dn_n2:(6) Here we dene Dn(~=4)Tr[(@S=@n)(@Sy=@n)]. Com- paring Eqs. (S7) and (6), we obtain Dn=0MsA nL; (7) where we replace the volume Vby the product of the cross-sectional area Aand the length Lof the AFM. We can express min the same manner, Dm=0MsA mL (8) withDm(~=4)Tr[(@S=@m)(@Sy=@m)]. Using Eqs. (7) and (8), we calculate the energy dissipation as a function of the length Land extract the damping param- etersn(m)via a linear least squares tting. Note thatthe above formalism can be generalized to include non- collinear AFM, such as DWs in AFMs, by introducing the position-dependent order parameters n(r) and m(r). It can also be extended for the AFMs containing more than two sublattices, which may not be collinear with one another [51]. For the latter case, one has to redene the proper order parameters instead of nandm[52]. First-principles calculations.| The above formalism is implemented using the rst-principles scattering calcu- lation and is applied here in studying the damping of metallic AFMs including PtMn, IrMn, PdMn and FeMn. The lattice constants and magnetic congurations are the same as in the reported rst-principles calculations [53]. Here we take tetragonal PtMn as an example to illustrate the computational details. A nite thickness ( L) of PtMn is connected to two semi-innite Au leads along (001) di- rection. The lattice constant of Au is made to match that of the aaxis of PtMn. The electronic structures are obtained self-consistently within the density functional theory implemented with a minimal basis of the tight- binding linear mun-tin orbitals (TB LMTOs) [54]. The magnetic moment of every Mn atom is 3.65 Band Pt atoms are not magnetized. To evaluate nandm, we rst construct a lateral 1010 supercell including 100 atoms per atomic layer in the scattering region, where the atoms are randomly dis- placed from their equilibrium lattice sites using a Gaus- sian distribution with the root-mean-square (RMS) dis- placement
[15, 55]. The value of
is chosen to repro- duce typical experimental resistivity of the corresponding bulk AFM. The scattering matrix Sare obtained using a rst-principles \wave-function matching" scheme that is also implemented with TB LMTOs [56] and its derivative is obtained by nite-dierence method [49]. Figure 1(a) shows the calculated energy pumping rate Dnof PtMn as a function of Lfornalong thecaxis with
=a= 0:049. The total pumping rate (solid sym- bols) increases linearly with increasing the volume of the AFM. A linear least squares tting yields n= (0:670:02)103, as plotted by the solid line. The nite intercept of the solid line corresponding to the interface- enhanced energy dissipation, which is essentially the spin pumping eect at the AFM jAu interface [57, 58]. The N eel order induced damping ncompletely results from spin-orbit coupling (SOC). If we articially turn SOC o, the calculated pumping rate is independent of the volume of the AFM indicating n= 0. This is because the spin space is decoupled from the real space without SOC and the energy is then invariant with respect to the direction ofn. The spin pumping eect is nearly unchanged by the SOC. The energy pumping rate Dmof PtMn with nalong thecaxis is plotted in Fig. 1(b), where we nd three important features. (1) The extracted value of m= 0:590:02, which is nearly 1000 times larger than n. (2) Turning SOC o only slightly increases the calculated3 0 5 10 15 20 25 30L (nm)0102030γDm/(µ0Ms A) (nm)0 0.5 1SOC Factor00.40.8αm103αn 0.020.030.04γDn/(µ0Ms A) (nm)PtMn1Mn2 m1 m2 m1 m2 ξSO=0 ξSO=0(a) (b) ξSO≠0 ξSO≠0abc FIG. 1. Calculated energy dissipation rate as a function of the length of PtMn due to variation of the order parameters n(a) and m(b).Ais the cross-sectional area of the lat- eral supercell. Arrows in each panels illustrate the dynamical modes of the order parameters. The empty symbols are cal- culated without spin-orbit interaction. The inset of panel (a) shows atomic structure of PtMn with collinear AFM order. The inset in (b) shows calculated nandmas a function of the scaled SOC strength. The factor 1 corresponds to the real SOC strength that is determined by the derivative of the self-consistent potentials. mindicating that SOC is not the main dissipative mech- anism ofm. The dierence between the solid and empty circles in Fig. 1(b) can be attributed to the SOC-induced variation of electronic structure near the Fermi level. To see more clearly the dierent in uence of SOC on mand n, we plot in the inset of Fig. 1(b) the calculated damp- ing parameters as a function of SOC strength. Indeed, as the SOC strength SOis articially tuned from its real value to zero, ndecreases dramatically and tends to vanish atSO= 0, whilemis less sensitive to SOthan n. (3) The intercepts of the solid and dashed lines are both vanishingly small indicating that this specic mode does not pump spin current into the nonmagnetic leads. The pumped spin current from an AFM generally reads Ipump s/n_n+m_m[58]. For the mode depicted in Fig. 1(b), one has _n= 0 and _mkmsuch that Ipump s = 0. To explore the disorder dependence of the damping pa- rametersnandm, we further perform the calculation by varying the RMS of atomic displacements
. Fig- ure 2(a) shows that the calculated resistivity increases monotonically with increasing
. The resistivity cwith nalongcaxis is lower than awithnalongaaxis. 0.51.01.52.0αn (10-3)(a)80160240ρ (µΩ cm)n//a(b)n//c4.6 5.4 6.2∆/a (10-2)01020AMR (%) 4.6 5.0 5.4 5.8 6.2∆/a (10-2)0.20.40.60.8αm4 6 8 10 12σ (105Ω-1 m-1)0.20.50.8αm(b)(c)FIG. 2. Calculated resistivity (a) and damping parameters n(b) andm(c) of PtMn as a function of the RMS of atomic displacements. The red squares and black circles are calculated with nalongaaxis andcaxis, respectively. The inset of (a) shows the calculated AMR. mis replotted as a function of conductivity in the inset of (c). The blue dashed line illustrates the linear dependence. The anisotropic magnetoresistance (AMR) dened by (ac)=cis about 10%, which slightly decreases with increasing
, as plotted in the inset of Fig. 2(a). The large AMR in PtMn is useful for experimental detection of the N eel order. The calculated AMR seems to be an order of magnitude larger than the reported values in lit- erature [59{61]. We may attribute the dierence to the surface scattering in thin-lm samples and other types of disorder that have been found to decrease the AMR of ferromagnetic metals and alloys [62]. nof PtMn plotted in Fig. 2(b) is of the order of 103, which is comparable with the magnitude of the Gilbert damping of ferromagnetic transition metals [2{4, 15]. For nalongaaxis,nshows a weak nonmonotonic depen- dence on disorder, while nfornalongcaxis increases monotonically. With the relativistic SOC, the electronic structure of an AFM depends on the orientation of n. When nvaries in time, the occupied energy bands may be lifted above the Fermi level. Then a longer relax- ation time (weaker disorder) gives rise to a larger energy dissipation, corresponding to the increase in nwith de- creasing
at small
. It is analogous to the intraband transitions accounting for the conductivity-like behavior of Gilbert damping at low temperature in the torque- correlation model [11, 12]. Suciently strong disorder4 renders the system isotropic and the variation of ndoes not lead to electronic excitation but scattering of conduc- tion electrons by disorder still dissipates energy into the lattice through SOC. The higher the scattering rate, the larger is the energy dissipation rate corresponding to the contribution of the interband transitions [11, 12]. There- fore,nshares the same physical origin as the Gilbert damping of metallic FMs. The value of mis about three orders of magnitude larger than nand it decreases monotonically with in- creasing the structural disorder, as shown in Fig. 2(c). This remarkable dierence can be attributed to the en- ergy involved in the dynamical motion of mandn. While the precession of nonly changes the magnetic anisotropy energy in an AFM, the variation of mchanges the ex- change energy that is in magnitude much larger than the magnetic anisotropy energy. Physically, mcan be understood in terms of spin pumping [63, 64] between the two sublattices of an AFM. The sublattice m2pumps a spin current that can be ab- sorbed by m1resulting in a damping torque exerted on m1as0m1[m1(m2_m2)]. Here0is a dimen- sionless parameter to describe the strength of the spin pumping. This torque can be simplied to be 0m1_m2 by neglecting the high-order terms of the total magne- tization m. In addition, the spin pumping by m1also contributes to the damping of the sublattice m1that is equivalent to a torque 0m1_m1exerted on m1. Tak- ing the inter-sublattice spin pumping into account, we are able to derive Eqs. (2) and (3) and obtain the damping parameters n=0=2 andm= (0+ 20)=2 [49]. Here 0is the intrinsic damping due to SOC for each sublat- tice. It is worth noting that the spin pumping strength within a metal is proportional to its conductivity [65{67]. We replotmas a function of conductivity in the inset of Fig. 2(c), where a general linear dependence is seen for bothnalongaaxis andcaxis. We list in Table I the calculated ,nandmfor typical metallic AFMs including PtMn, IrMn, PdMn and FeMn. For IrMn, mis only 10 times larger than n, whilemof the other three materials are about three orders of magnitude larger than their n. TABLE I. Calculated resistivity and damping parameters for the N eel order nalongaaxis andcaxis. AFM n( cm)n(103)m PtMnaaxis 119 5 1.60 0.02 0.49 0.02 caxis 108 4 0.67 0.02 0.59 0.02 IrMnaaxis 116 2 10.5 0.2 0.10 0.01 caxis 116 2 10.2 0.3 0.10 0.01 PdMnaaxis 120 8 0.16 0.02 1.1 0.10 caxis 121 8 1.30 0.10 1.30 0.10 FeMnaaxis 90 1 0.76 0.04 0.38 0.01 caxis 91 1 0.82 0.03 0.38 0.01 0 10 20 30 40Hext (kOe)00.040.080.12∆ω (THz)2.0 2.4ω (THz)-1.6 -1.2-Im χ (arb.units)Hext=20 kOe αm=αnαm=103αnm1 m2 m1 m2 HextHext ωL //// ωR ωL ωRFIG. 3. Linewidth of AFMR as a function of the external magnetic eld. The black dashed lines and red solid lines are calculated with m=nandm= 103n, respectively. Inset: the imaginary part of susceptibility as a function of the frequency for the external magnetic eld Hext= 20 kOe andm= 103n. The cartoons illustrate the corresponding dynamical modes. Here we use HE= 103kOe,HA= 5 kOe andn= 0:001. Antiferromagnetic resonance.| Keer and Kittel for- mulated antiferromagnetic resonance (AFMR) with- out damping [33] and determined the resonant fre- quencies that depend on the external eld Hext, ex- change eld HEand anisotropy eld HA,!res= h Hextp HA(2HE+HA)i . Here we follow their ap- proach, in which Hextis applied along the easy axis and the transverse components of m1andm2are supposed to be small. Taking both the intrinsic damping due to SOC and spin pumping between the two sublattices into account, we solve the dynamical equations of AFMR and nd the frequency-dependent susceptibility (!) that is dened by n?(!) =(!)
h?(!). Here n?andh? are the transverse components of the N eel order and mi- crowave eld, respectively. The imaginary part of the diagonal element of (!) withHext= 20 kOe is plotted in the inset of Fig. 3, where two resonance modes can be identied. The precessional modes for the positive ( !R) and negative frequency ( !L) are schematically depicted in Fig. 3. The linewidth of the AFMR
!can be deter- mined from the imaginary part of the (complex) eigen- frequency [68] by solving det j1(!)j= 0 and is plotted in Fig. 3 as a function of Hext. Without Hext, the two modes have the same linewidth. A nite external eld increases the linewidth of !Rand decreases that of !L, both linearly. By including the spin pumping between two sublattices, both the linewidth at Hext= 0 and the slope of
!as a function of Hextincrease by a factor of about 3.5. It indicates that the spin pumping eect between the two sublattices plays an important role in the magnetization dynamics of metallic AFMs.5 Conclusions.| We have generalized the scattering the- ory of magnetization dissipation in FMs to be applicable for AFMs. Using rst-principles scattering calculation, we nd the damping parameter accompanying the mo- tion of magnetization ( m) is generally much larger than that associated with the motion of the N eel order ( n) in metallic AFMs PtMn, IrMn, PdMn and FeMn. While narises from the spin-orbit interaction, mis mainly contributed by the spin pumping between the two sublat- tices in an AFM via exchange interaction. Taking AFMR as an example, we demonstrate that the linewidth can be signicantly enhanced by the giant value of m. Our nd- ings suggest that the magnetization dynamics of AFMs shall be revisited with the damping eect properly in- cluded. 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B 94, 064415 (2016). [68] A. Saib, Modeling and design of microwave devices based on ferromagnetic nanowires (Presses Universitaires du Louvain, 2004).1 Supplementary Material for \Mode-Dependent Damping in Metallic Antiferromagnets Due to Inter-Sublattice Spin Pumping" Qian Liu,1;H. Y. Yuan,2;Ke Xia,1;3and Zhe Yuan1;y 1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China 2Department of Physics, Southern University of Science and Technology of China, Shenzhen, Guangdong, 518055, China 3Synergetic Innovation Center for Quantum Eects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China In the Supplemental Material, we present the detailed derivation of the energy pumping arising from antiferromagnetic dynamics, the implementation of calculating the derivatives of scattering matrix and derivation of dynamic equations of nand mincluding the spin pumping between sublattices. DERIVATION OF ENERGY DISSIPATION IN ANTIFERROMAGNETIC DYNAMICS We consider a collinear antiferromagnet (AFM) with two sublattices, both of which have the magnetization Ms. The magnetization directions are denoted by the unit vectors m1andm2. Then we are able to dene the total magnetization m=m1+m2and the N eel order parameter n=m1m2. The dynamic equations of mandncan be written as [S1, S2] _m= (mhm+nhn) +mm_m+nn_n; (S1) _n= (mhn+nhm) +mn_m+nm_n: (S2) Herehmandhnare the eective elds acting on the total magnetization and the N eel order. Specically, if the free energy is written as F=0MsVE, where0is the vacuum permeability, Vis the volume of the AFM, and Eis a reduced free energy density, one has [S1] hm=E m;andhn=E n: (S3) In Eqs. (S1) and (S2), mandnare used to characterize the damping due to the variation of the magnetization and the N eel order, respectively. Ifmandnare the only time-varying parameters in the system, the energy dissipation can be represented by _E=_F=0MsV_E =0MsV _m
 E m +_n
 E n =0MsV(_m
hm+_n
hn): (S4) We then insert the dynamic Eqs. (S1) and (S2) into the above Eq. (S4) and obtain _E 0MsV= [ (mhm+nhn) +mm_m+nn_n]
hm +[ (mhn+nhm) +mn_m+nm_n]
hn = nhn
hm+ (mm_m+nn_n)
hm nhm
hn+ (mn_m+nm_n)
hn = (mm_m+nn_n)
hm+ (mn_m+nm_n)
hn =m(m_m
hm+n_m
hn) +n(n_n
hm+m_n
hn) =m(hmm
_m+hnn
_m) +n(hmn
_n+hnm
_n) =m(hmm+hnn)
_m+n(hmn+hnm)
_n =m1 (_mmm_mnn_n)
_m+n1 (_nmn_mnm_n)
_n =m _m2mn n_n
_m+n _n2nm n_m
_n =1 m_m2+n_n2
: (S5)2 Left Lead Right Lead Scattering Region !O!IOIm1 m2 FIG. S1. Schematic illustration of the scattering geometry that is used in the rst-principles calculations. Since both the left and right leads are semi-innite with periodic crystalline structure, the propagating (incoming and outgoing) Bloch states can be obtained by solving the Kohn-Sham equation self-consistently. Then the transmission and re ection coecients can be solved using the numerical technique called \wave function matching" [S3]. Therefore the energy dissipation during antiferromagnetic dynamics can be eventually obtained _E=0MsV m_m2+n_n2
: (S6) CALCULATING THE DERIVATIVE OF SCATTERING MATRIX Noting that the energy dissipation in a scattering geometry, i.e. the left lead{scattering region{the right lead (see Fig. S1), can be written in terms of the parametric pumping [S4] _E=~ 4Tr _S_Sy : (S7) HereSis the scattering matrix. Supposing only the magnetic order (=mor=n) of the system is varying in time, one can rewrite Eq. (S7) as _E=~ 4Tr@S @@Sy @ _2D_2: (S8) The quantity Dis generally a positive-denite and symmetric tensor [S5] with its elements dened by Dij =~ 4Tr@S @i@Sy @j : (S9) Noting that @S=@i,@Sy=@jand their product are all matrices, so we rewrite Eq. (S9) in terms of the specic matrix elements as Dij =~ 4X @S @i@Sy @j =~ 4X X @S @i@ Sy
 @j=~ 4X X @S @i@S @j : (S10) In particular, for i=j, we have the diagonal elements of D Dii =~ 4X ;@S @i2 ; (S11) which is a real number. All the remaining task is to numerically calculate the derivatives of the scattering matrix elements@S=@i. In the following, we take =nas an example and illustrate the calculation of @S=@i. Considering the N eel order along z-axis, i.e. m1=m2= ^zandn= 2^z, one can calculate the scattering matrix S(n). Then we add an innitesimal transverse component
n=^xonto the N eel order so that the new N eel order becomes n0= 2^z+^x. (In practice, we nd that the calculated results are well converged with in the range of 103{105.) Under such a magnetic conguration, we redo the scattering calculation to nd another scattering matrix S(n0). The derivatives of the matrix element Scan be obtained by @S @nx=S0 S : (S12)3 In the same manner, we can nd another scattering matrix S00atn00= 2^z+^yand consequently we have @S @ny=S00 S : (S13) Finally, we nd that the calculated o-diagonal elements Dxy n(m)andDyx n(m)are much smaller than the diagonal elements Dxx n(m)andDyy n(m). The latter two are nearly the same. So we take their average in practice, i.e. Dn= (Dxx n+Dyy n)=2 andDm= (Dxx m+Dyy m)=2. DYNAMICAL EQUATIONS WITH INTER-SUBLATTICE SPIN PUMPING We start from the coupled dynamical equations of an AFM with the sublattice index i= 1;2, _mi= mihi+0mi_mi: (S14) Herehiis the eective eld exerted on mi, which can be calculated from the functional derivative of the free energy Fas hi=1 0MsVF mi: (S15) 0is the damping parameter, which must be equal for m1andm2because of the permutation symmetry. Now we consider the spin pumping eect that discussed in the main text. The spin pumping by the sublattice m1contributes a dissipative torque 0m1_m1that is exerted on m1. Here0is a dimensionless parameter to quantify the magnitude of the inter-sublattice spin pumping. The pumped spin current by m1can be absorbed by m2resulting in a damping- like torque m2[m2(0m1_m1)]0m2_m1, which is exerted on m2. In the same manner, we can identify two torques due to the spin pumping of m2:0m1_m2exerted on m1and0m2_m2exerted on m2. Eventually, we obtain the coupled dynamical equations by including the inter-sublattice spin pumping as _m1= m1h1+ (0+0)m1_m1+0m1_m2; _m2= m2h2+ (0+0)m2_m2+0m2_m1: (S16) The above form of the dynamical equations can be rigorously derived using the Rayleigh functional to describe the dissipation [S6]. In the following, we rewrite Eq. (S16) into the dynamical equations of the total magnetization m=m1+m2and the N eel order n=m1m2. The eective eld hican be transformed as h1=1 0MsVF m1=1 0MsVF m@m @m1+F n@n @m1 =hm+hn; h2=1 0MsVF m2=1 0MsVF m@m @m2+F n@n @m2 =hmhn; (S17) where we have dened hm=1 0MsVF m; hn=1 0MsVF n: (S18) Then we nd _m=_m1+_m2= (m1h1+m2h2) + (0+0) (m1_m1+m2_m2) +0(m1_m2+m2_m1):(S19) Using Eq. (S17), the rst term in the right-hand side of Eq. (S19) can be simplied as (m1h1+m2h2) = m+n 2(hm+hn) +mn 2(hmhn) = (mhm+nhn):(S20)4 The second and the third terms in the right-hand side of Eq. (S19) can be simplied, respectively, as (0+0) (m1_m1+m2_m2) = (0+0)m+n 2_m+_n 2+mn 2_m_n 2 =0+0 2(m_m+n_n); (S21) and 0(m1_m2+m2_m1) =0m+n 2_m_n 2+mn 2_m+_n 2 =0(m_mn_n): (S22) Finally, Eq. (S19) is rewritten as _m= (mhm+nhn) +0 2+0 m_m+0 2n_n: (S23) The dynamical equation of the N eel order ncan be obtained in the same way _n= (mhn+nhm) +0 2+0 n_m+0 2m_n: (S24) Comparing Eqs. (S23) and (S24) with Eqs. (S1) and (S2), we can identify the relations of the damping parameters, i.e. m=0 2+0;andn=0 2: (S25) The above relations naturally show the spin pumping eect and is consistent with our rst-principles calculations. These authors contributed equally to this work. yCorresponding author: zyuan@bnu.edu.cn [S1] K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, \Phenomenology of current-induced dynamics in antiferromagnets," Phys. Rev. Lett. 106, 107206 (2011). [S2] E. V. Gomonay and V. M. Loktev, \Spintronics of antiferromagnetic systems (review article)," Low Temp. Phys. 40, 17 (2014). [S3] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W. Bauer, \First-principles scattering matrices for spin transport," Phys. Rev. B 73, 064420 (2006). [S4] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, \Optimal quantum pumps," Phys. Rev. Lett. 87, 236601 (2001). [S5] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, \Magnetization dissipation in ferromagnets from scattering theory," Phys. Rev. B 84, 054416 (2011). [S6] H. Y. Yuan, Qian Liu, Ke Xia, Zhe Yuan, and X. R. Wang, \Proper dissipative torques in antiferromagnetic dynamics," unpublished (2017).

1706.04670v2.Temperature_dependent_Gilbert_damping_of_Co2FeAl_thin_films_with_different_degree_of_atomic_order.pdf

1 Temperature -dependent Gilbert damping of Co 2FeAl thin films with different degree of atomic order Ankit Kumar1*, Fan Pan2,3, Sajid Husain4, Serkan Akansel1, Rimantas Brucas1, Lars Bergqvist2,3, Sujeet Chaudhary4, and Peter Svedlindh1# 1Department of Engineering Sciences, Uppsala University, Box 534, SE -751 21 Uppsala, Sweden 2Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Electrum 229, SE -16440 Kista, Sweden 3Swedish e -Science Research Center, KTH Roy al Institute of Technology, SE -10044 Stockholm, Sweden 4Department of Physics, Indian Institute of Technology Delhi, New Delhi -110016, India ABSTRACT Half-metallicity and low magnetic damping are perpetually sought for in spintronics materials and full He usler alloys in this respect provide outstanding properties . However, it is challenging to obtain the well -ordered half-metallic phase in as -deposited full Heusler alloys thin films and theory has struggled to establish a fundamentals understanding of the temperature dependent Gilbert damping in these systems. Here we present a study of the temperature dependent Gilbert damping of differently ordered as -deposited Co 2FeAl full Heusler alloy thin films. The sum of inter - and intraband electron scattering in conjunction with the finite electron lifetime in Bloch states govern the Gilbert damping for the well - ordered phase in contrast to the damping of partially -ordered and disordered phases which is governed by interband electronic scat tering alone. These results, especially the ultralow room temperature intrinsic damping observed for the well -ordered phase provide new fundamental insight s to the physical origin of the Gilbert damping in full Heusler alloy thin films. 2 INTRODUCTION The Co-based full Heusler alloys have gained massive attention over the last decade due to their high Curie temperature and half-metallicity; 100% spin polarization of the density of states at the Fermi level [1 -2]. The room temperature half- metallicity and lo w Gilbert damping make them ideal candidates for magnetoresistive and thermoelectric spintronic devices [3]. Co2FeAl (CFA), which is one of the most studied Co-based Heusler alloys , belongs to the 𝐹𝐹𝐹𝐹 3𝐹𝐹 space group, exhibits half-metallicity and a high C urie temperature (1000 K) [2, 4]. In CFA, half-metallicity is the result of hybridization between the d orbitals of Co and Fe. The d orbitals of Co hybridize resulting in bonding (2e g and 3t 2g) and non- bonding hybrids (2e u and 3t 1u). The bonding hybrids of Co further hybridise with the d orbitals of Fe yielding bonding and anti -bonding hybrids. However, the non-bonding hybrids of Co cannot hybridise with the d orbitals of Fe. The half-metallic gap arises from the separation of non-bonding states, i.e. the conduction band of e u hybrids and the valence band of t 1u hybrids [5, 6]. However, chemical or atomic disorder modifies the band hybridization and results in a reduc ed half-metallicity in CFA. The ordered phase of CFA is the L2 1 phase, which is half -metallic [7]. The partially ordered B2 phase forms when the Fe and Al atoms randomly share their sites, while the disordered phase forms when Co, Fe, and Al atoms randomly share all the sites [5-8]. These chemical disorders strongly influence the physical properties and result in additional states at the Fermi level therefore reducing the half- metallicity or spin polarization [7, 8]. It is challenging to obtain the ordered L2 1 phase of Heusler alloys in as-deposited films, which is expecte d to possess the lowest Gilbert damping as compared to the other phases [4, 9-11]. Therefore, in the last decade several attempts have been made to grow the ordered phase of CFA thin films employing different methods [4, 9- 13]. The most successful attempts used post -deposition annealing to reduce the anti -site disorder by a thermal activation process [4]. The observed value of the Gilbert damping for ordered thin films was found to lie in the range of 0.001-0.004 [7-13]. However, the requirement of post -deposition annealing might not be compatible with the process constraints of spintronics and CMOS devices. The annealing treatment requirement for the formation of the ordered phase can be circumvented by employing energy enhanced growth mechanisms such as io n beam sputtering where the sputtered species carry substantially larger energy, ~20 eV , compared to other deposition techniques [14, 15]. This higher energy of the sputtered species enhances the ad -atom mobility during coalescence of nuclei in the initial stage of the thin film growth, therefore enabling the formation of the ordered phase. Recently we have 3 reported growth of the ordered CFA phase on potentially advantageous Si substrate using ion beam sputtering. The samples deposited in the range of 300°C to 500°C substrate temperature exhibited nearly equivalent I(002)/I(004) Bragg diffraction intensity peak ratio, which confirms at least B2 order ed phase as it is difficult to identify the formation of the L2 1 phase only by X -ray diffraction analysis [16] . Different theoretical approaches have been employed to calculate the Gilbert damping in Co - based full Heusler alloys, including first principle calculations on the ba sis of (i) the torque correlation model [17], (ii) the fully relativistic Korringa -Kohn-Rostoker model in conjunction with the coherent potential approximation and the linear response formalism [8] , and (iii) an approach considering different exchange correlation effects using both the local spin density approximation including the Hubbard U and the local spin density approximation plus the dynamical mean field theory approximation [7]. However, very little is known about the temperature dependence of the Gilbert damping in differently ordered Co-based Heusler alloys and a unifying conse nsus between theoretical and experimental results is still lacking. In this study we report the growth of differently ordered phases, varying from disordered to well-ordered phases, of as -deposited CFA thin films grown on Si employing ion beam sputtering a nd subsequently the detailed temperature dependent measurements of the Gilbert damping. The observed increase in intrinsic Gilbert damping with decreasing temperature in the well -ordered sample is in contrast to the continuous decrease in intrinsic Gilbert damping with decreasing temperature observed for partially ordered and disordered phases. These results are satisfactorily explained by employing spin polarized relativistic Korringa -Kohn- Rostoker band structure calculations in combination with the local spin density approximation. SAMPLES & METHODS Thin films of CFA were deposited on Si substrates at various growth temperatures using ion beam sputtering system operating at 75W RF ion-source power ( 𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖). Details of the deposition process as well as st ructural and magnetic properties of the films have been reported elsewhere [16]. In the present work to study the temperature dependent Gilbert damping of differently ordered phases (L2 1 and B2) we have chosen CFA thin films deposited at 573K, 673K and 773K substrate temperature ( 𝑇𝑇𝑆𝑆) and the corresponding samples are named as LP573K, LP673K and LP773K, respectively. The sample thickness was kept constant at 50 nm and the samples were capped with a 4 nm thick Al layer. The capping layer protects the films by forming a 1.5 nm thin protective layer of Al 2O3. To obtain the A2 disordered CFA 4 phase, the thin film was deposited at 300K on Si employing 100W ion-source power, this sample is referred to as HP300K. Structural and magnetic properties of this film are presented in Ref. [18]. The absence of the (200) diffraction peak in the HP300K sample [18] reveals that this sample exhibits the A2 disordered structure. The appearance of the (200) pea k in the LP series samples clearly indicates at least formation of B2 order [16]. Employing the Webster model along with the analysis approach developed by Takakura et al. [19] we have calculated the degree of B2 ordering in the samples, S B2= �I200 I220⁄ I200full orderI220full order�� , where I200 I220⁄ is the experimentally obtained intensity ratio of the (200) and (220) diffractions and I200full orderI220full order⁄ is the theoretically calculated intensity ratio for fully ordered B2 structure in polycrystalline films [20]. The estimated values of SB2 for the LP573, LP673, and LP773 samples are found to be ∼ 90 %, 90% and 100%, respectively , as presented in Ref. [20]. The I200 I400⁄ ratio of the (200) and (400) diffraction peaks for all LP series samples is ∼ 30 %, which compares well with the theoretical value for perfect B2 order [21, 22]. Here it is important to note that the L21 ordering parameter, SL21, will take different values depending on the degree of B2 ordering. S L21 can be calculated from the I111 I220⁄ peak ratio in conjunction with the SB2 ordering parameter [19]. However, in the recorded grazing incident XRD spectra on the polycrystalline LP samples (see Fig. 1 of Ref. [16]) we did not observe the (111) peak. This could be attributed to the fact t hat theoretical intensity of this peak is only around two percent of the (220) principal peak. The appearance of this peak is typically observed in textured/columnar thicker films [19, 23 ]. Therefore, here using the experimental results of the Gilbert damping, Curie temperature and saturation magnetization, in particular employing the temperature dependence of the Gilbert damping that is very sensitive to the amount of site disorder in CFA films, and comparing with corresponding results obtained from first principle calculations, we provide a novel method for determining the type of crystallographic ordering in full Heusler alloy thin films. The observed values of the saturation magnetization ( µ0MS) and coercivity ( µ0Hci), taken from Refs. [16, 18] are presented in Table I. The temperature dependence of the magnetization was recorded in the high temperature region (300–1000K) using a vibrating sample magnetometer i n an external magnetic field of 𝜇𝜇 0𝐻𝐻=20 mT. An ELEXSYS EPR spectrometer from Bruker equipped with an X -band resonant cavity was used for angle dependent in-plane ferromagnetic resonance (FMR) measurements . For studying the 5 temperature dependent spin dynamics in the magnetic thin films, an in-house built out -of- plane FMR setup was used. The set up, using a Quantum Design Physical Properties Measurement System covers the temperature range 4 – 350 K and the magnetic field range ±9T. The system employs an Agilent N5227A PNA network analyser covering the frequency range 1 – 67 GHz and an in-house made coplanar waveguide. The layout of the system is shown in Fig. 1. The complex transmission coefficient ( 𝑆𝑆21) was recorded as a function of magnetic field for different frequencies in the range 9-20 GHz and different temperatures in the range 50-300 K. All FMR measurements were recorded keeping constant 5 dB power. To calculate the Gilbert damping, we have the used the torque –torque correlation model [7, 24], which includes both intra - and interband transitions. The electronic structure was obtained from the spin polarized relativistic Korringa -Kohn-Rostoker (SPR- KKR) band structure method [24, 25] and the local spin density approximation (LSDA) [26] was used for the exchange correlation potential. Relativistic effects were taken into account by solving the Dirac equation for the electronic states, and the atomic sphere approximation (ASA) was employed for the shape of potentials. The experimental bulk value of the lattice constant [27] was used. The angular momentum cut -off of 𝑙𝑙 𝑚𝑚𝑚𝑚𝑚𝑚 =4 was used in the mu ltiple -scattering expansion. A k-point grid consisting of ~1600 points in the irreducible Brillouin zone was employe d in the self -consistent calculation while a substantially more dense grid of ~60000 points was employe d for the Gilbert damping calculation. The exchange parameters 𝐽𝐽 𝑖𝑖𝑖𝑖 between the atomic magnetic moments were calculated using the magnetic force theorem implemented in the Liechtenstein -Katsnelson -Antropov-Gubanov (LKAG) formalism [28, 29] in order to construct a parametrized mod el Hamiltonian. For the B2 and L2 1 structures, the dominating exchange interactions were found to be between the Co and Fe atoms, while in A2 the Co-Fe and Fe -Fe interactions are of similar size. Finite temperature properties such as the temperature dependent magnetization was obtained by performing Metropolis Monte Carlo (MC) simulations [30] as implemented in the UppASD software [31, 32] using the parametrized Hamiltoni an. The coherent potential approximation (CPA) [33, 34] was ap plied not only for the treatment of the chemical disorder of the system, but also used to include the effects of quasi -static lattice displacement and spin fluctuations in the calculation of the temperature dependent Gilbert damping [35–37] on the basis of linear response theory [38]. RESULTS & DISCUSSION A. Magnetization vs. temperature measurements 6 Magnetization measurements were performed with the ambition to extract values for the Curie temperature ( 𝑇𝑇𝐶𝐶) of CFA films with different degree of atomic order; the results a re shown in Fig. 2(a). Defining 𝑇𝑇𝐶𝐶 as the inflection point in the magnetization vs. temperature curve, the observed values are found to be 810 K, 890 K and 900 K for the LP573K, LP773K and LP673K samples, respectively. The 𝑇𝑇𝐶𝐶 value for the HP300K sample is similar to the value obtained for LP573. Using the theoretically calculated exchange interactions, 𝑇𝑇𝐶𝐶 for different degree of atomic order in CFA varying from B2 to L2 1 can be calculated using MC simul ations. The volume was kept fixed as the degree of order varied between B2 and L2 1 and the data presented here represent the effects of differently ordered CFA phases. To obtain 𝑇𝑇𝐶𝐶 for the different phases, the occupancy of Fe atoms on the Heusler alloy 4a sites was varied from 50% to 100%, corresponding to changing the structure from B2 to L2 1. The estimated 𝑇𝑇𝐶𝐶 values , cf. Fig. 2 (b), monotonously increases from 𝑇𝑇 𝐶𝐶=810 K (B2) to 𝑇𝑇𝐶𝐶=950 K (L2 1). A direct comparison between experimental and calculated 𝑇𝑇𝐶𝐶 values is hampered by the high temperature (beyond 800K) induced structural transition from well -ordered to partially - ordered CFA phase which interferes with the magnetic transition [39, 40]. The irreversible nature of the recorded magnetization vs . temperature curve indicates a distortion of structure for the ordered phase during measurement , even though interface alloying at elevated temperature cannot be ruled out . The experimentally observe d 𝑇𝑇𝐶𝐶 values are presented in Table I. B. In-plane angle dependent FMR measurements In-plane angle dependent FMR measurements were performed at 9.8 GHz frequency for all samples; the resonance field 𝐻𝐻𝑟𝑟 vs. in -plane angle 𝜙𝜙𝐻𝐻 of the applied magnetic field is plotted in Fig. 3. The experimental results have been fitted using the expression [41], 𝑓𝑓= 𝑔𝑔∥𝜇𝜇𝐵𝐵𝜇𝜇0 ℎ��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−𝜙𝜙𝑀𝑀)+2𝐾𝐾𝑐𝑐 𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢 𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻− 𝜙𝜙𝑀𝑀)+ 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒+𝐾𝐾𝑐𝑐 2𝜇𝜇0𝑀𝑀𝑠𝑠(3+cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢 𝜇𝜇0𝑀𝑀𝑠𝑠cos2 (𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��12� , (1) where 𝑓𝑓 is resonance frequency , 𝜇𝜇𝐵𝐵 is the Bohr magneton and ℎ is Planck constant . 𝜙𝜙𝑀𝑀, 𝜙𝜙𝑢𝑢 and 𝜙𝜙𝑐𝑐 are the in -plane directions of the magnetization, uniaxial anisotropy and cubic anisotropy, respectively , with respect to the [100] direction of the Si substrate . 𝐻𝐻𝑢𝑢=2𝐾𝐾𝑢𝑢 𝜇𝜇0𝑀𝑀𝑠𝑠 and 𝐻𝐻𝑐𝑐=2𝐾𝐾𝑐𝑐 𝜇𝜇0𝑀𝑀𝑠𝑠 are the in-plane uniaxial and cubic anisotropy fields , respectively, and 𝐾𝐾𝑢𝑢 and 𝐾𝐾𝑐𝑐 7 are the uniaxial and cubic magnetic anisotrop y constant s, respectively, 𝑀𝑀𝑠𝑠 is the saturation magnetization and 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 is the effective magnetization . By considering 𝜙𝜙 𝐻𝐻 ∼ 𝜙𝜙𝑀𝑀, 𝐻𝐻𝑢𝑢 and 𝐻𝐻𝑐𝑐 <<𝐻𝐻𝑟𝑟<< 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, equation (1) can be simplified as: 𝐻𝐻𝑟𝑟=�ℎ𝑒𝑒 𝜇𝜇0𝑔𝑔∥𝜇𝜇𝐵𝐵�21 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒−2𝐾𝐾𝑐𝑐 𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝐻𝐻−𝜙𝜙𝑐𝑐)−2𝐾𝐾𝑢𝑢 𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝐻𝐻−𝜙𝜙𝑢𝑢) . (2) The extracted cubic anisotropy fields µ 0Hc ≤ 0.22mT are negligible for all the samples. The extracted in -plane Landé splitting factors g∥ and the uniaxial anisotropy fields µ0Hu are presented in T able I. The purpose of the angle dependent FMR measurements was only to investigate the symmetry of the in -plane magnetic anisotropy. Therefore, care was not taken to have the same in -plane orientation of the samples during angle dependent FMR measurements, which explains why the maxima appear at diffe rent angles for the different samples. C. Out-of-plane FMR measurements Field -sweep out -of-plane FMR measurements were performed at different constant temperatures in the range 50K – 300K and at different constant frequencies in the range of 9- 20 GHz. Figure 1(b) shows the amplitude of the complex transmission coefficient 𝑆𝑆21(10 GHz) vs. field measured for the LP673K thin film at different temperatures. The recorded FMR spectra were fitted using the equation [42], 𝑆𝑆21=𝑆𝑆�∆𝐻𝐻2��2 (𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐴𝐴�∆𝐻𝐻2��(𝐻𝐻−𝐻𝐻𝑟𝑟) (𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐷𝐷∙𝑡𝑡, (3) where 𝑆𝑆 represents the coefficient describing the transmitted microwave power, 𝐴𝐴 is used to describe a waveguide induced phase shift contribution which is, however, minute , 𝐻𝐻 is applied magnetic field, ∆𝐻𝐻 is the full-width of half maxim um, and 𝐷𝐷∙𝑡𝑡 describes the linear drift in time (𝑡𝑡) of the recorded signal. The extracted ∆ 𝐻𝐻 vs. frequency at different constant temperatures are shown in Fig. 4 for all the samples. For brevity only data at a few temperatures are plotted. The Gilbert damping was estimated using the equation [42 ], ∆𝐻𝐻=∆𝐻𝐻0+2ℎ𝛼𝛼𝑒𝑒 𝑔𝑔⊥𝜇𝜇𝐵𝐵𝜇𝜇0 (4) where ∆𝐻𝐻0 is the inhomogeneous line -width broadening, 𝛼𝛼 is the experimental Gilbert damping constant , and 𝑔𝑔⊥ is the Landé splitting factor measured employing out -of-plane FMR. The insets in the figures show the temperature dependence of 𝛼𝛼. The effective 8 magnetization ( 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) was estimated from the 𝑓𝑓 vs. 𝐻𝐻𝑟𝑟 curves using out -of-plane Kittel’s equation [43], 𝑓𝑓=𝑔𝑔⊥𝜇𝜇0𝜇𝜇𝐵𝐵 ℎ�𝐻𝐻𝑟𝑟−𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒�, (5) as shown in Fig. 5 . The temperature dependence of 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 and 𝜇𝜇0∆𝐻𝐻0 are shown as insets in each figure . The observed room temperature values of 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 are closely equal to the 𝜇𝜇0𝑀𝑀𝑠𝑠 values obtained from static magnetizat ion measurements, presented in T able I. The extracted values of g⊥ at different temperatures are within error limits constant for all samples. However , the difference between estimated values of g ∥ and g⊥ is ≤ 3%. This difference c ould stem from the limited frequency range used since these values are quite sensitive to the value of 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, and even a minute uncertainty in this quantity can result in the observed small difference between the g∥ and g⊥ values. To obtain the intrinsic Gilbert damping (𝛼𝛼 𝑖𝑖𝑖𝑖𝑖𝑖) all extrinsic contributions to the experimental 𝛼𝛼 value need to be subtracted. In metallic ferromagnets , the intrinsic Gilbert damping is mostly caused by electron magnon scattering, but several other extrinsic co ntributions can also contribute to the experimental value of the damping constant. One contribution is two - magnon scattering which is however minimized for the perpendicular geometry used in this study and therefore this contribution is disregarded [44]. Another contribution is spin- pumping into the capping layer as the LP573K, LP673K and LP773K samples are capped with 4 nm of Al that naturally forms a thin top layer consisting of Al2O3. Since spin pumping in low spin-orbit coupling materials with thickness less than the spin-diffusion length is quite small this contribution is also disregarded in all samples. However, the HP300K sample is capped with Ta and therefore a spin-pumping contribution have been subtracted from the experimental 𝛼𝛼 value ; 𝛼𝛼𝑠𝑠𝑠𝑠= 𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(with Ta capping )−𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(without capping )≈ 1×10−3. The third contribution arises from the inductive coupling between the precessi ng magnetization and the CPW , a reciprocal phenomenon of FMR, known as radiative damping 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 [45]. This damping is directly proportional to the magnetization and thickness of the thin films samples and therefore usually dominates in thicker and/or high magnetization samples. The l ast contribution is eddy current damping ( αeddy) caused by eddy current s in metallic ferromagnetic thin films [ 45, 46]. As per Faraday’s law the time varying magnetic flux density generate s an AC voltage in the metallic ferromagnetic layer and therefore result s in the eddy 9 current damping . Thi s damping is directly proportional to the square of the film thickness and inversely proportional to the resistivity of the sample [ 45]. In contrast to eddy -current damping, 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 is independent of the conductivity of the ferromagnetic layer, hence this damping mechanism is also operati ve in ferromagnetic insulators. Assuming a uniform magnetization of the sample the radiative damping can be expressed as [45], 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 𝜂𝜂𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿𝛿𝛿 2 𝑍𝑍0𝑤𝑤 , (6) where 𝛾𝛾=𝑔𝑔𝜇𝜇𝐵𝐵ℏ� is the gyromagnetic ratio, 𝑍𝑍0 = 50 Ω is the waveguide impedance, 𝑤𝑤 = 240 µm is the width of the waveguide, 𝜂𝜂 is a dimensionless parameter which accounts for the FMR mode profile and depends on boundary conditions, and 𝛿𝛿 and 𝑙𝑙 are the thickness and length of the sample on the waveguide, respectively. The strength of this inductive coupling depends on the inductance of the FMR mode which is determined by the waveguide width, sample length over waveguide, sample saturation magnetization and sample thickness. The dimensions of the LP573K, LP673K and LP773K samples were 6.3×6.3 mm2, while the dimensions of the HP300K sample were 4×4 mm2. Th e 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 damping was estimated experimentally as explained by Schoen et al. [45] by placing a 200 µm thick glass spacer between the waveguide and the sample , which decreases the radiative damping by more than one order magnitude as shown in Fig. 6(a). The measured radiative damping by placing the spacer between the waveguide and the LP773 sample, 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟=𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ𝑖𝑖𝑢𝑢𝑖𝑖 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟 −𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟≈ (2.36 ±0.10×10−3) − (1.57 ±0.20×10−3)= 0.79±0.22×10−3. The estimated value matches well with the calculated value using Eq. (6); 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 0.78 ×10−3. Our results are also analogous to previously reported results on radiative damping [45]. The estimated temperature dependent radiative damping values for all samples are shown in Fig. 6(b). Spin wave precession in ferromagnetic layers induces an AC current in the conducting ferromagnetic layer which results in eddy current damping. It can be expressed as [45, 46], 𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 = 𝐶𝐶𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿2 16 𝜌𝜌 , (7) where 𝜌𝜌 is the resistivity of the sample and 𝐶𝐶 accounts for the eddy current distribution in the sample ; the smaller the value of 𝐶𝐶 the larger is the localization of eddy currents in the sample. The measured resistivity values between 300 K to 50 K temperature range fall in the ranges 1.175 – 1.145 µΩ-m, 1 .055 – 1.034 µΩ -m, 1 .035 – 1.00 µΩ -m, and 1. 45 – 1.41 µΩ -m for the LP573K, LP673 , LP773 and HP300K samples, respect ively. The parameter 𝐶𝐶 was obtained 10 from thickness dependent experimental Gilbert damping constants measured for B2 ordered films, by line ar fitting of 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 vs. 𝛿𝛿2 keeping other parameters constant (cf. Fig. 6(c)). The fit to the data yield ed 𝐶𝐶 ≈ 0.5±0.1. These results are concurrent to those obtained for permalloy thin films [45]. Since the variations of the resistivity and magnetization for the samples are small , we have used the same 𝐶𝐶 value for the estimation of the eddy current damping in all the samples. The estimated temperature dependent values of the eddy current damping are presented in Fig. 6(d). All these contributions have been subtracted from the experimentally observed values of 𝛼𝛼. The estimated intrinsic Gilbert damping 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values so obtained are plotted in Fig. 7(a) for all samples. D. Theoretical results: first principle calculations The calculated temperature dependent intrinsic Gilbert damping for Co 2FeAl phases with different degree of atomic order are shown in Fig. 7(b). The temperature dependent Gilbert damping indicates that the lattice displacements and spin fluctuations contribute differently in the A2, B2 and L2 1 phases. The torque correlation model [47, 48] describes qualitatively two contributions to the Gilbert damping. The first one is the intraband scattering where the band index is always conserved. Since it has a linear dependence on the electron lifetime, in the low temperature regime this term increases rapid ly, it is also known as the conductivity like scattering. The second mechanism is due to interband transitions where the scattering occurs between bands with different indices. Opposite to the intraband scattering, the resistivity like interband scattering with an inverse depe ndence on the electron lifetime increases with increas ing temperature. The sum of the intra - and interband electron scattering contributions gives rise to a non-monotonic dependence of the Gilbert damping on temperature for the L2 1 structure. In contrast to the case for L2 1, only interband scattering is present in the A2 and B2 phases, which results in a monotonic increase of the intrinsic Gi lbert damping with increas ing temperature. This fact is also supported by a previous study [37 ] which showed that even a minute chemical disorder can inhibit the intraband scattering of the system. Our theoretical results manifest that the L2 1 phase has the lowest Gilbert damping around 4.6 × 10−4 at 300 K, and that the value for the B2 phase is only slightly larger at room temperature. According to the torque correlation model, the two main contributions to damping are the spin orbit coupling and the density of states (DOS) at the Fermi level [47 , 48]. Since the spin orbit strength is the same for the different phases it is enough to focus the discussion on the DOS 11 that provide s a qualitative explanation why damping is found lower in B2 and L2 1 structure s compare d to A2 structure. The DOS at the Fermi level of the B2 phase (24.1 states/Ry/f.u; f.u = formula unit ) is only slightly larger to that of the L2 1 phase (20.2 states/Ry/f.u.) , but both are significantly smaller than for the A2 phase (59.6 states/Ry/f.u.) as shown in Fig. 8. The gap in the mi nority spin channel of the DOS for the B2 and L2 1 phases indicate half- meta llicity, while the A2 phase is metallic. The atomically resolved spin polarized DOS indica tes that the Fermi -level states mostly have contr ibutions from Co and Fe atoms. For transition elements such as Fe and Ni, it has been reported that the intrinsic Gilbert damping increases significantly below 100K with decreas ing temperature [37]. The present electronic structure calculations were performed using Green’s functions, which do rely on a phenomenological relaxation time parameter, on the expense that the different contributions to damping cannot be separated eas ily. The reported results in Ref. [37] are by some means similar to our findings of the temperature dependent Gilbert damping in full Heusler alloy films with different degr ee of atomic order. The intermediate states of B2 and L2 1 are more close to the trend of B2 than L2 1, which indicates that even a tiny atomic orde r induced by the Fe and Al site disorder will inhibit the conductivity -like channel in the low temperature region. The theoretically calculated Gilbert damping constants are matching qualitatively with the experimentally observed 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values as shown in Fig. 7. However, the theoretically calculated 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 for the L2 1 phase increases rapidly below 100K, in co ntrast to the experimental results for the well -ordered CFA thin film (LP673K ) indicating that 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 saturates at low temperature. This discrepancy between the theoretical and experimental results can be understood taking into account the low temperature behaviour of the life time τ of Bloch states. The present theoretical model assum ed that the Gilbert damping has a linear dependence on the electron lifetime in intraband transitions which is however correct only in the limit of small lifetime, i.e., 𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏≪1, where q is the magnon wave vector and 𝑣𝑣𝐹𝐹 is the electron Fermi velocity. However, in the low temperature limit the lifetime 𝜏𝜏 increases and as a result of the anomalous skin effect the intrinsic Gilbert damping saturates 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖∝ tan−1𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏𝑞𝑞𝑣𝑣𝐹𝐹� at low temperature [37], which is evident from our experimental results. Remaining discrepancies between theoretical and experimental values of the intrinsic Gilbert damping might stem from the fact that the samples used in the present study are 12 polycrystalline and because of sample imperfections these fil ms exhibit significant inhomogenous line -width broadening due to superposition of local resonance fields. CONCLUSION In summary , we report temperature dependent FMR measurements on as -deposited Co 2FeAl thin films with different degree of atomic order. The degree of atomic ordering is established by comparing experimental and theoretical results for the temperature dependent intrinsic Gilbert damping constant. It is evidenced that the experimentally observed intrinsic Gilbert damping in samples with atomic disorder (A2 and B2 phase samples) decreases with decreasing temperature. In contrast, the atomically well -ordered sample, which we identify at least partial L21 phase, exhibits an intrinsic Gilbert damping constant that increases with decreasing temperat ure. These temperature dependent results are explained employing the torque correction model including interband transitions and both interband as well as intraband transitions for samples with atomic disorder and atomically ordered phases, respectively. ACKNOWLEDGEMENT This work is supported by the Knut and Alice Wallenberg (KAW) Foundation, Grant No. KAW 2012.0031 and from Göran Gustafssons Foundation (GGS), Grant No. GGS1403A. The computations were performed on resources provided by SNIC (Swedish National Infrastructure for Computing) at NSC (National Supercomputer Centre) in Linköping, Sweden. S. H. acknowledges the Department of Science and Technology India for providing the INSPIRE fellow (IF140093) grant. Daniel Hedlund is acknowledged for performing magnetization versus temperature measurements. Author Information Corresponding Authors E -mails: ankit.kumar@angstrom.uu.se , peter.svedlindh@angstrom.uu.se REFERENECS 1. S. Picozzi, A. Continenza, and A. J. Freeman, Phys. Rev. B. 69, 094423 (2004). 2. I. Galanakis, P. H. Dederichs, and N. Papanikolaou, P hys. Rev. B 66, 174429 (2002). 3. Z. Bai, L. Shen, G. Han, Y . P. Feng, Spin 02, 1230006 (2012). 4. M. Belmeguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor jr, C. Tuisan, F. 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(a) Layout of the in -house made VNA-based out -of-plane ferromagnetic resonance setup . (b) Out -plane ferromagnetic resonance spectra recorded for the well -ordered LP673K sample at different temperatures 𝑓𝑓=10 GHz . 17 Figure 2 Fig. 2. (a) Magnetization vs. temperature plots measured on the CFA films with different degree of atomic order. (b) Theoretically calculated magnetization vs. temperature curves for CFA phases with different degree of atomic order, where 50 % (100 %) Fe atoms on Heusler alloy 4a sites indicate B2 (L2 1) ordered phase, and the rest are intermediate B2 & L2 1 mixed ordered phases. 18 Figure 3 Fig. 3. Resonance field vs. in -plane orientation of the applied magnetic field of (a) 𝑇𝑇𝑆𝑆= 300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇 𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100 𝑊𝑊 deposited films. Red lines correspond to fits to the data using Eq. (1). 19 Figure 4 Fig. 4. Line-width vs. frequency of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100 𝑊𝑊 deposited samples. Red lines correspond to fits to the data to extract the experimental Gilbert damping constant and inhomogeneous line -width. Respective insets show the experimentally determined temperature dependent Gilbert damping constants. 20 Figure 5 Fig. 5. Frequency vs. applied field of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆= 400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100 𝑊𝑊 deposited samples. Red lines correspond to Kittel’s fits to the data. Respective insets show the temperature dependent effective magnetization a nd inhomogeneous line -width broadening values. 21 Figure 6 Fig. 6. (a) Linewidth vs. frequency with and without a glass spacer between the waveguide and the sample. Red lines correspond to fits using Eq. (4). (b) Temperature dependent values of the radiative damping using Eq. (6). The lines are guide to the eye. (c) 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 vs 𝛿𝛿2. The red line corresponds to a fit using Eq. (7) to extract the value of the correction factor 𝐶𝐶. (d) Temperature dependent values of eddy current dampi ng using Eq. (7). The lines are guide to the eye. 22 Figure 7 Fig. 7. Experimental (a) and theoretical (b) results for the temperature dependent intrinsic Gilbert damping constant for CFA samples with different degree of atomic order . The B2 & L21 mixed phase corresponds to the 75 % occupancy of Fe atoms on the Heusler alloy 4a sites. The lines are guide to the eye. 23 Figure 8 Fig. 8. Total and atom -resolved spin polarized density of states plots for various compositional CFA phases; (a) A2, (b) B2 and (c) L2 1.

1403.3914v2.Interpolating_local_constants_in_families.pdf

arXiv:1403.3914v2 [math.NT] 20 Aug 2015INTERPOLATING LOCAL CONSTANTS IN FAMILIES GILBERT MOSS 1.Introduction LetG=GLn(F), letkbe an algebraically closed field of characteristic ℓ, and W(k) its ring of Witt vectors. By an ℓ-adic family of representations we mean an A[G]-moduleVwhereAis a commutative W(k)-algebra with unit; then each point pofAgives aκ(p)[G]-moduleV⊗Aκ(p) whereκ(p) denotes the residue field at p. In [EH12], Emerton and Helm conjecture a local Langlands correspo ndence for ℓ- adicfamiliesofadmissiblerepresentations. ToanycontinuousGaloisr epresentation ρ:GF→GLn(A), they conjecturally associate an admissible smooth A[G]-module π(ρ), which interpolates the local Langlands correspondence for poin tsA→κwith κcharacteristic zero. They prove that any A[G]-module which is subject to this interpolation property and a short list of representation-theore tic conditions (see [EH12, Thm 6.2.1]) must be unique. In [Hel12b], Helm further investigates the structure of π(ρ) by taking the list of representation-theoretic conditions in [EH12, Thm 6.2.1] as a start ing point for the theory of “co-Whittaker” A[G]-modules (see Section 2.5 below for the definitions). Using this theory, he is able to reformulate the conjecture in terms of the existence of a certain homomorphism between the integral Bernstein center and a universal deformation ring ([Hel12b, Thm 7.8]). Roughly speaking, representations of GLn(F) overCare completely determined by data involving only local constants ([Hen93]), and in particular th e bijections of the classical local Langlands correspondence are uniquely dete rmined using L- and epsilon-factors (see, for example, [Jia13]). However, L- and epsilon-factors are absent from the the local Langlands correspondence in families. Th us it is natural to ask whether it is possible to attach L- and epsilon-factors to an ℓ-adic family such asπ(ρ) as in [EH12], or more generally any co-Whittaker A[G]-module, in a way that interpolates the L- and epsilon factors at each point. OverC,L-factorsL(π,X) arise as the greatest common denominator of the zeta integralsZ(W,X;j) of a representation πasWvaries over the space W(π,ψ) of Whittakerfunctions (seeSections2.2, 3.1fordefinitions). Epsilon- factorsǫ(π,X,ψ) are the constant of proportionality (i.e. not depending on W) in a functional equation relating the modified zeta integralZ(W,X) L(π,X)to its pre-composition with a Fourier transform. Here, the formal variable Xreplaces the complex variable q−(s+n−1 2)appearingin [JPSS79] and otherliterature, and weconsiderthese objects as formal series. It appears difficult to construct L-factors in a way compatible with arbitrary change of coefficients. To see this, consider the following simple exam ple: letq≡ 1 modℓ, and letχ1,χ2:F×→W(k)×be smooth characters such that χ1is Date: August 17, 2021. 12 GILBERT MOSS unramifiedbut χ2isramified, andsuchthat χ1≡χ2modℓ. Followingtheclassical procedure (see for example [BH06, 23.2]) for finding a generator of the fractional ideal of zeta integrals, we get L(χi,X)∈W(k)(X) and find that L(χ1,X) = 1 1−χ1(̟F)X, andL(χ2,X) = 1. Now let Abe the Noetherian local ring {(a,b)∈ W(k)×W(k) :a≡bmodℓ}, which has two characteristic zero points p1,p2and a maximal ideal ℓA. Letπbe theA[F×]-moduleA, with the action of F×given byx·(a,b) = (χ1(x)a,χ2(x)b). Interpolating L(χ1,X) andL(χ2,X) would mean findinganelement L(π,X)inA[[X]][X−1] suchthatL(π,X)≡L(χi,X) modℓfor i= 1,2, but such a task is impossible because L(χ1,X) andL(χ2,X) are different modℓ. On the otherhand, zeta integralsthemselvesseem to be much more well-behaved with respect to specialization. Classically, zeta integrals form elemen ts of the quo- tient field C(X) ofC[X,X−1]. Our first result is identifying, for more arbitrary coefficient rings A, the correct fraction ring in which our naive generalization of zeta factors will live: Theorem 1.1. SupposeAis aNoetherian W(k)-algebra. Let Sbe the multiplicative subset ofA[X,X−1]consisting of polynomials whose first and last coefficients ar e units. Then if Vis a co-Whittaker A[G]-module,Z(W,X;j)lies in the fraction ringS−1(A[X,X−1])for allW∈ W(V,ψ)and for0≤j≤n−2. The proof of rationality in the setting of representations over a fie ld relies on a useful decomposition of a Whittaker function into “finite” functio ns ([JPSS79, Prop 2.2]). In the setting of rings, such a structure theorem is lack ing, but certain elements of its proof can be translated into a question about the fin iteness of the (n−1)st Bernstein-Zelevinsky derivative. This finiteness property, c ombined with a simple translation property of the zeta integrals, yields Theorem 1 .1 (see§3.2). Classically, zeta integrals satisfy a functional equation which does n ot involve dividingbythe L-factor. Theconstantofproportionalityin thisfunctionalequat ion is called the gamma-factorand equals ǫ(π,X,ψ)L(πι,1 qnX) L(π,X), when theL-factormakes sense. Our second main result is that gamma-factors interpolate in ℓ-adic families (see§4.1 for details on the notation): Theorem 1.2. SupposeAis a Noetherian W(k)-algebra and suppose Vis a prim- itive co-Whittaker A[G]-module. Then there exists a unique element γ(V,X,ψ)of S−1(A[X,X−1])such that Z(W,X;j)γ(V,X,ψ) =Z(/tildewidew′W,1 qnX;n−2−j) for anyW∈ W(V,ψ)and for any 0≤j≤n−2. ToproveTheorem1.2weusethe theoryoftheintegralBernsteinc entertoreduce to the characteristic zero case of [JPSS79]. The question of interpolating local constants in ℓ-adic families has been inves- tigated in a simple case by Vigneras in [Vig00]. For supercuspidal repre sentations ofGL2(F) overQℓ, Vigneras notes in [Vig00] that it is known that epsilon factors define elements of Zℓ, and proves that for two supercuspidal integral representa- tions to be congruent modulo ℓit is necessary and sufficient that they have epsilon factors which are congruent modulo ℓ(we call a representation with coefficients in a local field Eintegral if it stabilizes an OE-lattice). The classical epsilon and gamma factors are equal in the supercuspidal case, so when the s pecialization of anINTERPOLATING LOCAL CONSTANTS IN FAMILIES 3 ℓ-adic family at a characteristic zero point is supercuspidal, the gamm a factor we construct in this paper specializes to the epsilon factor of [JPSS79, Vig00]. Since two representations V1,V2overOEwhich are congruent mod mEdefine a family V1×VV2over the connected W(k)-algebra OE×kEOE, Theorems 1.1 and 1.2 give the following corollary (implying the “necessary” part of [Vig00]): Corollary 1.3. LetKdenote the fraction field of W(k). Ifπandπ′are absolutely irreducible integral representations of GLn(F)over a coefficient field Ewhich is a finite extension of K, then: (1)γ(π,X,ψ)andγ(π′,X,ψ)have coefficients in the fraction ring S−1(OE[X,X−1]). (2) IfmEis the maximal ideal of OE, andπ≡π′modmE, thenγ(π,X,ψ)≡ γ(π′,X,ψ) modmE. The question of extending the theory of zeta integrals to the ℓ-modular setting has been investigated in [M ´12], and very recently in [KM14] for the Rankin-Selberg integrals. The question of deforming local constants over polynom ial rings over Chas been investigated by Cogdell and Piatetski-Shapiro in [CPS10], an d the techniques of this paper owe much to those in [CPS10]. Analogous to the results of Bernstein and Deligne in [BD84] for RepC(G), Helm shows in [Hel12a, Thm 10.8] that the category RepW(k)(G) has a decomposition into full subcategories known as blocks. Our third main result is cons tructing for each block a gamma factor which is universal in the sense that it gives rise via specialization to the gamma factor for any co-Whittaker module in th at block. We will now state this result more precisely. Each block of the category RepW(k)(G) corresponds to a primitive idempotent in the Bernstein center Z, which is defined as the ring of endomorphisms of the identity functor. It is a commutative ring whose elements consist of collections of compatible endomorphisms of every object, each such endomorph ism commuting with all morphisms. Choosing a primitive idempotent eofZ, the ringeZis the center of the subcategory e·RepW(k)(G) of representations satisfying eV=V. The ringeZhas an interpretation as the ring of regular functions on an affine alg ebraic variety over W(k), whosek-points are in bijection with the set of unramified twists of a fixed conjugacy class of supercuspidal supports in Repk(G). See [Hel12a] for details. In [Hel12b], Helm determines a “universal co-Whittaker m odule” with coefficientsin eZ, denoted hereby eW, whichgivesrisetoanyco-Whittakermodule via specialization (see Proposition 2.31 below). By applying our theory of zeta integrals to eWwe get a gamma factor which is universal in the same sense: Theorem 1.4. SupposeAis any Noetherian W(k)-algebra, and suppose Vis a primitive co-Whittaker A[G]-module. Then there is a primitive idempotent e, a homomorphism fV:eZ →A, and an element Γ(eW,X,ψ)∈S−1(eZ[X,X−1]) such thatγ(V,X,ψ) =fV(Γ(eW,X,ψ)). Interpolating gamma factors of pairs may be the next step in obtain ing a local conversetheorem for ℓ-adic families. By capturing the interpolation property, fami- lies ofgammafactorsmight givean alternativecharacterizationoft he co-Whittaker moduleπ(ρ) appearing in the local Langlands correspondence in families. The author would like to thank his advisor David Helm for suggesting th is prob- lem and for his invaluable guidance, Keenan Kidwell for his helpful conv ersations, and Peter Scholze for his helpful questions and comments at the MS RI summer4 GILBERT MOSS school on New Geometric Techniques in Number Theory in 2013. He wo uld also like to thank the referee for her/his very helpful comments and su ggestions. 1.1.Notation and Conventions. LetFbe a finite extension of Qp, letqbe the order of its residue field, and let kbe an algebraically closed field of characteristic ℓ, whereℓ/ne}ationslash=pis an odd prime. Denote by W(k) the ring of Witt vectors over k. The assumption that ℓis odd is made so that W(k) contains a square root ofq. Whenℓ= 2 all the arguments presented will remain valid, after possibly adjoining a square root of qtoW(k). The letter GorGnwill always denote the groupGLn(F). Throughout the paper Awill be a Noetherian commutative ring which is aW(k)-algebra, with additional properties in various sections, and κ(p) will denote the residue field of a prime ideal pofA. For any locally profinite group H, RepA(H) denotes the category of smooth representations of Hover the ring A, i.e.A[H]-modules for which every element is stabilized by an open subgroup of H. Even when this category is not mentioned, all representations are presumed to be smooth. When Hisa closedsubgroupof G, wedefine the non-normalizedinduction functor IndG H(resp. c-IndG H) : RepA(H)→RepA(G) sendingτto the smooth part oftheA[G]-module, underrighttranslation, offunctions(resp. functionsc ompactly supported modulo H)f:G→τsuch thatf(hg) =τ(h)f(g),h∈H,g∈G. The integral Bernstein center of [Hel12a] (see the discussion prec eding Theorem 1.4) will alwaysbe denoted by Z. IfVis in RepA(G), then it is also in RepW(k)(G), and we frequently use the Bernstein decomposition of RepW(k)(G) to interpret properties of V. IfAhas a nontrivial ideal I, thenI·Vis anA[H]-submodule of V, which shows that most content would be missing if we developed the representat ion theory of RepA(H) around the notion of irreducible objects, or simple A[H]-modules. Thus conditions appear throughout the paper which in the traditional se tting are implied by irreducibility: Definition 1.5. VinRepA(H)will be called (1)Schurif the natural map A→EndA[G](V)is an isomorphism; (2)G-finiteif it is finitely generated as an A[G]-module. (3)primitive if there exists a primitive idempotent ein the Bernstein center Z such thateV=V. We say a ring is connected if it has connected spectrum or, equivalen tly, no nontrivial idempotents, for example any local ring or integral doma in. Note that if Ais connected, Corollary2.32implies all co-Whittaker A[G]-modules are primitive. Denote by Nnthe subgroup of Gnconsisting of all unipotent upper-triangular matrices. Let ψ:F→W(k)×be an additive character of Fwith kerψ=p. Then ψdefines a character on any subgroup of Nn(F) by (u)i,j/mapsto→ψ(u1,2+···+un−1,n); we abusively denote this character by ψas well. IfHis asubgroupnormalizedby anothersubgroupgroup K, andθis a character of the group H, denote by θkthe character given by θk(h) =θ(khk−1) forh∈H, k∈K. ForVin RepA(H), denote by VH,θthe quotient V/V(H,θ) whereV(H,θ) is the sub-A-module generated by elements of the form hv−θ(h)vforh∈Hand v∈V; it isK-stable ifθk=θ,k∈K. Given a standard Levi subgroup M⊂GnINTERPOLATING LOCAL CONSTANTS IN FAMILIES 5 with unipotent radical U, and1the trivial character, we denote by JMthe non- normalized Jacquet functor RepA(G)→RepA(M) :V/mapsto→VU,1. For eachm≤n, letGmdenoteGLm(F) and embed it in Gvia (Gm0 0In−m). We let{1}=P1⊂ ··· ⊂Pndenote the mirabolic subgroups of G1⊂ ··· ⊂Gn, wherePmis given by/braceleftbig (gm−1x 0 1) :gm−1∈Gm−1, x∈Fm−1/bracerightbig . We also have the unipotent upper triangular subgroup UmofPmgiven by/braceleftbig (Im−1x 0 1) :x∈Fm−1/bracerightbig . In particular, Um≃Fm−1andPm=UmGm−1. Note that this is different from the groups N(r) defined in Proposition 2.3. SinceGncontains a compact open subgroup whose pro-order is invertible in W(k), there exists a unique (for that choice of subgroup) normalized H aar mea- sure, defining integration on the space C∞ c(G,A) of smooth compactly supported functionsG→A([Vig96, I.2.3]). 2.Representation Theoretic Background 2.1.Co-invariants and Derivatives. Asin[EH12,BZ77], wedefinethefollowing functors with respect to the character ψ. Φ+:RepA(Pn−1)→RepA(Pn) Ψ+:Rep(Gn−1)→Rep(Pn) V/mapsto→c-IndPn Pn−1UnV(Unacts viaψ)V/mapsto→V(Unacts trivially) ˆΦ+:Rep(Pn−1)→Rep(Pn) Ψ−:Rep(Pn)→Rep(Gn−1) V/mapsto→IndPn Pn−1UnV V /mapsto→V/V(Un,1) Φ−:Rep(Pn)→Rep(Pn−1) V/mapsto→V/V(Un,ψ) Note that we give these functors the same names as the ones origin ally defined in [BZ76], but we use the non-normalized induction functors, as in [BZ7 7, EH12], because they are simpler for our purposes. As observed in [EH12], t hese functors retain the basic adjointness properties proved in [BZ77, §3.2]. This is because the methods of proof in [BZ76, BZ77] use properties of l-sheaves which carry over to the setting of smooth A[G]-modules where Ais a Noetherian W(k)-algebra. Proposition 2.1 ([EH12],3.1.3) .(1) The functors Ψ−,Ψ+,Φ−,Φ+,ˆΦ+are ex- act. (2)Φ+is left adjoint to Φ−,Ψ−is left adjoint to Ψ+, andΦ−is left adjoint to ˆΦ+. (3)Ψ−Φ+= Φ−Ψ+= 0 (4)Ψ−Ψ+,Φ−ˆΦ+, andΦ−Φ+are naturally isomorphic to the identity functor. (5) For each VinRep(Pn)we have an exact sequence 0→Φ+Φ−(V)→V→Ψ+Ψ−(V)→0. (6) (Commutativity with Tensor Product) If Mis anA-module and FisΨ−,Ψ+, Φ−,Φ+, orˆΦ+, we haveF(V⊗AM)∼=F(V)⊗AM. For 1≤m≤nwe define the mth derivative functor (−)(m):= Ψ−(Φ−)m−1: Rep(Pn)→Rep(Gn−m).6 GILBERT MOSS This gives a functor Rep( Gn)→Rep(Gn−m) by first restricting representations to Pnand then applying ( −)(m); this functor is also denoted ( −)(m). The zero’th de- rivativefunctor ( −)(0)is the identity. We candescribe the derivativefunctor ( −)(m) more explicitly by using the following lemma on the transitivity of coinvar iants: Lemma 2.2 ([BZ76]§2.32).LetHbe a locally profinite group, θa character of H, andVa representation of H. SupposeH1,H2are subgroups of Hsuch that H1H2=HandH1normalizes H2. Then/parenleftig VH2,θ|H2/parenrightig H1,θ|H1=VH,θ. DefineN(r) to be the group of matrices whose first rcolumns are those of the identity matrix, and whose last n−rcolumns are those of elements of Nn(recall thatNnis the group of unipotent upper triangular matrices). For 2 ≤r≤nwe haveUrN(r) =N(r−1) andUrnormalizes N(r). AsN(r) is contained in Nn, we defineψonN(r) via its superdiagonal entries. We can also define a character /tildewideψon N(r) slightly differently from the usual definition: /tildewideψwill be given as usual via ψ on the last n−r−1 superdiagonal entries, but trivially on the ( r,r+1) entry, i.e. /tildewideψ(x) :=ψ(0+xr+1,r+2+···+xn−1,n) forx∈N(r). The functors (Φ−)mand (−)(m), defined above, can be described more explicitly. Letm=n−r. By applying Lemma 2.2 repeatedly with H1=Ur, andH2= N(r−1), we get Proposition 2.3 ([Vig96] III.1.8) .(1)(Φ−)mVequals the module of coinvariants V/V(N(n−m),ψ). (2)V(m)equals the module of coinvariants V/V(N(n−m),/tildewideψ). In particular, if m=n, this gives V(n)=V/V(Nn,ψ). Note that V(n)is simply anA-module. 2.2.Whittaker and Kirillov Functions. The character ψ:Nn→A×defines a representationof NnintheA-moduleA, whichwealsodenoteby ψ. ByProposition 2.3 we have Hom A(V(n),A) = Hom Nn(V,ψ). Definition 2.4. ForVinRepA(Gn), we say that Vis of Whittaker type if V(n) is free of rank one as an A-module. As in [EH12, Def 3.1.8] , ifAis a field we refer to representations of Whittaker type as generic. IfVis of Whittaker type, Hom Nn(V,ψ) is free of rank one, so we may choose a generator λ. The image of λunder the Frobenius reciprocity isomorphism HomNn(V,ψ)∼→HomGn(V,IndGn Nnψ) is the map v/mapsto→WvwhereWv(g) =λ(gv). TheA[G]-module formed by the image of the map v/mapsto→Wvis independent of the choice ofλ. Definition 2.5. The image of the homomorphism V→IndGn Nnψis called the space of Whittaker functions of Vand is denoted W(V,ψ)or justW. Choosing a generator of V(n)and allowing Nnto act via ψ, we get an iso- morphismV(n)∼→ψ. Composing this with the natural quotient map V→V(n) gives anNn-equivariant map V→ψ, which is a generator λ. Note that the map V→ W(V,ψ) is surjective but not necessarily an isomorphism, unlike the setting ofirreducible generic representations with coefficients in a field. Different A[G]- modules of Whittaker type can have the same space of Whittaker fu nctions:INTERPOLATING LOCAL CONSTANTS IN FAMILIES 7 Lemma 2.6. SupposeV′,VinRepA(G)are of Whittaker type, and suppose there is aG-equivariant homomorphism α:V′→Vsuch thatα(n): (V′)(n)→V(n) is an isomorphism. Then W(V′,ψ)is the subrepresentation of W(V,ψ)given by W(α(V′),ψ). Proof.Letq′:V′→V′/V′(Nn,ψ) andq:V→V/V(Nn,ψ) be the quotient maps. Choosing a generator for V(n)gives isomorphisms η,η′such that the following diagram commutes. Vq>V(n)η>A V′α∨ q′ >(V′)(n)α(n) ∨ η′> Givenv′∈V′we get Wα(v′)(g) =η(q(gαv′)) =η((q◦α)(gv′)) =η′(q′(gv′)) =Wv′(g), g∈G. This shows W(V′,ψ) =W(α(V′),ψ)⊂ W(V,ψ). /square IfVin RepA(Gn) is Whittaker type and v∈V, we will denote by Wv|Pnthe restriction of the function Wvto the subgroup Pn⊂Gn. Definition 2.7. The image of the Pn-equivariant homomorphism V→IndPn Nnψ: v/mapsto→Wv|Pnis called the Kirillov functions of Vand is denoted K(V,ψ)or justK. The following properties of the Kirillov functions are well known for Re pC(G), but we will need them for RepA(G): Proposition 2.8. LetVbe of Whittaker type in RepA(Pn), and choose a generator ofV(n)in order to identify V(n)withA. Then the following hold: (1)(Φ+)n−1V(n)= c-IndPn Nnψand(ˆΦ+)n−1V(n)= IndPn Nnψ. (2) The composition (Φ+)n−1V(n)→V→IndPn Nnψdiffers from the inclusion c-IndPn Nnψ֒→IndPn Nnψby multiplication with an element of A×. (3) The Kirillov functions K(V,ψ)contains c-IndPn Nnψas a sub-A[Pn]-module. Proof.The proof in [BZ76] Proposition 5.12 (g) works to prove (1) in this con text. LetS= (Φ+)n−1V(n). There is an embedding S→Vby Proposition 2.1 (5); denote by tthe composition S→V→Indψ. Thent(n):S(n)→Indψ(n) is a nonzero homomorphism between free rank one A-modules, hence given by multiplication with an element aofA. By Proposition 2.1 (6), For any maximal idealmofA,t(n)⊗κ(m) must be an isomorphism because it is a nonzero element of Homκ(m)((S(V)⊗κ(m))(n),(Indψ⊗κ(m))(n)) =κ(m). Thusais nonzero in κ(m) for allm, hence a unit, so t(n)is an isomorphism. On the other hand there is the natural embedding c-Ind ψ→Indψ, which we will denote s. Sinces(n)is an isomorphism by [BZ77, Prop 3.2 (f)], we have s(n)=ut(n) for someu∈A×. Thus, if K:= ker(s−ut) thenK(n)=S(V)(n)=V(n), whence Hom P(S(V)/K,Indψ)∼=HomA((S(V)/K)(n),A) = Hom A({0},A) = 0, which implies s−ut≡0. To prove (3), note that since K(V,ψ) is defined to be the image of the map V→IndPn Nnψ, this follows from (2). /square8 GILBERT MOSS Definition 2.9 ([EH12],§3.1 ).IfVis inRep(Pn), the image of the natural inclu- sion(Φ+)n−1V(n)→Vis called the Schwartz functions of Vand is denoted S(V). ForVinRep(Gn)we also denote by S(V)the Schwartz functions of Vrestricted toPn. We can ask how the functor Φ−is reflected in the Kirillov space of a represen- tation. First we observe that Φ−commutes with the functor K: Lemma 2.10. For0≤m≤n, theA[Pm]-modules K((Φ−)n−mV,ψ)and (Φ−)n−mK(V,ψ)are identical. Proof.The image of the Pn−m-submodule V(N(m),ψ) in the map V→ Kequals the submodule K(N(m),ψ). The lemma then follows from Proposition 2.3 /square Following [CPS10], we can explicitly describe the effect of the functor Φ−on the Kirillov functions K. Recall that K(Un,ψ) denotes the A-submodule generated by {uW−ψ(u)W:u∈Un, W∈ K}and Φ−K:=K/K(Un,ψ). Proposition 2.11 ([CPS10] Prop 1.1) . K(Un,ψ) ={W∈ K:W≡0on the subgroup Pn−1⊂Pn}. Proof.The proof of [CPS10, Prop 1.1] carries over in this setting. It utilizes the Jacquet-Langlandscriterion for an element vof a representation Vto be in the sub- spaceV(Uni,ψ), which remains valid over more general coefficient rings Abecause all integrals are finite sums. /square Thus Φ−has the same effect as restriction of functions to the subgroup Pn−1 insidePn: Φ−K∼=/braceleftbig W/parenleftbigp0 0 1/parenrightbig :W∈ W(V,ψ), p∈Pn−1/bracerightbig . By applying for each m= 1,...,n−2 the argument of [CPS10, Prop 1.1] to the Pn−m+1representation /braceleftig W/parenleftig p0 0Im−1/parenrightig :W∈ W(V,ψ), p∈Pn−m+1/bracerightig instead of to K, we can describe (Φ−)mK: Corollary 2.12. Form= 1,...,n−1, (Φ−)mK∼=/braceleftbig W/parenleftbigp0 0Im/parenrightbig :W∈ W(V,ψ), p∈Pn−m/bracerightbig . 2.3.Partial Derivatives. Given a product H1×H2of subgroups of G, and a characterψon the unipotent upper triangular elements of H2, we can define “par- tial” versionsof the functors Φ±, Ψ±as follows: given Vin RepA(H1×H2), restrict it to a representation of H1={1}×H2, then apply the functor Φ±or Ψ±, and observethat H1×{1}acts naturally on the result, since it commutes with {1}×H2.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 9 More precisely: Φ+,2:RepA(Gn−m×Pm−1)→RepA(Gn−m×Pm) V/mapsto→c-IndGn−m×Pm Gn−m×Pm−1Um(V), with {1}×Umacting viaψ ˆΦ+,2:Rep(Gn−m×Pm−1)→Rep(Gn−m×Pm) V/mapsto→c-IndGn−m×Pm Gn−m×Pm−1Um(V) Φ−,2:Rep(Gn−m×Pm)→Rep(Gn−m×Pm−1) V/mapsto→V/V({1}×Um,ψ) Ψ+,2:Rep(Gn−m×Gm−1)→Rep(Gn−m×Pm) V/mapsto→V({1}×Umacts trivially) Ψ−,2:Rep(Gn−m×Pm)→Rep(Gn−m×Gm−1) V/mapsto→V/V({1}×Um,1) BecauseH1×{1}commutes with {1}×H2, we immediately get Lemma 2.13. The analogue of Proposition 2.1 (1)-(6) holds for Φ+,2,ˆΦ+,2,Φ−,2, Ψ+,2, andΨ−,2. Definition 2.14. We define the functor (−)(0,m): RepA(Gn−m×Gm)→ RepA(Gn−m)to be the composition Ψ−,2(Φ−,2)m−1. The proof of the following Proposition holds for W(k)-algebrasA: Proposition 2.15 ([Zel80] Prop 6.7, [Vig96] III.1.8) .LetM=Gn−m×Gm. For0≤m≤nthem’th derivative functor (−)(m)is the composition of the Jacquet functor JM: Rep(Gn)→Rep(Gn−m×Gm)with the functor (−)(0,m): RepA(Gn−m×Gm)→RepA(Gn−m). Lemma 2.16. LetVbe inRepA(Gn−m×Gm). ThenVcontains an A-submodule isomorphic to V(0,m). Proof.The image of the natural embedding (Φ+,2)m−1Ψ+,2(V(0,m))→V, which is given by Proposition 2.13 (5), will be denoted S0,2(V). By Proposition 2.13 (4), the natural surjection V→V(0,m)restricts to a surjection S0,2(V)→V(0,m). By Proposition 2.13 (6), the map of A-modules S0,2(V)→V(0,m)arises from the map (Φ+,2)m−1Ψ+,2(A)→Aby tensoring over AwithV(0,m). Take the A-submodule generated by any element of (Φ+,2)m−1Ψ+,2(A) that maps to the identity in A; then tensor with V(0,m). /square 2.4.Finiteness Results. In this subsection we gathercertain finiteness results in- volving derivatives, most of which are well-known when Ais a field of characteristic zero. LetHbeanytopologicalgroupcontainingadecreasingsequence {Hi}i≥0ofopen subgroups whose pro-order is invertible in A, and which forms a neighborhood base of the identity in H. IfVis a smooth A[H]-module we may define a projection πi:V→VHi:v/mapsto→/integraltext Hihvfor a Haar measure on HiwhereHihas total measure 1. TheA-submodules Vi:= ker(πi)∩VHi+1then satisfy/circleplustext iVi=V. Lemma 2.17 ([EH12] Lemma 2.1.5,2.1.6) .A smoothA[H]-moduleVis admissible if and only if each A-moduleViis finitely generated. In particular, quotients of admissibleA[H]-modules by A[H]-submodules are admissible.10 GILBERT MOSS Thus the following version of the Nakayama lemma applies to admissible A[H]- modules: Lemma2.18 ([EH12]Lemma3.1.9) .LetAbe a Noetherian local ring with maximal idealm, and suppose that Mis a submodule of a direct sum of finitely generated A-modules. If M/mMis finite dimensional then Mis finitely generated over A. IfVis admissible, then it is G-finite if and only if V/mVisG-finite. To see this, takeS⊂V/mVan (A/m)[H]-generating set, let Wbe theA[H]-span of a lift to V. SinceV/Wis admissible, we can apply Nakayama to each factor ( V/W)ito concludeV/W= 0. Proposition 2.19 ([EH12] 3.1.7) .Letκbe aW(k)-algebra which is a field, and Van absolutely irreducible admissible representation of Gn. ThenV(n)is zero or one-dimensional over κ, and is one-dimensional if and only if Vis cuspidal. Proposition 2.20 ([Vig96] II.5.10(b)) .Letκbe aW(k)-algebra which is a field. IfVis aκ[G]-module, then Vis admissible and G-finite if and only if Vis finite length over κ[G]. Proof.SupposeVis admissible and G-finite. Ifκwere algebraically closed of char- acteristic zero (resp. characteristic ℓ), this is [BZ77, 4.1] (resp. [Vig96, II.5.10(b)]). Otherwise, let κbe an algebraic closure, then V⊗κis finite length, so Vis finite length. IfVis finite length, so is V⊗κκ. Overan algebraicallyclosed field ofcharacteris- tic different from p, irreducible representations are admissible ([BZ77, 3.25],[Vig96, II.2.8]). Since admissibility is preserved under taking extensions V⊗κbeing finite length implies it is admissible, hence Vis admissible. Thus we can reduce prov- ingG-finiteness to proving that, given any exact sequence of admissible objects, 0→W0→V→W1→0 whereW0andW1areG-finite, then VisG-finite. But there is a compact open subgroup Usuch thatW0andW1are generated by WU 0 andWU 1, respectively. It follows that that Vis generated by VU. /square Lemma 2.21. Letκbe aW(k)-algebra which is a field. If Vis an absolutely irreducible κ[Gn]-module, then for m≥0,V(m)is finite length as a κ[Gn−m]- module. Proof.We follow [Vig96, III.1.10]. Given j,kpositive integers, let M=Gj×Gk and letP=MNbe the associated standard parabolic subgroup. Given τin Repκ(Gj) andσin Repκ(Gk), we define τ×σto be the normalized induction c-IndP(δ1/2 N(σ⊗τ)) in Repκ(Gj+k), whereδNdenotes the modulus character of N(for the definition of δNsee [BZ77, 1.7]). There exists a multiset {π1,...,π r} of irreducible cuspidals such that V⊂π1× ··· ×πr. The Liebniz formula for derivatives says that ( π1×π2)(t)has a filtration whose successive quotients are π(t−i) 1×π(i) 2. Its proof, given in [BZ77, §7], carries over in this generality. Then V(m)⊂(π1× ··· ×πr)(m), which is finite length by induction, using Proposition 2.19 combined with the Liebniz formula. /square Proposition 2.22 ([Hel12b] Prop 9.15) .LetMbe a standard Levi subgroup of G. IfVinRepA(G)is admissible and primitive, then JMVinRepA(M)is admissible. Corollary 2.23. IfAis a local Noetherian W(k)-algebra and Vis admissible and G-finite, then V(m)is admissible and G-finite for 0≤m≤n.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 11 Proof.LetM=Gn−m×Gm. By Proposition 2.15, V(m)= (JMV)(0,m), so by Lemma 2.16, there is an embedding V(m)→JMV. Admissibility and G-finiteness meanVis generated over A[G] by vectors in VKfor some compact open subgroup K. SinceVKis finite over A,eVKis nonzero for only a finite set of primitive idempotents eofthe Bernstein center, so eV/ne}ationslash= 0for at most finitely many primitive idempotents eof the integral Bernstein center. Therefore, Proposition 2.22 ap plies, showingV(m)embeds in an admissible module. Thus by Lemma 2.18, we are reduced to proving the result for V:=V/mV. SinceVis admissible and G-finite, andA/mischaracteristic ℓ, Lemma2.20shows Visfinite length, thereforeitfollows from Lemma 2.21 that V(m)is finite length. Applying Lemma 2.20 once more, we have the result. /square Loosely speaking, the ( n−1)st derivative describes the restriction of a Gn- representation to a G1-representation (see Corollary 2.12). The next result shows that this restriction intertwines a finite set of characters: Theorem 2.24. IfAis a localW(k)-algebra and VinRepA(G)is admissible and G-finite, then V(n−1)is finitely generated as an A-module. Proof.ByLemma2.18andCorollary2.23itissufficienttoshowthat V(n−1)isfinite over the residue field κ. We know V(n−1)isG-finite and admissible by Corollary 2.23, hence finite length asa κ[G1]-module by Proposition2.20. Since G1is abelian, all composition factors are 1-dimensional, so V(n−1)being finite length implies it is finite dimensional over κ. /square Since the hypotheses of being admissible and G-finite are preserved under local- ization by Proposition 2.1 (6), we can go beyond the local situation: Corollary 2.25. LetAbe a Noetherian W(k)-algebra and suppose that Vis ad- missible and G-finite. Then for every pinSpecA,V(n−1) pis finitely generated as anAp-module. 2.5.Co-Whittaker A[G]-Modules. In this subsection we define co-Whittaker representations and show that every admissible A[G]-moduleVof Whittaker type contains a canonical co-Whittaker subrepresentation. Definition 2.26 ([Hel12b] 3.3) .Letκbe a field of characteristic different from p. An admissible smooth object UinRepκ(G)is said to have essentially AIG dual if it is finite length as a κ[G]-module, its cosocle cos(U)is absolutely irreducible generic, andcos(U)(n)=U(n)(the cosocle of a module is its largest semisimple quotient) . This condition is equivalent to U(n)being one-dimensional over κand having the property that W(n)/ne}ationslash= 0 for any nonzero quotient κ[G]-moduleW(see [EH12, Lemma 6.3.5] for details). Definition 2.27 ([Hel12b] 6.1) .An objectVinRepA(G)is said to be co-Whittaker if it is admissible, of Whittaker type, and V⊗Aκ(p)has essentially AIG dual for eachp. Proposition 2.28 ([Hel12b] Prop 6.2) .LetVbe a co-Whittaker A[G]-module. Then the natural map A→EndA[G](V)is an isomorphism.12 GILBERT MOSS Lemma 2.29. SupposeVis admissible of Whittaker type and, for all primes p, any non-generic quotient of V⊗κ(p)equals zero. Then Vis generated over A[G] by a single element. Proof.Letxbe a generator of V(n), and let ˜x∈Vbe a lift of x. IfV′is the A[G]-submodule of Vgenerated by ˜ x, then (V/V′)(n)= 0. Since any non-generic quotient of V⊗κ(p) equals zero, ( V/V′)⊗κ(p) = 0 for all p. SinceV/V′is admissible, we can apply Lemma 2.18 over the local rings Apto conclude V/V′is finitely generated, then apply ordinary Nakayama to conclude it is ze ro. /square Thus every co-Whittaker module is admissible, Whittaker type, G-finite (in fact G-cyclic), and Schur, so satisfies the hypotheses of Theorem 3.5, b elow. Moreover, every admissible Whittaker type representation contains a canonic al co-Whittaker submodule: Proposition 2.30. LetVinRepA(G)be admissible of Whittaker type. Then the sub-A[G]-module T:= ker(V→/productdisplay {U⊂V: (V/U)(n)=0}V/U) is co-Whittaker. Proof.(V/T)(n)= 0 soTis Whittaker type. Since Vis admissible so is T. Let pbe a prime ideal and let T:=T⊗κ(p). We show that cos( T) is absolutely irreducible and generic. By its definition, cos( T) =/circleplustext jWjwithWjan irre- ducibleκ(p)[G]-module. Since the map T→/circleplustext jWjis a surjection and ( −)(n) is exact and additive, the map ( T)(n)→/circleplustext jW(n) jis also a surjection. Hence dimκ(p)(/circleplustext jW(n) j)≤dimκ(p)(T(n)). SinceTis Whittaker type and T(n)=T(n)is nonzero, there can only be one jsuch thatW(n) jis potentially nonzero. On the other hand, suppose some W(n) jwere zero, then Wjis a quotient appearing in the target of the map V→/productdisplay {U⊂V: (V /U)(n)=0}V/U, hence as a quotient of Tit would have to be zero, a contradiction. Hence precisely oneWjis nonzero. Now applying [EH12, 6.3.4] with Abeingκ(p) andVbeing cos(T), we have that End G(cos(T))∼=κ(p) hence absolutely irreducible. It also shows that cos( T)(n)=W(n) j/ne}ationslash= 0. Hence T(n)= cos(T)(n). By Lemma 2.29, Tis κ(p)[G]-cyclic; since it is admissible, it is finite length by Lemma 2.20. /square 2.6.The Integral Bernstein Center. IfAis a Noetherian W(k)-algebra and Vis anA[G]-module, then in particular Vis aW(k)[G]-module, so we use the Bernstein decomposition of RepW(k)(G) to studyV. LetWbe theW(k)[G]-module c-IndGn Nnψ. Ifeis a primitive idempotent of Z, the representation eWlies in the block eRepW(k)(G), and we may view it as an object in the category RepeZ(G). With respect to extending scalars from eZtoA, the module eWis “universal” in the following sense: Proposition 2.31 ([Hel12b] Thm 6.3) .LetAbe a Noetherian eZ-algebra. Then eW⊗eZAis a co-Whittaker A[G]-module. Conversely, if Vis a primitive co- WhittakerA[G]module in the block eRepW(k)(G), andAis aneZ-algebra viaINTERPOLATING LOCAL CONSTANTS IN FAMILIES 13 fV:eZ →A, then there is a surjection α:W⊗A,fVA→Vsuch thatα(n): (W⊗A,fVA)(n)→V(n)is an isomorphism. If we assume Ahas connected spectrum (i.e. no nontrivial idempotents), then the mapfV:Z →Awould factor through a map eZ →Afor some primitive idempotent e, hence: Corollary 2.32. IfAis a connected Noetherian W(k)-algebra and Vis co- Whittaker, then Vmust be primitive for some primitive idempotent e. Remark 2.33. Theorems 1.1, 1.2, and 1.4 remain true if the hypothesis that Vis primitive is replaced with the hypothesis that Ais connected. 3.Zeta Integrals In this section we use the representation theory of Section 2 to de fine zeta inte- grals and investigate their properties. 3.1.Definition of the Zeta Integrals. We first propose a definition of the zeta integral which is the analog of that in [JPSS79], and then check that t he definition makes sense. Definition 3.1. ForW∈ W(V,ψ)and0≤j≤n−2, letXbe an indeterminate and define Z(W,X;j) =/summationdisplay m∈Z(qn−1X)m/integraldisplay x∈Fj/integraldisplay a∈UFW/bracketleftbigg/parenleftbigg ̟ma0 0 x Ij0 0 0In−j−1/parenrightbigg/bracketrightbigg d×adx, andZ(W,X) =Z(W,X;0) We first show that Z(W,X;0) defines an element of A[[X]][X−1]. Lemma 3.2. LetWbe any element of IndG Nnψ. Then there exists an integer N <0such thatW(a0 0In−1)is zero for vF(a)< N. Moreover if Wis compactly supported modulo Nn, then there exists an integer L >0such thatW(a0 0In−1)is zero forvF(a)>L Proof.There is an integer jsuch that/parenleftbigg 1pj0 0 1 0 0 0In−2/parenrightbigg stabilizesW. Forxinpj, we have W/parenleftiga0 0 0 1 0 0 0In−2/parenrightig =W/parenleftig/parenleftiga0 0 0 1 0 0 0In−2/parenrightig/parenleftig1x0 0 1 0 0 0In−2/parenrightig/parenrightig =ψ/parenleftig1ax0 0 1 0 0 0In−2/parenrightig W/parenleftiga0 0 0 1 0 0 0In−2/parenrightig WhenevervF(a) is negative enough that axlands outside of ker ψ=p, we get that ψ/parenleftig1ax0 0 1 0 0 0In−2/parenrightig is a nontrivial p-power root of unity ζinW(k), hence 1 −ζis the lift of something nonzero in the residue field k, and defines an element of W(k)×. This shows that W(a0 0In−1) = 0. /square Just as in [JPSS79], the next two lemmas show that Z(W,X;j) defines an ele- ment ofA[[X]][X−1] when 0<j <n−2 by reducing it to the case of Z(W,X;0). Lemma 3.3 ([JPSS79] Lemma4.1.5) .LetHbe a function on G, locally fixed under right translation by G, and satisfying H(ng) =ψ(n)H(g)forg∈G,n∈Nn. Then the support of the function on Fjgiven by x/mapsto→H/bracketleftbigg/parenleftbigga0 0 x Ij0 0 0In−j−1/parenrightbigg/bracketrightbigg14 GILBERT MOSS is contained in a compact set independent of a∈F×. Corollary 3.4. Ifρdenotes right translation (ρ(g)φ)(x) =φ(xg), andUis the unipotent radical of the standard parabolic subgroup of typ e(1,n−1), then there is a finite set of elements u1,...,u rofUsuch that Z(W,X;j) =r/summationdisplay i=1Z(ρ(tui)W,X;0) for anyW∈IndG Nnψ. In [JPSS79], the zeta integrals form elements of the field C((X)) and it is proved that in fact they are elements of the subfield C(X) of rational functions. Whereas C((X)) (resp. C(X)) is the fraction field of the domain C[[X]] (resp. C[X,X−1]), our ringsA[[X]][X−1] andA[X,X−1] are not in general domains. The first main result of this paper is determining the appropriate fraction ring of A[X,X−1] in which the zeta integrals Z(W,X;j) live: Theorem 3.5. SupposeAis aNoetherian W(k)-algebra. Let Sbe the multiplicative subset ofA[X,X−1]consisting of polynomials whose first and last coefficients ar e units. Then if Vis admissible, Whittaker type, and G-finite, then Z(W,X;j)lies inS−1A[X,X−1]for allWinW(V,ψ)for0≤j≤n−2. In particular, the result holds if Vis co-Whittaker, as in Theorem 1.1. The proof of Theorem 3.5 will occupy the remainder of this section. The key idea is that the zeta integrals Z(W,X) are completely determined by the values W/parenleftbiga0 0In−1/parenrightbig for a∈F×, and asWranges over W(V,ψ), the set of these values is equivalent to the data of the P2-representation (Φ−)n−2K. Determining the rationality of Z(W,X) will then reduce to a finiteness result for the quotient K(n−1), or more generally for V(n−1). 3.2.Proof of Rationality. Denote byτthe righttranslationrepresentationof G1 onK(n−1). LetBbe the commutative A-subalgebra of End A(K(n−1)) generated byτ(̟) andτ(̟−1), where̟is a uniformizer of F. It follows from Corollary 2.25 that K(n−1) pis finitely generated over Ap. For every pof SpecA, the inclusion Bp⊂End(K(n−1))p֒→End(K(n−1) p), showsBpis finitely generated as an Ap- module. Lemma 3.6. Bis finitely generated as an A-module. Proof.Bis the image of the map A[X,X−1]→EndA(K(n−1)) sendingXtoτ(̟). Bpis the image of the localized map Ap[X,X−1]→/parenleftbig EndA(K(n−1))/parenrightbig p, which is finitely generated. Thus for every p,τ(̟) andτ(̟−1) satisfy monic polynomials sp(X),tp(X) with coefficients in Ap. Sincespandtphave finitely many coefficients there exists a global section fp/∈psuch thatsp(X),tp(X) lie inAfp[X]. The open subsetsD(fp) cover Spec Aand we can take a finite subset {f1,...,f n} ⊂ {fp} such that (fi) = 1. Since τ(̟) andτ(̟−1) satisfy monic polynomials over Afi, we have thatBfiis finitely generated over Afifor eachi. It follows that Bis finitely generated over A. /square SinceBis finitely generated over A,τ(̟) andτ(̟−1) satisfy monic polynomials c0+c1X+...cr−1Xr−1+Xrandb0+b1X+···+bs−1Xs−1+Xsrespectively.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 15 The degrees randsare nonzero because τ(̟) andτ(̟−1) are units in B. Adding these together we have 0 =τ(̟)−s+bs−1τ(̟)−s+1+···+b0+c0+...cr−1τ(̟)r−1+τ(̟)r, henceτ(̟) satisfies a Laurent polynomial whose first and last coefficients are units. The final ingredient in proving rationality is the following transformat ion prop- erty. Lemma 3.7. Z(̟nW,X) =X−nZ(W,X)for anyW∈ W(V,ψ), and any integer n. Proof of Lemma. Denotebybmthecoefficient/integraltext UFW(̟mu)d×u. ThenZ(̟nW,X) is/summationtext m∈ZXmbm+n, which can be rewritten X−nZ(W,X). /square Deducing Theorem 3.5. The representation K(n−1)is formed by restricting the right translation representation on (Φ−)n−2KfromP2toG1, then taking the quo- tient by the G1-stable submodule (Φ−)n−2K(U2,1). By Corollary 2.12, the right translation representation on (Φ−)n−2Kis given by translations of the restricted Kirillov functions W|(x0 0I), denotedW(x) for short. As an endomorphism of the quotientmodule K(n−1),τ(̟) satisfiesapolynomial Xn−an−1Xn−1−···−a1X−a0 (in fact we can take a0to be−1). Hence for any restricted Kirillov function W(x) we have ̟nW(x) =n−1/summationdisplay i=0ai̟iW(x)+W1(x), for some element W1of ((Φ−)n−2K)(U2,1). Therefore we get a relation Z(̟nW,X) =n−1/summationdisplay i=0aiZ(̟iW,X)+Z1(X) withZ1(X) being a Laurent polynomial by Lemma 3.2. Using Lemma 3.7, then multiplying through by Xnand rearranging we get Z(W,X)(1−/summationtextn−1 i=0aiXn−i) = XnZ1(X) which completes the proof since a0is a unit. /square 4.Functional Equation and Gamma Factor 4.1.Contragredient Whittaker Functions. There isan analogueofthe contra- gredient which is reflected on the level of Whittaker functions by a t ransform/tildewidest(−); the functional equation will relate the zeta integral of Wto that of its transform. We will need the following two matrices: w= 0···0 1 0··· −1 0 ... (−1)n−1···0 0 , w′= (−1)n0···0 0 0 ···(−1)n−2 ...... 0 (−1)0···0 For any element Wof IndG Nnψ, define the transform /tildewiderWofWas/tildewiderW(g) :=W(wgι), wheregι:=tg−1. Observation 4.1. IfVis of Whittaker type, then for v∈V,/tildewiderWvis an element of IndG Nψbecause /tildewiderW(ng) =W(w(ng)ι) =W(wnιw−1wgι) =ψ(wnιw−1)W(wgι) =ψ(n)/tildewiderW(g).16 GILBERT MOSS ThusZ(/tildewidew′W,X;j) lands inA[[X]][X−1] by Lemma 3.2. In this section we state the second main result and recover the rationality properties of Se ction 2.2 for Z(/tildewidew′W,X;j). The second main result is as follows: Theorem 4.2. SupposeAis a Noetherian W(k)-algebra, and suppose Vin RepA(G)is co-Whittaker and primitive. Let Sdenote the multiplicative subset of Theorem 3.5. Then there exists a unique element γ(V,X,ψ)ofS−1A[X,X−1]such that for any W∈ W(V,ψ), Z(W,X;j)γ(V,X,ψ) =Z(/tildewidew′W,1 qnX;n−2−j) for0≤j≤n−2. The proof of Theorem 4.2 is in Section 5. We now verify that Z(/tildewidew′W,1 qnX;j) always lives in S−1A[X,X−1]. Proposition 4.3. SupposeVinRepA(G)is admissible, Whittaker type, G-finite, Schur, and primitive. Let Vιdenote the smooth A[G]-module whose underlying A- module isVand whoseG-action is given by g·v=gιv. ThenVιis also admissible, Whittaker type, G-finite, Schur, and primitive. Proof.Consider the map Hom Nn(V,ψ)→HomNn(Vι,ψ) given by λ/mapsto→/tildewideλ, where /tildewideλ:x/mapsto→λ(wx). We have /tildewideλ(n·v) =λ(wnιw−1wv) =ψ(n)/tildewideλ(v), which shows /tildewideλindeed defines an element of Hom Nn(Vι,ψ). Sincew2= (−1)n−1In, it is an isomorphism of A-modules. Admissibility, G-finiteness, and Schur-ness all hold for Vιsinceg/mapsto→gιis a topological automorphism of the group G. SinceVis Schur, Amust be connected, hence Vmust be primitive since it is Schur. /square In particular, ( Vι)(n)=Vι/Vι(Nn,ψ) is free of rank one and we may define (/tildewiderW)v(g) =/tildewideλ(gιv) and take W(Vι,ψ) to be the A-module {(/tildewiderW)v:v∈Vι}as before. Note that this is precisely the same as {/tildewide(Wv) :v∈V}. We record this simple observation as a Lemma: Lemma 4.4. Ifλis a generator of HomNn(V,ψ)then/tildewideλ:x/mapsto→λ(wx)is a generator ofHomNn(Vι,ψ)and defines W(Vι,ψ). There is an isomorphism of G-modules W(V,ψ)→ W(Vι,ψ)sendingWto/tildewiderW. Thus all the hypotheses for the results of the previous sections, in particular Theorem 3.5, apply to Vιwhenever they apply to V, so we get Z(/tildewidew′W,X;j) is inS−1A[X,X−1]. Now we can make the substition1 qnXforXin the ratio of polynomials Z(/tildewidew′W,X;j)to getZ(/tildewidew′W,1 qnX;j). It will againbe in S−1A[X,X−1] because this process swaps the first and last coefficients in the den ominator (and q is a unit in Asinceqis relatively prime to ℓ). 4.2.Zeta Integrals and Tensor Products. The goal of this subsection is to check that the formation of zeta integrals commutes with change o f base ring A. For anyf:A→B, denote by ψA⊗Bthe free rank one B-module with action given by the character f◦ψ. The group action on V⊗ABis given by acting in the first factor. Let idenote the map V→V⊗AB. Proposition 2.1 (6), gives the following lemma. Lemma 4.5. (1) IfVis of Whittaker type, so is V⊗AB.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 17 (2) Letλgenerate HomA[N](V,ψ)as anA-module. Then λ⊗idis a generator of HomB[N](V⊗B,ψ⊗B). (3) LetWv⊗b(g) := (f◦λ)(gv)⊗bdefine elements of W(V⊗B,ψ⊗B). Then f◦Wv=Wi(v)for anyv∈V. From the definition of integration given in §1.1, it follows that if Φ kis the characteristic function of some Hk, then/integraltext (f◦Φk)d(f◦µ×) = (f◦µ×)(Hk) = f/parenleftbig/integraltext Φkd(µ×)/parenrightbig . It follows from the definitions that ( f◦/tildewiderW)(x) =/tildewiderf◦W(x). Corollary 4.6. LetFdenote the map of formal Laurentseries rings A[[X]][X−1]→ B[[X]][X−1]induced by f, then we have F(Z(Wv,X;j)) =Z(f◦W,X;j) =Z(Wi(v),X;j) (1) F/parenleftig Z(/tildewidew′W,X;j)/parenrightig =Z(f◦/tildewidew′W,X;j) =Z(/tildewiderw′(f◦W),X;j) (2) for anyWinW(V,ψ), and for 0≤j≤n−2. The next proposition follows from the linearity of the zeta integrals a nd the transform/tildewidest(−). Proposition 4.7. Suppose there is an element γ(V,X,ψ)inA[[X]][X−1]satisfying a functional equation as in Theorem 4.2 for all Wv∈ W(V,ψ). Then the element F(γ(V,X,ψ))∈B[[X]][X−1]satisfies the functional equation for all W∈ W(V⊗ B,ψ⊗B). 4.3.Construction of the Gamma Factor. We define the gamma factor to be what it must in order to satisfy the functional equation of Theorem 4.2 for a single, particularly simple Whittaker function W0. We seek a W0such thatZ(W,X;0) is a unit inS−1A[X,X−1]. By Proposition2.8 and Lemma 2.10, we have that c-IndP2 U2ψ⊂(Φ−)n−2K. Since c-IndP2 U2ψis isomorphic to C∞ c(F×,A) via restriction to G1(recall that C∞ c(F×,A) denotes the locally constant compactly supported functions F×→A), we find the following: Proposition 4.8. SupposeVinRepA(G)is of Whittaker type. Then the charac- teristic function of U1 Fis realized as a restricted Whittaker function W0(g0 0In−1)for someW0inW(V,ψ). From now on, the symbol W0will denote a choice of element in W(V,ψ) whose restriction to (g0 0In−1) is the characteristic function of U1 F. ThenZ(W0,X) is/integraltext UFW1(a0 0 1)d×a=µ×(U1 F) = 1. Since we want our gamma factor to satisfy the functional equation for W0, we are left with no choice: Definition 4.9 (The Gamma Factor) .LetAbe any Noetherian W(k)-algebra and suppose VinRepA(G)is of Whittaker type. We define the gamma factor ofVwith respect to ψto be the element of A[[X]][X−1]given byγ(V,X,ψ) := Z(/tildewiderw′W0,1 qnX;n−2). WhenVis co-Whittaker and primitive the uniqueness of this gamma factor will follow from the functional equation: if γandγ′both satisfy the functional equation for all Whittaker functions, then γ=Z(/tildewiderw′W0,1 qnX,n−2) =γ′. In particular for such representations our construction of the gamma factor doe s not depend on the choice ofW0.18 GILBERT MOSS 4.4.Functional Equation for Characteristic Zero Points. If the residue field κ(p) ofphas characteristic zero, the reduction modulo pofZ(W,X;j) forms an element of κ(p)(X). Asκ(p) is an uncountable algebraically closed field of char- acteristic zero, we may fix an embedding C֒→κ(p). The proof of [JPSS83, Thm 2.7(iii)(2)] (which occurs in [JPSS83, §2.11]) carries over verbatim to the setting whereπandπ′are admissible, Whittaker type, G-finite representations over any field containing C, hence for representations over κ(p). Thus the reduction modulo pof Ψ(W,X;j) is precisely the integral Ψ( s,W;j) of [JPSS79, §4.1], after replac- ing the complex variable q−(s+n−1 2)with the indeterminate X, and there exists a unique element, which we will call γp(s,V⊗κ(p),ψ), inκ(p)(q−s) such that for all W∈ W(Vp⊗κ(p),ψp) and for all j≥0, Ψ(1−s,/tildewidew′W;n−2−j) =γp(s,V⊗κ(p),ψp)Ψ(s,W,j). The changeofvariable s/mapsto→1−scan be re-written as −(s+n−1 2)/mapsto→(s+n−1 2)−n, so writing the functional equation in terms of Xwe have shown the following Lemma: Lemma 4.10. SupposeVis admissible of Whittaker type, and G-finite. For each primepofAwith residue characteristic zero, there exists a unique ele mentγp(V⊗ κ(p),X,ψp)inκ(p)(X)such that for all WinW(V⊗κ(p),ψp)and for0≤j≤n−2 we have Z(/tildewidew′W,1 qnX;n−2−j) =γp(X,V⊗κ(p),ψp)Z(W,X;j). Moreover,γp(V⊗κ(p),X,ψp) =γ(V,X,ψ) modpby uniqueness in [JPSS79] . 4.5.Proof of Functional Equation When Ais Reduced and ℓ-torsion Free. In the case that Ais reduced and ℓ-torsion free as a W(k)-algebra, we get a slightly stronger result than that of Theorem 4.2. Theorem 4.11. IfAis a Noetherian W(k)-algebra and Ais reduced and ℓ-torsion free, then the conclusion of Theorem 4.2 holds for any VinRepA(G)which is G-finite, and admissible of Whittaker type. Proof.Letpbe any characteristic zero prime, and let fp:A→κ(p) be reduction modulop. Corollary 4.6 and Lemma 4.10 tell us that fp/parenleftbigg γ(V,X,ψ)Z(W,X)−Z(/tildewidew′W,1 qnX;n−2)/parenrightbigg = 0 foranyWinW(V,ψ), not justW0. This shows that the difference γ(V,X,ψ)Z(W,X)−Z(/tildewidew′W,1 qnX;n−2) is in the intersection of all characteristic zero primes of A. WhenAis reduced its zero divisors are the union of its minimal primes, so it is ℓ-torsion free if and only if all minimal primes have residue characteristic zero. Thus when Ais reduced and ℓ-torsion free, the intersection of all characteristic zero primes o fAequals zero, so the functional equation holds for any WinW(V,ψ). We now prove uniqueness. If there were another element γ′satisfying the same property, it would satisfy the functional equation in κ(p) for allWi(v)by reduction, so it satisfies the functional equation for all WinW(V⊗κ(p),ψp). But uniqueness in Lemma 4.10 then shows fp(γ(V,X,ψ)−γ′) = 0 for all characteristic zero primes pofA. Again, this means γ′=γ(V,X,ψ).INTERPOLATING LOCAL CONSTANTS IN FAMILIES 19 We get rationality by observing that whenever Vis admissible of Whittaker type, it has a canonical co-Whittaker submodule Tby Proposition 2.30, which is primitive if Vis primitive. Since γ(T,X,ψ) satisfies the functional equation for all WinW(T,ψ), we must have γ(T,X,ψ) =γ(V,X,ψ) by the construction of the gamma factor. But γ(T,X,ψ) is inS−1A[X,X−1] by Theorem 3.5, which holds for primitive co-Whittaker modules. /square 5.Universal Gamma Factors WhenVis primitive and co-Whittaker, we can remove the hypothesis that A is reduced and ℓ-torsion free by specializing the gamma factor for the universal co-Whittaker module eW. Theorem 5.1 ([Hel12a] Thm 12.1) .Any block eZof the Bernstein center of RepW(k)(G)is a finitely generated (hence Noetherian), reduced, ℓ-torsion free W(k)-algebra. By Proposition 2.31, eWis co-Whittaker, and since it is clearly primitive, all the hypotheses of Proposition 4.11 are satisfied. Hence (Thm 4.11) there exists a unique gamma factor in S−1(eZ[X,X−1]), which we will denote Γ( eW,X,ψ), satisfying the functional equation for all WinW(eW,ψ). Proof of Theorem 4.2. SinceVis primitive and co-Whittaker, there is a (unique) primitive idempotent eofZand a ring homomorphism fV:eZ →EndG(V)∼→A, and a surjection of A[G]-moduleseW⊗fVA→Vpreserving the top derivative, so thatfV(Γ(eW,X,ψ)) =γ(eW⊗fVA,X,ψ). Since Γ( eW,X,ψ) satisfies the functional equation for all WinW(eW,ψ), we can apply Proposition 4.7 again to concludethat γ(eW⊗A,X,ψ)satisfiesthe functionalequationforall WinW(eW⊗ A, ψ). SinceeW⊗Ahas a surjection onto Vpreserving the top derivative, Lemma 2.6 tells us that W(V,ψ) =W(eW⊗A,ψ). The functional equation shows that Definition 4.9 gives a unique gamma factor, hence γ(V,X,ψ) =γ(eW⊗A,X,ψ); it satisfies the functional equation for all WinW(V,ψ). Note that since Γ( eW,X,ψ) is inS−1(eZ[X,X−1]), its image in fVis in the corresponding fraction ring of A[X,X−1]. This proves Theorem 4.2. /square We can extend the uniqueness and rationality result to a larger class of rep- resentations, though with a weaker functional equation coming on ly from the co- Whittaker case: Corollary 5.2. LetVbe admissible, primitive, of Whittaker type and let Tbe its canonical co-Whittaker submodule. Then there exists a u nique gamma factor γ(V,X,ψ)inS−1(A[X,X−1])which equals γ(T,X,ψ), and satisfies the functional equation for all WinW(T,ψ). Proof.WhenVis admissible of Whittaker type it has a canonical co-Whittaker sub by Proposition 2.30. We have just shown that its gamma factor γ(T,X,ψ) satisfies the functional equation for all WinW(T,ψ). Applying Proposition 2.6 withα:T→Vbeing the inclusion map, we conclude that W(T,ψ)⊂ W(V,ψ). The coefficients of the series Z(/tildewiderw′W0,1 qnX;n−2) in Definition 4.9 are determined byG-translates of the Whittaker function W0, so occurs already in W(T,ψ), so by definitionγ(T,X,ψ) =γ(V,X,ψ). In particular γ(V,X,ψ) lies inS−1A[X,X−1] and satisfies the functional equation for all WinW(T,ψ). /square20 GILBERT MOSS We can make precise the sense in which we have created a universal g amma factor: Corollary 5.3. SupposeAis a Noetherian W(k)-algebra, and suppose Vis a co-Whittaker A[G]-module in the subcategory eRepW(k)(G)ofRepW(k)(G). Then there is a homomorphism fV:eZ →AandfV(Γ(eW,X,ψ))equals the unique γ(V,X,ψ)satisfying a functional equation for all WinW(V,ψ). Again,wecanbroadentheclassofrepresentationsatthecostof amorerestrictive functional equation: Theorem 5.4. SupposeAis any Noetherian W(k)-algebra, and suppose Vis an admissibleA[G]-module of Whittaker type in the subcategory eRepW(k)(G). Then there is a homomorphism fV:eZ →Aand the gamma factor of Corollary 5.2 equalsfV(Γ(eW,X,ψ)). Proof.We define fVto be the homomorphism eZ →EndG(T)∼→AwhereT is the canonical co-Whittaker submodule of Proposition 2.30. Since Tlies in eRepW(k)(G),eW⊗fVAsurjectsonto T, andwehave fV(Γ(eW,X,ψ) =γ(T,X,ψ) (Prop2.6), andsince Tinjects into V(with topderivativepreserved),againbyProp 2.6, we have fV(Γ(eW,X,ψ)) =γ(eW⊗eZ,fVA,X,ψ) =γ(T,X,ψ) =γ(V,X,ψ). /square References [BD84] J. Bernstein and P. Deligne. Le “centre” de Bernstein .Representations des groups re- ductifs sur un corps local, Traveux en cours , 1984. [BH06] Colin Bushnell and Guy Henniart. The Local Langlands Conjecture for GL(2). Berlin Heidelberg: Springer-Verlag, 2006. [BZ76] I. N. Bernshtein and A. V. Zelevinskii. Representati ons of the group GL(n,F) where F is a nonarchimedean local field. Russian Math. Surveys 31:3 (1976), 1-68 , 1976. [BZ77] I. N. Bernshtein and A. V. Zelevinskii. Induced repre sentations of reductive p-adic groups. I. Annales scientifiques de l’ ´E.N.S., 1977. [CPS10] James Cogdell and I.I. Piatetski-Shapiro. Derivat ives and L-functions for GL(n).The Heritage of B. Moishezon, IMCP , 2010. [EH12] Matthew Emerton and David Helm. The local Langlands c orrespondence for GL(n) in families. Ann. Sci. E.N.S. , 2012. [Hel12a] David Helm. The Bernstein center of the category of smooth W(k)[GLn(F)]-modules. arxiv:1201.1874 , 2012. [Hel12b] David Helm. Whittaker models and the integral Bern stein center for GL(n). arXiv:1210.1789 , 2012. [Hen93] Guy Henniart. Caracterisation de la correspondanc e de langlands locale par les facteurs ǫde paires. Inventiones Mathematicae , 1993. [Jia13] Dihua Jiang. On the local Langlands conjecture and r elated problems over p-adic local fields.Proceedings of the 6th International Congress of Chinese Ma thematicians , 2013. [JPSS79] Herve Jacquet, Ilja Iosifovitch Piatetski-Shapi ro, and Joseph Shalika. Automorphic forms on GL(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. [KM14] Robert Kurinczuk and Nadir Matringe. Rankin-selber g local factors modulo ℓ. arXiv:1408.5252v2 , 2014. [M´12] Alberto M´ ınguez. Fonctions Zˆ eta ℓ-modulaires. Nagoya Math. J. , 208, 2012. [Vig96] Marie-France Vigneras. Representations ℓ-modulaires d’un groupe reductif p-adique avec ℓdifferent de p. Boston: Birkhauser, 1996.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 21 [Vig00] Marie-France Vigneras. Congruences modulo ℓbetween ǫfactors for cuspidal represen- tations of GL(2).Journal de theorie des nombres de Bordeaux , 2000. [Zel80] A. V. Zelevinsky. Induced representations of reduc tivep-adic groups. II. on irreducible representations of GL(n).Annales scientifiques de l’ ´E.N.S., 1980.

1803.01280v2.Optimization_of_Time_Resolved_Magneto_optical_Kerr_Effect_Signals_for_Magnetization_Dynamics_Measurements.pdf

Optimization of Time -Resolv ed Magneto -optical Kerr Effect S ignals for Magnetization Dynamics Measurements Dustin M. Lattery1, Delin Zhang2, Jie Zhu1, Paul Crowell3, Jian-Ping Wang2 and Xiaojia Wang1* 1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA 2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA 3School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA *Corresponding authors: wang4940@umn.edu Abstract: Recently magnetic storage and magnetic memory have shifted towards the use of magnetic thin films with perpendicular magnetic anisotropy (PMA). Understanding the magnetic damping in these ma terials is crucial, but normal Ferromagnetic Resonance (FMR) measurements face some limitations. The desire to quantify the damping in materials with PMA has resulted in the adoption of Time -Resolved Magneto -optical Kerr Effect (TR -MOKE) measurements. In t his paper, we discuss the angle and field dependent signals in TR -MOKE, and utilize a numerical algorithm based on the Landau -Lifshitz -Gilbert (LLG) equation to provide information on the optimal conditions to run TR -MOKE measu rements . I. INTRODUCTION Spintronics utilizing perpendicular magneti c anisotropy (PMA) are very promising for the advancement of computer memory, logic, and storage. Due to the time scale of magnetic switching in these devices (~ 1 ns), it is crucial to understand the ultrafast dy namic magnetization, which behave according to the Landau -Lifshitz -Gilbe rt (LLG) equation. The application of this equation to understand magnetization dynamics requires knowledge of the magnetic anisotropy and the Gilbert damping (α). While anisotropy can be determined through magnetostatic measurements, extracting α requires measurements that can capture the dynamic magnetization at time scales faster than magnetic switching. To date, the most common method to do this is through frequency domain measureme nts of ferr omagnetic resonance ( FMR ). By measuring the resonance frequency and linewidth as a function of field, FMR can probe both the magnetic anisotropy a nd Gilbert damping . As spintronic applications begin to use materials with large PMA, the use of another technique, time -resolved magneto -optical Kerr effect (TR -MOKE), has increased. This technique (which is essentially a time -domain FMR measurement technique) is able to measure at higher resonance frequencies and external fields, which allows ex tremely hard mag netic materials to be measured . There are many papers discussing TR -MOKE measurements for measuring the Gilbert Damping. Most of these papers utilize similar polar MOKE measurement techniques, but there is often a large variation in both the Hext range for measurements and in the angle of external field. While some papers utilize in -plane external field because of its well -understood frequency dependence, others choose to apply the field at a chosen angle away from the surface normal . It has been theorized and shown in measurements that the process of applying the field at some angle between 0 and 90° is beneficial to increase the TR -MOKE signal amplitude, but the explanations as to why this occurs are lacking . In this paper, we aim to discuss why the signal depends on the angle of external field and calculate the optimal angle for conducting TR -MOKE measurements of damping on magnetic materials with PMA. II. FINITE DIFFERENCE METHOD LANDAU -LIFSHITZ -GILBERT EQUATIONS Simulations in this work utilize a finite difference approach to solve the LLG equation (Eq. 1) with an explicit solution for the magnetization vector ( M) as a function of time following the forward Euler method. eff sdd dt M dt MMM H M (1) where M is the magneti zation vector with a magnitude of Ms (the saturation magnetization), γ is the gyromagnetic ratio, Heff is the effective magnetic field, and α is the Gilbert damping parameter. The vector Heff is determined by taking the gradient of the magnetic free energy density ( F) with respect to the magnetization direction ( eff FM H ). The scalar quantity F is the summation of contributions from Zeeman energy (from the external magnetic field, Hext), perpendicular uniaxial magnetic anisotropy ( Ku), and the demagnetizing field (assuming the sample is a magnetic thin film). While Eqn. 1 is often used to describe magneto -dynamics due to the use of α, it is not conducive to numerical solutions of this ordinary differential equation. To simplify the development of computational algorithms, it is preferential to utilize the Landau -Lifshitz equation (Eq. 2). eff eff 2 s'd dt M MM H M M H . (2) The coefficients in Eq. 2 can be related to the previously defined constants in Eqs. 3 and 4 [1]. 2' 1 (3) s'M (4) In equilibrium, M is parallel to Heff, and so the magnetization does not precess . If the magnetization is removed from the equilibrium direction, it will begin precessing around the equilibrium direction, finally damping towards equilibrium at a rate determined by the magnitude of α (shown in Fig. 1) . Figure 1 . A three -dimensional representation of the magnetization vector ( M) precessing around the equilibrium direction ( θ) displayed on the surface of a sphere of radius Ms. The equilibrium direction is controlled by the magnitude and direction ( θH) of the external magnetic field vec tor (Hext). The change in the z-component of magnetization (Δ Mz) is proportional to the TR -MOKE signal. To initiate precession, a thermal demagnetization process is applied , emulating TR-MOKE measurements. For TR -MOKE measurements, a “pump” laser pulse increases the temperature at an ultrafast time scale, causing a thermal demagnetization (a decrease in Ms caused by temperature) [2, 3] . This thermal demagnetization temporarily moves the equilibrium direction causing the magnetization to begin precession, which is continued even when Ms has recovered to its original state. Here, the demagnetization process is treated as a step decrease in Ms that lasts for 2.5 ps before an instant recovery to the initial value. All signal analysis discussed in this work is following the recovery of Ms. For polar MOKE measurements, the projected magnetization in the z -direction ( Mz, through -plane magnetization ) is proportional to the Kerr rotation [4]. The projection of Mz in time during precession will appear is a decaying sinusoid ( sin exp /zM t t t ), which is also captured by TR -MOKE measurements. The amplitude of the precession will greatly depend on the applied field magnitude and angle, which is also carried into TR -MOKE signal. By analyzing the precession as a function of field and angle, the precession amplitude (delta Mz) can be extracted. Figure 2 shows the process of extracting the amplitude as a function of angle for two different regions of magnetic field. Tracking this signal amplitude as a function of θH, reveals that the precession (and thus the signal) will be maximized for a certain θH as shown in Fig. 2(b). Maximizing the oscillation implies that it will be beneficial to maximize the “magnetic torque” term (M × Heff, which prefers a large angle between M and Heff), but it also important to factor in that TR -MOKE measures the projection of the magnetization along the z-direction (which prefers θ = 90°). Because of this, the value of θH,MAX requires weighing inputs from both the magnetic torque and the z-direction projection of magnetization. Figure 2. For specific conditions, the LLG simulation will produce a time -dependent magnetization vector. The difference between the maximum and mini mum of the z-component of magnetization in time (Δ Mz) provides information about the strength of the TR -MOKE signal. These simulations are conducted for a range of θH resulting in the curves in (b). The trend of signal with increasing θH also depends on th e magnitude of the external field relative to Hk,eff, as shown by the black ( Hk,eff < Hext) and red ( Hk,eff > Hext) lines. Depending on whether the field ratio ( Hext/Hk,eff) the angular dependence on magnitude will drastically change. For Hext<Hk,eff, the magnetization will be in equilibrium between the perpendicular direction and the in -plane direction (0 ≤ θ ≤ 90°). Maximizing the magnetic torque and projection in the z -direction in these cases will cause Hext applied in -plane (θH = 90°) to be the optimal setup [shown by the red line in Fig. 4(b)]. Once Hext exceeds Hkeff, the Stoner -Wolfarth minimum energy model predicts that the magnetization will approach the direction of external field, but never align (except along θH = 0 or 90 °). Because these two directions will have no magnetic torque, there should be no magnetic precession, and thus there will be amplitude minima at these extremes. Between these two angles, the two effects for optimizing signal will complete, leading to a n amplit ude maximum at an angle that depends on the size of the ratio. Figure 3 shows a contour plot of the dependence on signal amplitude as a function of both the magnitude of Hext and θH. The highest amplitude of precession will occur near Hk,eff when the field is applied in the film plane. If the external field is greater than Hk,eff, it is beneficial to conduct the measurement at an angle that is out of the film plane. To reveal this trend, the dotted red line in Fig. 3 indicates the angle of maximum signal ( θH,MAX ) as a function at specified field ratios. Note that the curve does not follow the gradient of signal vs. θH and Hext. This is due to the definition of θH,MAX as the value of θH that maximizes signal for a given Hext , instead of a maximization of signal with both parameters. Based on these results, measurement conditions can be tuned to maximize the signal based on the field ratio. For example, if the maximum strength of the magnetic field is only 2 Hk,eff, then it would be beneficial to set θH > 60°. Furthermore, measurements conducted at a constant field and a varied magnetic field angle, should not necessarily conduct the measurement at the highest possible Hext if the goal is to maximize SNR. Figure 3. A contour plot of the relative signal size as a function of field ratio ( Hext/Hk,eff) and θH where a value of “1” indicates the maximum possible signal. The dotted line shows the θH where the signal is maximized at a specific field ratio. For field -swept measurements, (where the angle is held constant and the field is swept) Fig. 4 should provide a simple guide for maximizing signals (a summary of θH,MAX in Fig. 3). To further assist in the design of TR -MOKE signals to maximize SNR, we suggest a simplified estimation for the determination of th e amplitude of TR -MOKE signal. Equation 5 predicts the precession amplitude based on the equilibrium direction ( θ, from Fig. 1) and the external field angle. The magnitude of Hext is integrated into Eq. 5 through the θ through Eq. 6 whic h provides the mini mum energy condition. H ssin sinzM M (5) ext H k,eff2 sin sin 2HH (6) This simplified expression is based on the product of the two components for signal maximization previously discussed: the projection of the magnetization in the z -direction , sin , and the magnetic torque, H sin . While the simplified expression presented in Eq. 2 cannot capture all the details of a more complex LLG simulation, it is more than accurate enough for an initial estimate of θH,MAX , as shown by the comparison in Fig. 4. Figure 4. The trend of θH,MAX at a given field ratio. The open circles indicate results from the LLG simulation discussed in Section I, while the red curve is the simplified model from Eq. 5. III. COMPARING SIMULATION RESULTS TO TR -MOKE MEASUREMENTS To verify the precited results for the m aximum TR -MOKE signal amplitude, a series of measurements were conducted on a 300 °C post -annealed W/CoFeB/MgO film (see our previous publication for more information ). After conducting measurements, the thermal background was subtracted leaving purely the decaying sinusoidal term. The oscillation amplitude from measurement was calculated as shown in Fig. 2a. Results from four values of Hext and six value s of θH are summarized in Fig. 5. Figure 5. Normalized TR -MOKE oscillation amplitudes directly for a W/CoFeB/MgO when Hext is 4, 6, 8, and 10 kOe. The open red circles show the measurement data (a line between points is provided to guide the eye) while the black curves indicate the results from the LLG simulations for a material with Hk,eff ≈ 6 kOe. Comparisons between the trends predicted simula tions and measurement results show remarkable agreement. As expected, the signal amplitude decreases with increasing angle for Hext < Hk,eff (Hk,eff ≈ 6 kOe ) and decreases with increasing angle for Hext > Hk,eff. These measurements can even capture the predicted peak of amplitude at nearly the same θH for fields near Hk,eff. For the 6 kOe measurements, there is a slight deviation in the amount of decay in signal strength for decreasing θH (simulations predict a s lower decrease). This is most likely due to an inhomogeneous broadening effect (i.e. the Hk,eff in the sample has a distribution of values) leading to a deviation from theory near Hk,eff. While the θH in the setup used in this experiment was limited, these results verify that the excellent agreement between simulation and measurement. IV. CONCLUSION In conclusion, we utilized a numerical approach to calculate the dynamic response of magnetization to a demagnetization process. We find that the size of the magnetic precession, and thus the size of the TR -MOKE signal depends on the angle and amplitude of the external field (relative to Hk,eff). To verify the results of these simulations, we conducted measurements on a W/CoFeB/MgO sample with perpendicular magnetic anisotropy. The results of the measurements show that the magnitude of the TR -MOKE signal shows good agreement with our prediction. These results should assist to m aximize the SNR in TR-MOKE measurements. ACKNOWLEDGEMENTS This work is supported by C -SPIN (award #: 2013 -MA-2381) , one of six centers of STARnet, a Semiconductor Research Corporation progra m, sponsored by MARCO and DARPA. REFERENCES [1] Iida, S., 1963, "The difference between gilbert's and landau -lifshitz's equations," Journal of Physics and Chemistry of Solids, 24(5), pp. 625 -630. [2] van Kampen, M., Jozsa, C., Ko hlhepp, J. T., LeClair, P., Lagae, L., de Jonge, W. J. M., and Koopmans, B., 2002, "All -Optical Probe of Coherent Spin Waves," Physical Review Letters, 88(22), p. 227201. [3] Zhu, J., Wu, X., Lattery, D. M., Zheng, W., and Wang, X., 2017, "The Ultrafast Laser Pump - Probe Technique for Thermal Characterization of Materials With Micro/Nanostructures," Nanoscale and Microscale Thermophysical Engineering, 21(3), pp. 177 -198. [4] You, C. -Y., and Shin, S. -C., 1998, "Generalized analytic formulae for magneto -optical Kerr effects," Journal of Applied Physics, 84(1), pp. 541 -546.

2401.12022v1.Damping_Enhanced_Magnon_Transmission.pdf

Damping-Enhanced Magnon Transmission Xiyin Ye,1Ke Xia,2Gerrit E. W. Bauer,3, 4and Tao Yu1,∗ 1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2School of Physics, Southeast University, Jiangsu 211189, China 3WPI-AIMR and Institute for Materials Research and CSRN, Tohoku University, Sendai 980-8577, Japan 4Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing 100190, China (Dated: January 23, 2024) The inevitable Gilbert damping in magnetization dynamics is usually regarded as detrimental to spin transport. Here we demonstrate in a ferromagnetic-insulator–normal-metal heterostructure that the strong momentum dependence and chirality of the eddy-current-induced damping causes also beneficial scattering properties. Here we show that a potential barrier that reflects magnon wave packets becomes transparent in the presence of a metallic cap layer, but only in one direction. We formulate the unidirectional transmission in terms of a generalized group velocity with an imaginary component and the magnon skin effect. This trick to turn presumably harmful dissipation into useful functionalities should be useful for future quantum magnonic devices. Introduction .—Magnonic devices save power by ex- ploiting the collective excitations of the magnetic or- der, i.e., spin waves or their quanta, magnons, for non- reciprocal communication, reprogrammable logics, and non-volatile memory functionalities [1–10]. The possibil- ity to modulate magnon states and their transport in fer- romagnets by normal metals or superconductors brings functionalities to spintronics [11–14], quantum informa- tion [15–21], and topological materials [22, 23]. The pre- diction of inductive magnon frequency shifts by supercon- ducting gates on magnetic insulators [24–30] have been experimentally confirmed [31]. Normal metals are not equally efficient in gating magnons [32–35], but the stray fields of magnetically driven “eddy currents” [36–43] sig- nificantly brake the magnetization dynamics [36]. The intrinsic Gilbert damping seems to be detrimental to transport since it suppresses the magnon propagation length. However, in high-quality magnets such as yt- trium iron garnet (YIG) films, this is not such an issue since the magnon mobility is often limited by other scat- tering processes such as two-magnon scattering by disor- der, and measurements can be carried out in far smaller length scales. Natural and artificial potential barriers are impor- tant instruments in electronics and magnonics by confin- ing and controlling the information carriers. They may guide magnon transport [31, 44], act as magnonic logic gate [45], induce magnon entanglement [18, 46], and help detecting exotic magnon properties [47–50]. In the lin- ear transport regime, the transmission of electrons and magnons through an obstacle has always been assumed to be symmetric, i.e., the same for a wave or particle coming from either side. In this Letter, we address the counter-intuitive ef- fect that the strong momentum-dependent eddy-current- induced damping by a normal metal overlayer as shown in Fig. 1 may help surmount obstacles such as mag- netic inhomogeneities [51], artificial potential barriers formed by surface scratches [52], or dc-current carryingwires [46]. Here we focus on the band edges of magnetic films that are much thinner than the extinction length of the Damon-Eshbach surface states in thick slabs and are therefore not chiral. Instead, the effect therefore origi- nates from the Oersted fields generated by the eddy cur- rents in the overlayer that act in only half of the recip- rocal space [7] and causes magnon accumulations at the sample edges or magnon skin effect [8, 9]. The trans- mission through a barrier that is small and symmetric for magnons with opposite wave numbers in an uncov- ered sample becomes unidirectional with the assistance of dissipative eddy currents. FIG. 1. Ferromagnetic insulator-normal metal heterostruc- ture. An in-plane external magnetic field H0orients the magnetization at an angle θwith the ˆz-direction. The yellow sheet between the normal metal and ferromagnetic insulator indicates suppression of the exchange interaction and conven- tional spin pumping. Model and non-perturbation theory .—We consider the ferromagnetic insulator (FI)-normal metal (NM) het- erostructure with thickness 2 dFanddMand an in-plane magnetic field H0in Fig. 1. The saturated equilib- rium magnetization Msmakes an angle θwith the ˆz- direction such that the torques exerted by the external and anisotropy fields cancel. For convenience, we set θ= 0 in the following discussion and defer results forarXiv:2401.12022v1 [cond-mat.mes-hall] 22 Jan 20242 finite θto the Supplemental Material (SM) [53]. We gen- eralize a previous adiabatic theory [7, 36] to the full elec- trodynamics of the system by self-consistently solving the Maxwell equations coupled with the linearized Landau- Lifshitz (LL) equations and Ohm’s Law. This treatment becomes exact in the limit of an instantaneous response of the metal electrons and high-quality ultrathin mag- netic films. The driving force is an externally generated spatiotem- poral magnetization dynamics M(r, t) =M(r, ω)e−iωtat frequency ω. According to Maxwell’s theory, the electric fieldEobeys the wave equation ∇2E(r, ω)+k2 0E(r, ω) = −iωµ0JM, where the wave number k0=ω√µ0ε0,µ0(ε0) is the vacuum permeability (permittivity), and JM= ∇×Mis the “magnetization current” [54]. Disregarding the intrinsic Gilbert damping, the LL equation iωM=−µ0γM×Heff[M] (1) governs the magnetization dynamics in the FI, where γ is the gyromagnetic ratio. The effective magnetic field Heff[M] =−δF[M]/δM(r), where the free energy Fis a functional of the magnetization. It includes the static fieldH0, the dipolar field Hd, and (in the FI) the ex- change field Hex=αex∇2Mthat depends on the spin- wave stiffness αex. In the presence of the NM layer, Heff[M] also contains the Oersted magnetic fields gen- erated by the “eddy” currents J=σE, where the elec- trical conductivity σis real. This defines a closed self- consistency problem that we solve numerically. We consider a thin FI film with constant Ms= (0,0, Ms). The transverse fluctuations M(r, ω) = (Mx(k, ω), My(k, ω),0)eik·rwith in-plane wave vectors k= (0, ky, kz) are small precessions with iMx(k, ω) = akMy(k, ω), where the complex ellipticity akbecomes unity for circular motion. The electric-field modes outside the magnet are plane waves with wave numbers km=p ω2µ0ε0+iωµ0σ, where σ= 0 in the absence of an NM layer. The continu- ity of electric and magnetic fields provides the interface boundary conditions. The field in the FI Eη={x,y,z}(−dF⩽x⩽dF) =E(0) η(−dF⩽x⩽dF) +RkE(0) η(x=dF)e−iAk(x−dF) is now modified by the reflection coefficient Rk= A2 k−B2 k
eiBkdM− A2 k−B2 k
e−iBkdM (Ak−Bk)2eiBkdM−(Ak+Bk)2e−iBkdM,(2) where E(0)is the solution of Eq. (1) inside the FI without the NM cap [53], Ak=p k2 0−k2, and Bk=p k2m−k2. The reflection is isotropic and strongly depends on the wave vector. Naturally, Rk= 0 when dM= 0. On the other hand, when |k|= 0, the electric field cannot escape the FI, since the reflection is total with Rk=−1.A corollary of Maxwell’s equation—Faraday’s Law— reads in frequency space iωµ0[Hd(r, t) +M(r, t)] =∇ × E(r, t). When the magnetization of sufficiently thin mag- netic films is uniform, the Zeeman interaction is propor- tional to the spatial average Hdover the film thickness. Referring to SM for details [53], we find Hd,x= −Rk 4A2 kdFak(e2iAkdF−1)2(−iAkak+ky) +i 2AkdF(e2iAkdF−1) Mx≡ζx(k)Mx, Hd,y=" −Rk 4iAkdF(e2iAkdF−1)2 −ky iAkak+k2 y A2 k+ 2! +k2 y A2 k−k2 y A2 k1 2iAkdF(e2iAkdF−1)# My≡ζy(k)My. By substitution into the LL equation (1), the spin wave eigenfrequencies and ellipticities become ω(k) =µ0γq (˜H0−ζx(k)Ms)(˜H0−ζy(k)Ms),(3a) ak=q (˜H0−ζy(k)Ms)/(˜H0−ζx(k)Ms), (3b) where ˜H0=H0+αexk2Ms. Imω(k)̸= 0 because of the Joule heating due to the eddy currents in the cap layer. Chiral damping and frequency shifts .—The stray elec- tric fields of spin waves propagating perpendicular to the magnetization are chiral, i.e., they depend on their prop- agation direction by a hand rule. When kz= 0,E=Ezˆz is along the equilibrium magnetization and Ez∝My is complex only for positive ky. We illustrate the re- sults of the self-consistent calculations for dF= 100 nm, dM= 500 nm, conductivity σ= 6.0×107(Ω·m)−1 for copper at room temperature [55], applied magnetic field µ0H0= 0.02 T, µ0Ms= 0.178 T, the exchange stiffness αex= 3×10−16m2for YIG [56], and γ= 1.77×1011(s·T)−1. The presence of the NM cap lay- ers shifts the relative phases between the stray electric fields and that of the generating spin waves. We focus here on the wave numbers ky=±1µm−1in Fig. 2(a) [Fig. 2(b)] at which the electric field is in-phase (out- of-phase) with the transverse magnetization Myˆy. The response to an in-phase (out-of-phase) electric field is dis- sipative (reactive). Both components decay in the FI and the vacuum as ∝1/|k|. In the NM, the in-phase compo- nent is screened only in the metal region on the scale of a skin depth λ=p 2/(ωµ0σ)∼1.5µm at ω= 11 GHz. The out-of-phase electric field, on the other hand, cre- ates only a reactive response and is therefore symmetric above and below the metallic film. Also in this case the damping is modulated for constant Gilbert damping by the associated spin wave frequency shift in Fig. 2(b), an effect that cannot be captured by the adiabatic approxi- mation [7, 36].3 FIG. 2. The system responds strongly to a phase difference between the spin waves and their wave vector-dependent ac electric stray fields E. ReEcauses damping [(a)] and Im E a frequency shift [(b)]. Im Ezgoverns the spin wave vector dependence of the chiral damping [(c)]. (d) illustrates the strong ky-dependence of the damping of the lowest standing spin wave for Cu thicknesses dM={50,100,200,500}nm. (e) shows the real and imaginary parts of the reflection coefficient Rkthat causes the frequency shifts plotted in (f). The chirality of the radiated electric field controls the backaction of the NM layer that modifies the magnon dispersion in a chiral fashion. Figure 2(c) illustrates the strong wave vector-dependent damping coefficient αeff(k) =|Imωk|/Reωk. Spin waves propagating in the positive ˆy-direction decay much faster than those along the negative direction, while the damping for positive and negative kzis the same. According to Fig. 2(d), the calculated damping for kz= 0 in Fig. 2(c) increases (de- creases) with the thickness of the Cu (YIG) film. The enhancement of the damping saturates for NM thick- nesses dN>1/p k2+ 1/λ2, depending on the skin depth (λ∼1.5µm) and the wave number 1 /kof the electric field. Moreover, the Kittel mode at k= 0 in Fig. 2(e) is not affected by the metal at all because the reflection coefficient Rk=−1, which implies that the dynamics of the FI and metal fully decouple. Indeed, recent ex- periments do not find a frequency shift of the FMR by a superconducting overlayer [57, 58]. The additional damp- ing by eddy currents reported by Ref. [39] is caused bythe width of the exciting coplanar waveguide, a finite-size effect that we do not address here. The real part of Rkin Fig. 2(e) causes an in-phase Oersted magnetic field that chirally shifts the spin wave frequencies by as much as ∼1 GHz, see Fig. 2(f). Refer- ence [59] indeed reports a frequency shift of perpendicular standing spin wave modes in Bi-YIG films in the presence of thin metallic overlayers. The predicted effects differ strongly from those caused by spin pumping due to the interface exchange coupling αsp= (ℏγ/M sdF)Reg↑↓, where g↑↓is the interfacial spin mixing conductance [60]. αspdoes not depend on the thickness of the metal and vanishes like 1 /dF. The fre- quency shift scales like Im g↑↓/dFand is very small even for very thin magnetic layers. In contrast, the eddy current-induced damping is non-monotonic, scaling like ∝dFwhen 2 kdF≪1, vanishing for much thicker mag- netic layers, and reaching a maximum at dF∼2λ. Unidirectional transmission of wave packets through a potential barrier .—The transmission of a wave packet im- pinging from the left or right at a conventional potential barrier is the same [61]. In the presence of a metal cap, this does not hold for magnons in thin magnetic films. Before turning to the potential scattering in this model, we have to address the effect of the edges. When magnons propagate in the negative direction without damping but decay quickly when propagating in the op- posite one, those reflected at the left boundary of the sample accumulate, which is a non-Hermitian skin ef- fect [62–65]. We substantiate this conclusion by nu- merical calculations for a two-dimensional square lat- tice model with ˆ mi= (1 /√ N)P kˆmkeik·ri, where ˆ mk is the annihilation operator of magnons with frequency ωkfrom Eq. (3a) and ilabels the sites and Nis the number of sites. The Hamiltonian in the real space ˆH0=P ijtjiˆm† jˆmi, where tji= (1/N)P kℏωkeik·(rj−ri)is a hopping amplitude between possibly distant sites iand jand the summation is over the first Brillouin zone. With a coarse-grained lattice constant of ay=az= 0.1µm the reciprocal lattice vector 2 π/ay,zis much larger than the magnon modes of interest (refer to the SM [53] for de- tails). When the frequencies ωkare complex, the Hamil- tonian is non-Hermitian, i.e.,tji̸=t∗ ij. Figure 3(a) shows the winding path of the real and imaginary eigenfrequencies with wave number. In the interval ky= [−25,25]µm−1and an applied magnetic field parallel to the boundary with θ= 0, the complex component is hysteretic, indicating localization of modes at opposite boundaries. Figure 3(b)-(c) show the average spatial distributions W(r) = (1 /Nm)PNm l=1|ϕl(r)|2ofNm lowest-frequency eigenstates ϕl(r) for ky∈[−1,1]µm−1 andkz∈[−1,1]µm−1. When the static magnetic field aligns with the sample boundary z-axis, i.e.θ= 0 in Fig. 3(b), the magnons tend to accumulate at the left4 edge. In the antiparallel configurations θ=π[Fig. 3(c)], the magnons aggregate at the right. In the noncollinear configuration with θ=π/4 [Fig. 3(d)], the maxima shifts to the upper-left corner. While Wis an average, we also illustrate the localization of individual low-frequency modes in SM [53]. FIG. 3. The magnon skin effect caused by chiral damping. (a) Complex spectral winding under periodic boundary con- ditions when kyevolves from −25 to 25 µm−1forθ= 0. (b)- (d) corresponds to the edge or corner aggregations of magnon eigenstates for other magnetic configurations θ∈ {0, π, π/ 4}. We now illustrate the effect of square potential barri- ers of width dand height u0,ˆV(y) =u0[Θ(y+d/2)− Θ(y−d/2)], where Θ( x) is the Heaviside step function, on the magnon transmission along ˆy(⊥Ms). With in- coming ⟨y|k0⟩=eik0y, the scattered states |ψs⟩obey the Lippmann-Schwinger formula [66] |ψs⟩=|k0⟩+1 iℏ∂t−ˆH0+i0+ˆV|ψs⟩. (4) where ˆH0=P kℏωkˆm† kˆmkis the magnon Hamiltonian for an extended film. The transmitted waves read ⟨y|ψs⟩=T+(k0)eik0y,{y, k0}>0 T−(k0)eik0y,{y, k0}<0. (5) In the weak scattering limit |u0d| ≪ | ℏvk0|, T±(k0) = 1±iℏvk0 u0d−vk0 2|vk0|−1 ≈1∓iu0d ℏvk0,(6) where vk0=∂ωk/∂k|k=k0ˆyis a generalized group ve- locity that dissipation renders complex. The imaginary part of the group velocity and transmission amplitudes depend on the direction of the incoming wave: D±(k0) =|T±(k0)|2≈1±2Imu0d ℏvk0 . (7)For example, with u0/ℏ= 30.5 GHz, d= 0.1µm,k0= ±0.8µm−1,vk0>0= (2.32 + 0 .52i) km/s and vk0<0= −(2.64 + 0 .16i) km/s lead to T+(k0>0)≈0.6 while T−(k0<0)≈0.9, so even in the weak scattering limit the NM cap layer significantly and asymmetrically reduces the transmission probability. We can assess the strong scattering regime with |u0d|≳ |ℏvk0|by numerical calculations but find dramatic ef- fects on the time evolution of a real-space spin-wave packet as launched, e.g., by a current pulse in a mi- crowave stripline. We adopt a Gaussian shape Ψ( r,0) = e−(r−r0)2/(2η2)eiq0·rcentered at r0with a width η≫ay,z that envelopes a plane wave with wave vector q0and ˆV(r) = u0f(r) with either f(|y−˜y0|< d) = 1 or f(|z−˜z0|< d) = 1, where ˜ y0and ˜z0are the center of the barriers. According to Schr¨ odinger’s equation Ψ( r, t) = eiˆHt/ℏΨ(r, t= 0) with ˆH=ˆH0+ˆV(r). Numerical results in Fig. 4(a) and (b) u0d≪ |ℏvk0|agree with perturbation theory (7) in the weak scattering regime. However, when |ℏvk0|≲u0dand|Im(v−k0)| ≪ | Im(vk0)|≲|Re(v±k0)| the transmission and unidirectionality becomes almost perfect. Figure 4(c) and (d) show a nearly unidirectional transmission of the wave packet through the potential barrier for the Damon-Eshbach configuration q0⊥Ms; it is transparent for spin waves impinging from the left, but opaque for those from the right. In the calculations, q0=q(0) yˆywith q(0) y=±5µm−1andη= 3µm≫d. The potential barrier is peaked with d=ay,z= 0.1µm and its height u0/ℏ= 15 GHz is relatively weak (the regular on-site energy ∼13 GHz). Also, dM= 50 nm anddF= 20 nm. The results are insensitive to the de- tailed parameter values (see SM [53]). The red and blue curves are the incident and reflected wave packets, re- spectively. When q(0) y<0, the barrier does not affect the wave packet that propagates freely through the poten- tial barrier and accumulates on the left edge [Fig. 4(c)]. When q(0) y>0, as shown in Fig. 4(d), the barrier reflects the wave packet nearly completely, which we associate again with the skin effect since these magnons cannot ac- cumulate on the right side. The unidirectional transmis- sion is therefore a non-local phase-coherent phenomenon that involves the wave function of the entire sample. Since we find the skin effect to be crucial, its absence in waves propagating in the ˆz-direction must affect the transport over the barrier. Indeed, our calculations in Fig. 4(e) and (f) find strong reflection for both propaga- tion directions, even when reducing the barrier height by an order of magnitude to u0= 1.5 GHz (see SM [53]). Discussion and conclusion .—In conclusion, we cal- culate the chiral damping, chiral frequency shift, and anomalous transport of magnonic modes in ferromag- netic films with NM cap layers beyond the adiabatic ap- proximations. We predict anomalous unidirectional spin transport over potential barriers. This effect is rooted in the non-Hermitian magnon skin effect and reflects the5 FIG. 4. Calculated transmissions [(a) and (b)] and time evolu- tion of spin-wave packets in the presence of a potential barrier at the origin when q0⊥Ms[(c) and (d)] and q0∥Ms[(e) and (f)], where Msand the applied magnetic field are parallel to the sample edge with θ= 0. The red and blue curves rep- resent, respectively, the incident and scattered wave packets with propagation directions indicated by arrows. global response of the entire system to a local perturba- tion. 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0808.3923v1.Gilbert_Damping_in_Conducting_Ferromagnets_II__Model_Tests_of_the_Torque_Correlation_Formula.pdf

arXiv:0808.3923v1 [cond-mat.mtrl-sci] 28 Aug 2008Gilbert Damping in Conducting Ferromagnets II: Model Tests of the Torque-Correlation Formula Ion Garate and Allan MacDonald Department of Physics, The University of Texas at Austin, Au stin TX 78712 (Dated: October 29, 2018) We report on a study of Gilbert damping due to particle-hole p air excitations in conducting ferromagnets. We focus on a toy two-band model and on a four-b and spherical model which provides an approximate description of ferromagnetic (Ga,Mn)As. Th ese models are sufficiently simple that disorder-ladder-sum vertex corrections to the long-wavel ength spin-spin response function can be summed to all orders. An important objective of this study is to assess the reliability of practical approximate expressions which can be combined with electro nic structure calculations to estimate Gilbert damping in more complex systems. PACS numbers: I. INTRODUCTION The key role of the Gilbert parameter αGin current- driven1and precessional2magnetization reversal has led to a renewed interest in this important magnetic ma- terial parameter. The theoretical foundations which re- late Gilbert damping to the transversespin-spinresponse function of the ferromagnet have been in place for some time3,4. It has nevertheless been difficult to predict trends as a function of temperature and across mate- rials systems, partly because damping depends on the strength and nature of the disorder in a manner that re- quires a more detailed characterization than is normally available. Two groups have recently5reported success- ful applications to transition metal ferromagets of the torque-correlation formula4,5,6forαG. This formula has the important advantage that its application requires knowledge only of the band structure, including its spin- orbit coupling, and of Bloch state lifetimes. The torque- correlation formula is physically transparent and can be applied with relative ease in combination with modern spin-density-functional-theory7(SDFT) electronic struc- ture calculations. In this paper we compare the pre- dictions of the torque correlation formula with Kubo- formula self-consistent-Born-approximation results for two different relatively simple model systems, an ar- tificial two-band model of a ferromagnet with Rashba spin-orbit interactions and a four-band model which cap- tures the essential physics of (III,Mn)V ferromagnetic semiconductors8. The self-consistent Born approxima- tion theory for αGrequires that ladder-diagram vertex corrections be included in the transverse spin-spin re- sponse function. Since the Born approximation is ex- act for weak scattering, we can use this comparison to assess the reliability of the simpler and more practical torque-correlationformula. Weconcludethat the torque- correlationformulaisaccuratewhentheGilbertdamping is dominated by intra-band excitations of the transition metal Fermi sea, but that it can be inaccurate when it is dominated by inter-band excitations. Our paper is organized as follows. In Section II we ex- plain how we evaluate the transverse spin-spin responsefunction for simple model ferromagnets. Section III dis- cusses our result for the two-band Rashba model while Section IV summarizes our findings for the four-band (III,Mn)V model. We conclude in Section V with a sum- mary of our results and recommended best practices for the use of the torque-correlation formula. II. GILBERT DAMPING AND TRANSVERSE SPIN RESPONSE FUNCTION A. Realistic SDFT vs.s-d and p-d models We view the two-band s−dand four band p−dmod- els studied in this paper as toy models which capture the essential features of metallic magnetism in systems that are, at least in principle9, more realistically described using SDFT. The s−dandp−dmodels correspond to the limit of ab initio SDFT in which i) the majority spind-bands are completely full and the minority spin d-bands completely empty, ii) hybridization between s orpandd-bands is relatively weak, and iii) there is ex- change coupling between dandsorpmoments. In a recent paper we have proposed the following expression for the Gilbert-damping contribution from particle-hole excitations in SDFT bands: αG=1 S0∂ωIm[˜χQP x,x] (1) where ˜χQP x,xis a response-function which describes how the quasiparticle bands change in response to a spatially smooth variation in magnetization orientation and S0is the total spin. Specifically, ˜χQP α,β(ω) =/summationdisplay ijfj−fi ωij−ω−iη∝an}bracketle{tj|sα∆0(/vector r)|i∝an}bracketri}ht∝an}bracketle{ti|sβ∆0(/vector r)|j∝an}bracketri}ht. (2) whereαandβlabel the xandytransverse spin direc- tions and the easy direction for the magnetization is as- summed to be the ˆ zdirection. In Eq.( 2) |i∝an}bracketri}ht,fiandωij are Kohn-Sham eigenspinors, Fermi factors, and eigenen- ergy differences respectively, sαis a spin operator, and2 ∆0(/vector r) is the difference between the majorityspin and mi- nority spin exchange-correlation potential. In the s−d andp−dmodels ∆ 0(/vector r) is replaced by a phenomeno- logical constant, which we denote by ∆ 0below. With ∆0(/vector r) replaced by a constant ˜ χQP x,xreduces to a standard spin-response function for non-interacting quasiparticles in a possibly spin-dependent random static external po- tential. The evaluation of this quantity, and in particu- lar the low-frequency limit in which we are interested, is non-trivial only because disorder plays an essential role. B. Disorder Perturbation Theory We start by writing the transverse spin response func- tion of a disordered metallic ferromagnet in the Matsub-ara formalism, ˜χQP xx(iω) =−V∆2 0 β/summationdisplay ωnP(iωn,iωn+iω) (3) where the minus sign originates from fermionic statistics, Vis the volume of the system and P(iωn,iωn+iω)≡/integraldisplaydDk (2π)DΛα,β(iωn,iωn+iω;k)Gβ(iωn+iω,k)sx β,α(k)Gα(iωn,k). (4) In Eq. ( 4) |αk∝an}bracketri}htis a band eigenstate at momentum k,Dis the dimensionality of the system, sx α,β(k) =∝an}bracketle{tαk|sx|βk∝an}bracketri}ht is the spin-flip matrix element, Λ α,β(k) is its vertex-corrected counterpart (see below), and Gα(iωn,k) =/bracketleftbigg iωn+EF−Ek,α+i1 2τk,αsign(ωn)/bracketrightbigg−1 . (5) We have included disorder within the Born approximation by incorpora ting a finite lifetime τfor the quasiparticles and by allowing for vertex corrections at one of the spin vertices. α,kΛβ,kβ,k α,ksxβ,k k' α,kΛ α'k''β /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 + = FIG. 1: Dyson equation for the renormalized vertex of the tra nsverse spin-spin response function. The dotted line denot es impurity scattering. The vertex function in Eq.( 4) obeys the Dyson equation (Fig. ( 1)): Λα,β(iωn,iωn+iω;k) =sx α,β(k)+ +/integraldisplaydDk′ (2π)Dua(k−k′)sa α,α′(k,k′)Gα′(iωn,k′)Λα′,β′(iωn,iωn+iω;k′)Gβ′(iωn+iω,k′)sa β′,β(k′,k),(6) whereua(q)≡naV2a(q)(a= 0,x,y,z),nais the den- sity of scatterers, Va(q) is the scattering potential (di- mensions: (energy) ×(volume)) and the overline stands for disorder averaging10,11. Ward’s identity requires thatua(q) andτk,αbe related via the Fermi’s golden rule: 1 ταk= 2π/integraldisplay k′ua(k−k′)/summationdisplay α′sa α,α′sa α′,αδ(Ekα−Ek′α′),(7) where/integraltext k≡/integraltext dDk/(2π)D. In this paper we restrict ourselves to spin-independent ( a= 0) disorder and3 spin-dependent disorder oriented along the equilibrium- exchange-field direction( a=z)12. Performing theconventional13integration around the branch cuts of P, we obtain ˜χQP xx(iω) =V∆2 0/integraldisplay∞ −∞dǫ 2πif(ǫ)[P(ǫ+iδ,ǫ+iω)−P(ǫ−iδ,ǫ+iω)+P(ǫ−iω,ǫ+iδ)−P(ǫ−iω,ǫ−iδ)] (8) wheref(ǫ) is the Fermi function. Next, we perform an analytical continuation iω→ω+iηand take the imag- inary part of the resulting retarded response function. Assuming low temperatures, this yields αG=∆2 0 2πs0{Re[P(−iδ,iδ)]−Re[P(iδ,+iδ)]} =∆2 0 2πs0Re(PA,R−PR,R) (9) wheres0=S0/V, PR(A),R=/integraldisplay kΛR(A),R α,β(k)GR β(0,k)sx β,α(k)GR(A) α(0,k) (10) andGR(A)(0,k) is the retarded (advanced) Green’s func- tion at the Fermi energy. The principal difficulty of Eq.( 9) resides in solving the Dyson equation for the ver- tex function. We first discuss our method of solution in general terms before turning in Sections III and IV to its application to the s−dandp−dmodels. C. Evaluation of Impurity Vertex Corrections for Multi-Band Models Eq.( 6) encodes disorder-induced diffusive correlations between itinerant carriers, and is an integral equation of considerable complexity. Fortunately, it is possible to transform it into a relatively simple algebraic equation, provided that the impurity potentials are short-rangedin real space.Referring back at Eq.( 6) it is clear that the solution of the Dysonequationwouldbe trivialifthevertexfunction wasindependent ofmomentum. That is certainlynot the case in general, because the matrix elements of the spin operators may be momentum dependent. Yet, for short- range scatterers the entire momentum dependence of the vertex matrix elements comes from the eigenstates alone: sa α,α′(k,k′) =/summationdisplay m,m′∝an}bracketle{tαk|m∝an}bracketri}ht∝an}bracketle{tm′|α′k′∝an}bracketri}htsa m,m′(11) This property motivates our solution strategy which characterizes the momentum dependence of the vertex function by expanding it in terms of the eigenstates of sz (sxorsybases would work equally well): Λα,β(k) =∝an}bracketle{tαk|Λ|βk∝an}bracketri}ht =/summationdisplay m,m′∝an}bracketle{tαk|m∝an}bracketri}htΛm,m′∝an}bracketle{tm′|βk∝an}bracketri}ht(12) where|m∝an}bracketri}htisaneigenstateof sz, witheigenvalue m. Plug- ging Eqs.( 11) and ( 12) into Eq.( 6) demonstrates that, as expected, Λ m,m′isindependent of momentum. After cancelling common factors from both sides of the result- ing expressionand using ∂qua(q) = 0 (a= 0,z)we arrive at ΛR(A),R m,m′=sx m,m′+/summationdisplay l,l′UR(A),R m,m′:l,l′ΛR(A),R l,l′ (13) where UR(A),R m,m′:l,l′≡/parenleftbig u0+uzmm′/parenrightbig/integraldisplay k∝an}bracketle{tm|αk∝an}bracketri}htGR(A) α(0,k)∝an}bracketle{tαk|l∝an}bracketri}ht∝an}bracketle{tl′|βk∝an}bracketri}htGR β(0,k)∝an}bracketle{tβk|m′∝an}bracketri}ht (14) Eqs. ( 12),(13)and(14)provideasolutionforthevertex function that is significantly easier to analyse than the original Dyson equation.III. GILBERT DAMPING FOR A MAGNETIC 2DEG The first model we consider is a two-dimensional elec- trongas(2DEG)model withferromagnetismandRashba spin-orbit interactions. We refer to this as the magnetic4 2DEG (M2DEG) model. This toy model is almost never even approximately realistic14, but a theoretical study of its properties will prove useful in a number of ways. First, it is conducive to a fully analytical evaluation of the Gilbert damping, which will allow us to precisely un- derstand the role of different actors. Second, it enables us to explain in simple terms why higher order vertex corrections are significant when there is spin-orbit inter- action in the band structure. Third, the Gilbert damping of a M2DEG has qualitative features similar to those of (Ga,Mn)As. The band Hamiltonian of the M2DEG model is H=k2 2m+bk·σ (15) wherebk= (−λky,λkx,∆0), ∆0is the difference be- tween majority and minority spin exchange-correlation potentials, λis the strength of the Rashba SO couplingand/vector σ= 2/vector sis avectorofPaulimatrices. Thecorrespond- ing eigenvalues and eigenstates are E±,k=k2 2m±/radicalBig ∆2 0+λ2k2 (16) |αk∝an}bracketri}ht=e−iszφe−isyθ|α∝an}bracketri}ht (17) where φ=−tan−1(kx/ky) and θ= cos−1(∆0//radicalbig ∆2 0+λ2k2) are the spinor angles and α=±is the band index. It follows that ∝an}bracketle{tm|α,k∝an}bracketri}ht=∝an}bracketle{tm|e−iszφe−isyθ|α∝an}bracketri}ht =e−imφdm,α(θ) (18) wheredm,α=∝an}bracketle{tm|e−isyθ|α∝an}bracketri}htis a Wigner function for J=1/2 angular momentum15. With these simple spinors, the azimuthal integral in Eq.( 14) can be performed an- alytically to obtain UR(A),R m,m′:l,l′=δm−m′,l−l′(u0+uzmm′)/summationdisplay α,β/integraldisplaydkk 2πdmαGR(A) α(k)dlα(θ)dm′β(θ)GR β(k)dl′β(θ), (19) where the Kronecker delta reflects the conservation of the angular momentum along z, owing to the azimuthal symmetry of the problem. In Eq.( 19) dm,m′=/parenleftbigg cos(θ/2)−sin(θ/2) sin(θ/2) cos(θ/2)/parenrightbigg ,(20) and the retarded and advanced Green’s functions are GR(A) +=1 −ξk−bk+(−)iγ+ GR(A) −=1 −ξk+bk+(−)iγ−, (21) whereξk=k2−k2 F 2m,bk=/radicalbig ∆2 0+λ2k2, andγ±is (half)the golden-rulescatteringrate ofthe band quasiparticles. In addition, Eq. ( 13) is readily inverted to yield ΛR(A),R +,+= ΛR(A),R −,−= 0 ΛR(A),R +,−=1 21 1−UR(A),R +,−:+,− ΛR(A),R −,+=1 21 1−UR(A),R −,+:−,+(22) In order to make further progress analytically we as- sumethat (∆ 0,λkF,γ)<< EF=k2 F/2m. It then follows thatγ+≃γ−≡γand that γ=πN2Du0+πN2Duz 4≡ γ0+γz. Eqs. ( 19) and ( 20) combine to give UR,R −,+:−,+=UR,R +,−:+,−= 0 UA,R −,+:−,+= (γ0−γz)/bracketleftbiggi −b+iγcos4/parenleftbiggθ 2/parenrightbigg +i b+iγsin4/parenleftbiggθ 2/parenrightbigg +2 γcos2/parenleftbiggθ 2/parenrightbigg sin2/parenleftbiggθ 2/parenrightbigg/bracketrightbigg UA,R +,−:+,−= (UA,R −,+:−,+)⋆(23) whereb≃/radicalbig λ2k2 F+∆2 0and cosθ≃∆0/b. The first and second terms in square brackets in Eq.( 23) emerge from inter-band transitions ( α∝ne}ationslash=βin Eq. ( 19)), while the last term stems from intra-band transitions ( α=β).Amusingly, Uvanishes when the spin-dependent scatter- ing rate equals the Coulomb scattering rate ( γz=γ0); in this particular instance vertex corrections are completely absent. On the other hand, when γz= 0 and b << γ5 we have UA,R −,+:−,+≃UA,R +,−:+,−≃1, implying that vertex corrections strongly enhance Gilbert damping (recall Eq. ( 22)). We will discuss the role of vertexcorrectionsmore fully below. 0.00 0.10 0.20 0.30 0.40 1/(εFτ0)0.0000.0050.0100.015αG∆0=0.3 εF ; λ kF = 0 ; u0=3 uz no vertex corrections 1st vertex correction all vertex corrections FIG. 2: M2DEG : Gilbert damping in the absence of spin- orbit coupling. When the intrinsic spin-orbit interaction is small, the 1st vertex correction is sufficient for the evalua- tion of Gilbert damping, provided that the ferromagnet’s ex - change splitting is large compared to the lifetime-broaden ing of the quasiparticle energies. For more disordered ferroma g- nets (EFτ0<5 in this figure) higher order vertex corrections begin to matter. In either case vertex corrections are signi fi- cant. In this figure 1 /τ0stands for the scattering rate off spin- independent impurities, defined as a two-band average at the Fermi energy, and the spin-dependent and spin-independent impurity strengths are chosen to satisfy u0= 3uz. After evaluating Λ( k) from Eqs. ( 12),( 22)and ( 23), the last step is to compute PR(A),R=/integraldisplay kΛR(A),R α,β(k)sx β,α(k)GR(A) α(k)GR β(k).(24) SinceweareassumingthattheFermienergyisthelargest energy scale, the integrand in Eq. ( 24) is sharply peaked at the Fermi surface, leading to PR,R≃0. In the case of spin-independent scatterers ( γz= 0→γ=γ0), tedious but straightforward algebra takes us to αG(uz= 0) =N2D∆2 0 4s0γ0(λ2k2 F)(b2+∆2 0+2γ2 0) (b2+∆2 0)2+4∆2 0γ2 0.(25) Eq. (29) agrees with results published in the recent literature16. We note that αG(uz= 0) vanishes in the absence of SO interactions, as expected. It is illustrative to expand Eq. ( 25) in the b >> γ 0regime: αG(uz= 0)≃N2D∆2 0 2s0/bracketleftbiggλ2k2 F 2(b2+∆2 0)1 γ0+λ4k4 F (b2+∆2 0)3γ0/bracketrightbigg (26) which displays intra-band ( ∼γ−1 0) and inter-band ( ∼ γ0) contributions separately. The intra-band damp- ing is due to the dependence of band eigenenergies on0.00 0.05 0.10 0.15 0.20 1/(εFτ0)0.00.20.40.60.81.0αG∆0=εF/3 ; λ kF=1.2 εF ; u0=3 uz no vertex corrections 1st vertex correction all vertex corrections FIG.3:M2DEG :Gilbert dampingforstrongSOinteractions (λkF= 1.2EF≃4∆0). In this case higher order vertex cor- rections matter (up to 20 %) even at low disorder. This sug- gests that higher order vertex corrections will be importan t in real ferromagnetic semiconductors because their intrin sic SO interactions are generally stronger than their exchange splittings. 0.00 0.10 0.20 0.30 0.40 0.50 1/(εFτ0)0.0000.0050.0100.0150.020αG∆0=0.3 εF ; λkF=∆0/5 ; u0=3 uz intra−band inter−band total FIG. 4: M2DEG : Gilbert damping for moderate SO inter- actions ( λkF= 0.2∆0). In this case there is a crossover be- tween the intra-band dominated and the inter-band domi- nated regimes, which gives rise to a non-monotonic depen- dence of Gilbert damping on disorder strength. The stronger the intrinsic SO relative to the exchange field, the higher th e value of disorder at which the crossover occurs. This is why the damping is monotonically increasing with disorder in Fi g. ( 2) and monotonically decreasing in Fig. ( 3). magnetization orientation, the breathing Fermi surface effect4which produces more damping when the band- quasiparticles scatter infrequently because the popula- tion distribution moves further from equilibrium. The intra-band contribution to damping therefore tends to scale with the conductivity. For stronger disorder, the inter-band term in which scattering relaxes spin-6 orientations takes over and αGis proportional to the resistivity. Insofar as phonon-scattering can be treated as elastic, the Gilbert damping will often show a non- monotonic temperature dependence with the intra-band mechanism dominating at low-temperatures when the conductivity is largeand the inter-band mechanism dom- inatingathigh-temperatureswhentheresistivityislarge. For completeness, we also present analytic results for the case γ=γzin theb >> γ zregime: αG(u0= 0)≃N2D∆2 0 2s0/bracketleftbigg1 γzλ2k2 F 6b2−2∆2 0+γz3b4+6b2∆2 0−∆4 0 (3b2−∆2 0)3/bracketrightbigg (27) This expression illustrates that spin-orbit (SO) interac- tions in the band structure are a necessary condition for the intra-band transition contribution to αG. The in- terband contribution survives in absence of SO as long as the disorder potential is spin-dependent. Interband scatteringis possiblefor spin-dependent disorderbecause majority and minority spin states on the Fermi surface are not orthogonal when their potentials are not identi- cal. Note incidentally the contrast between Eq.( 26) and Eq. ( 27): in the former the inter-bandcoefficient is most suppressed at weak intrinsic SO interaction while in the latter it is the intra-band coefficient which gets weakest for small λkF. More general cases relaxing the (∆ 0,λkF,γ)<< E F assumption must be studied numerically; the results are collected in Figs. ( 2), ( 3) and ( 4). Fig ( 2) highlights the inadequacy of completely neglecting vertex correc- tions in the limit of weak spin-orbit interaction; the in- clusion of the the leading order vertex correction largely solves the problem. However, Fig. ( 2) and ( 3) together indicate that higher order vertex corrections are notice- able when disorder or spin-orbit coupling is strong. In the light of the preceding discussion the monotonic de- cay in Fig.( 3) may appear surprising because the inter- band contribution presumably increases with γ. Yet, this argument is strictly correct only for weakly spin- orbit coupled systems, where the crossover betwen inter- band and intra-band dominated regimes occurs at low disorder. For strongly spin-orbit coupled systems the crossover may take place at a scattering rate that is (i) beyondexperimentalrelevanceand/or(ii)largerthanthe band-splitting, in which case the inter-band contribution behaves much like its intra-band partner, i.e. O(1/γ). Non-monotonic behavior is restored when the spin-orbit splitting is weaker, as shown in Fig. ( 4). Finally, our analysis opens an opportunity to quan- tify the importance of higher order impurity vertex- corrections. Kohno, Shibata and Tatara11claim that the bare vertex along with the firstvertex correction fully captures the Gilbert damping of a ferromagnet, provided that ∆ 0τ >>1. To first order in Uthe vertex function is ΛR(A),R m,m′=sx m,m′+/summationdisplay ll′UR(A),R m,m′:l,l′sx l,l′(28)Takingγ=γzfor simplicity, we indeed get lim λ→0αG≃Aγ+O(γ2) A(1) A(∞)= 1 (29) whereA(1) contains the first vertex correction only, and A(∞) includes all vertex corrections. However, the state of affairs changes after turning on the intrinsic SO inter- action, whereupon Eq. ( 29) transforms into αG(λ∝ne}ationslash= 0)≃Bγ+C1 γ B(1) B(∞)=∆2 0(3b2−∆2 0)3(3b2+∆2 0) 4b6(3b4+6b2∆2 0−∆4 0) C(1) C(∞)=(b2+∆2 0)(3b2−∆2 0) 4b4(30) When ∆ 0<< λk F,bothintra-bandand inter-bandratios show a significant deviation from unity17, to which they converge as λ→0. In order to understand this behavior, let us look back at Eq. ( 22). There, we can formally ex- pand the vertex function as Λ =1 2/summationtext∞ n=0Un, where the n-th order term stems from the n-th vertex correction. From Eq. ( 23) we find that when λ= 0,Un∼O(γn) and thus n≥2 vertex corrections will not matter for the Gilbert damping, which is O(γ)18whenEF>> γ. In contrast, when λ∝ne}ationslash= 0 the intra-band term in Eq. ( 23) is no longer zero, and consequently allpowers of Ucon- tainO(γ0) andO(γ1) terms. In other words, all vertices contribute to O(1/γ) andO(γ) in the Gilbert damping, especially if λkF/∆0is not small. This conclusion should prove valid beyond the realm of the M2DEG because it relies only on the mantra “intra-band ∼O(1/γ); inter- band∼O(γ)”. Our expectation that higher order vertex correctionsbe importantin (Ga,Mn)As will be confirmed numerically in the next section. IV. GILBERT DAMPING FOR (Ga,Mn)As (Ga,Mn)As and other (III,Mn)V ferromagnets are like transition metals in that their magnetism is carried mainly by d-orbitals, but unlike transition metals in that neither majority nor minority spin d-orbitals are present at the Fermi energy. The orbitals at the Fermi energy are very similar to the states near the top of the valence band states of the host (III,V) semiconductor, although they are of course weakly hybridized with the minority and majority spin d-orbitals. For this reason the elec- tronic structure of (III,Mn)V ferromagnets is extremely simple and can be described reasonably accurately with the phenomenologicalmodel whichwe employin this sec- tion. Becausethe top ofthe valence band in (III,V) semi- conductors is split by spin-orbit interactions, spin-orbit coupling plays a dominant role in the bands of these fer- romagnets. An important consequence of the strong SO7 interaction in the band structure is that diffusive vertex corrections influence αGsignificantly at allorders; this is the central idea of this section. Using a p-d mean-field theory model8for the ferro- magnetic groundstate and afour-band sphericalmodel19 for the host semiconductor band structure, Ga 1−xMnxAs may be described by H=1 2m/bracketleftbigg/parenleftbigg γ1+5 2γ2/parenrightbigg k2−2γ3(k·s)2/bracketrightbigg +∆0sz,(31) wheresis the spin operator projected onto the J=3/2 total angular momentum subspace at the top of the va- lence band and {γ1= 6.98,γ2=γ3= 2.5}are the Lut- tinger parameters for the spherical-band approximation to GaAs. In addition, ∆ 0=JpdSNMnis the exchange field,Jpd= 55meVnm3is the p-d exchange coupling, S= 5/2 is the spin of the Mn ions, NMn= 4x/a3is the density of Mn ions, and a= 0.565nm is the lattice constant of GaAs. 0.00 0.10 0.20 0.30 0.40 1/(εFτ0)0.0000.0200.0400.060αGp=0.6 nm−3 (εF=500 meV) ; x=0.04 ; u0=3 uz no vertex corrections 1st vertex correction all vertex corrections FIG. 5: GaMnAs : Higher order vertex corrections make a significant contributionto Gilbert damping, dueto theprom i- nent spin-orbit interaction in the band structure of GaAs. xis the Mn fraction, and pis the hole concentration that determines the Fermi energy EF. In this figure, the spin- independent impurity strength u0was taken to be 3 times larger than the magnetic impurity strength uz. 1/τ0corre- sponds to the scattering rate off Coulomb impurities and is evaluated as a four-band average at the Fermi energy. The ∆ 0= 0 eigenstates of this model are |˜α,k∝an}bracketri}ht=e−iszφe−isyθ|˜α∝an}bracketri}ht (32) where|˜α∝an}bracketri}htis an eigenstate of szwith eigenvalue ˜ α. Un- fortunately, the analytical form of the ∆ 0∝ne}ationslash= 0 eigen- states is unknown. Nevertheless, since the exchange field preserves the azimuthal symmetry of the problem, the φ-dependence of the full eigenstates |αk∝an}bracketri}htwill be iden- tical to that of Eq. ( 32). This observation leads to Um,m′:l,l′∝δm−m′,l−l′, which simplifies Eq. ( 14). αG canbe calculatednumericallyfollowingthe stepsdetailed0.00 0.10 0.20 0.30 0.40 1/(εFτ0)0.0000.0020.0040.0060.0080.010αGp=0.2 nm−3 (εF=240 meV) ; x=0.08 ; u0=3 uz intra−band inter−band total γ3=γ2=0.5 FIG. 6: GaMnAs : When the spin-orbit splitting is reduced (in this case by reducing the hole density to 0 .2nm−3and artificially taking γ3= 0.5), the crossover between inter- and intra-band dominated regimes produces a non-monotonic shape of the Gilbert damping, much like in Fig. ( 4). When eitherγ2orpis made larger or xis reduced, we recover the monotonic decay of Fig.( 5). in the previous sections; the results are summarized in Figs. ( 5) and ( 6). Note that vertex corrections mod- erately increase the damping rate, as in the case of a M2DEG model with strong spin-orbit interactions. Fig. ( 5) underlines both the importance of higher order ver- tex corrections in (Ga,Mn)As and the monotonic decay of the damping as a function of scattering rate. The lat- ter signals the supremacy of the intra-band contribution to damping, accentuated at larger hole concentrations. Hadtheintrinsicspin-orbitinteractionbeensubstantially weaker20,αGwould have traced a non-monotonic curve as shown in Fig. ( 6). The degree to which the intraband breathing Fermi surface model effect dominates depends on the details of the band-structure and can be influ- enced by corrections to the spherical model which we haveadoptedheretosimplifythevertex-correctioncalcu- lation. The close correspondence between Figs. ( 5)-( 6) and Figs. ( 3)-( 4) reveals the success of the M2DEG as a versatile gateway for realistic models and justifies the extensive attention devoted to it in this paper and elsewhere. V. ASSESSMENT OF THE TORQUE-CORRELATION FORMULA Thus far we have evaluated the Gilbert damping for a M2DEG model and a (Ga,Mn)As model using the (bare) spin-flip vertex ∝an}bracketle{tα,k|sx|β,k∝an}bracketri}htand its renormal- ized counterpart ∝an}bracketle{tα,k|Λ|β,k∝an}bracketri}ht. The vertex corrected re- sults are expected to be exact for 1 /τsmall compared to the Fermi energy. For practical reasons, state-of-the- art band-structure calculations5forgo impurity vertex8 corrections altogether and instead employ the torque- correlation matrix element, which we shall denote as ∝an}bracketle{tα,k|K|β,k∝an}bracketri}ht(see below for an explicit expression). In this section we compare damping rates calculated using sx α,βvertices with those calculated using Kα,βvertices. We also compare both results with the exact damping rates obtained by using Λ α,β. The ensuing discussion overlaps with and extends our recent preprint6. We shall begin by introducing the following identity4: ∝an}bracketle{tα,k|sx|β,k∝an}bracketri}ht=i∝an}bracketle{tα,k|[sz,sy]|β,k∝an}bracketri}ht =i ∆0(Ek,α−Ek,β)∝an}bracketle{tα,k|sy|β,k∝an}bracketri}ht −i ∆0∝an}bracketle{tα,k|[Hso,sy]|β,k∝an}bracketri}ht.(33) In Eq. ( 33) we have decomposed the mean-field quasi- particle Hamiltonian into a sum of spin-independent, ex- change spin-splitting, and other spin-dependent terms: H=Hkin+Hso+Hex, whereHkinis the kinetic (spin- independent) part, Hex= ∆0szis the exchange spin- splitting term and Hsois the piece that contains the in- trinsic spin-orbit interaction. The last term on the right hand side of Eq. ( 33) is the torque-correlation matrix element used in band structure computations: ∝an}bracketle{tα,k|K|β,k∝an}bracketri}ht ≡ −i ∆0∝an}bracketle{tα,k|[Hso,sy]|β,k∝an}bracketri}ht.(34) Eq. ( 33) allows us to make a few general remarks on the relation between the spin-flip and torque-correlation matrix elements. For intra-band matrix elements, one immediately finds that sx α,α=Kα,αand hence the two approaches agree. For inter-band matrix elements the agreement between sx α,βandKα,βshould be nearly iden- tical when the first term in the final form of Eq.( 33) is small, i.e.when21(Ek,α−Ek,β)<<∆0. Since this requirement cannot be satisfied in the M2DEG, we ex- pect that the inter-band contributions from Kandsx will always differ significantly in this model. More typi- calmodels,likethefour-bandmodelfor(Ga,Mn)As, have band crossings at a discrete set of k-points, in the neigh- borhood of which Kα,β≃sx α,β. The relative weight of these crossing points in the overall Gilbert damping de- pends on a variety of factors. First, in order to make an impact they must be located within a shell of thick- ness 1/τaround the Fermi surface. Second, the contri- bution to damping from those special points must out- weigh that from the remaining k-points in the shell; this might be the case for instance in materials with weak spin-orbit interaction and weak disorder, where the con- tribution from the crossing points would go like τ(large) while the contribution from points far from the cross- ings would be ∼1/τ(small). Only if these two con- ditions are fulfilled should one expect good agreement between the inter-band contribution from spin-flip and torque-correlationformulas. When vertexcorrectionsare included, of course, the same result should be obtained using either form for the matrix element, since all matrixelements are between essentially degenerate electronic states when disorder is treated non-perturbatively6,16. 0.00 0.10 0.20 0.30 0.40 0.50 1/(εFτ)0.0000.0020.0040.0060.0080.010αG∆0=0.8 εF ; λ kF=0.05 εF ; uz =0 K sx Λ FIG. 7:M2DEG : Comparison of Gilbert damping predicted using spin-flip and torque matrix element formulas, as well a s the exact vertex corrected result. In this figure the intrins ic spin-orbit interaction is relatively weak ( λkF= 0.05EF≃ 0.06∆0) and we have taken uz= 0. The torque correla- tionformula does notdistinguish between spin-dependenta nd spin-independent disorder. 0.00 0.10 0.20 0.30 0.40 0.50 1/(εFτ)0.000.050.100.150.20αG∆0=0.1 εF ; λ kF=0.5 εF ; uz=0 K Λ sx FIG. 8:M2DEG : Comparison of Gilbert damping predicted using spin-flip and torque matrix element formulas, as well a s the exact vertex corrected result. In this figure the intrins ic spin-orbit interaction is relatively strong ( λkF= 0.5EF= 5∆0) and we have taken uz= 0 In the remining part of this section we shall focus on a more quantitative comparison between the different for- mulas. For the M2DEG it is straightforward to evaluate αGanalytically using Kinstead of sxand neglecting ver- tex corrections; we obtain αK G=N2D∆0 8s0/bracketleftBigg λ2k2 F b2∆0 γ+/parenleftbiggλ2k2 F ∆0b/parenrightbigg2γ∆0 γ2+b2/bracketrightBigg (35)9 where we assumed ( γ,λkF,∆0)<< ǫF. By compar- ing Eq. ( 35) with the exact expression Eq. ( 25), we find that the intra-band parts are in excellent agreement when ∆ 0<< λk F, i.e. when vertex corrections are rela- tively unimportant. In contrast, the inter-band parts dif- fer markedly regardless of the vertex corrections. These trendsarecapturedby Figs. (7) and( 8), which compare the Gilbert damping obtained from sx,Kand Λ matrix elements. Fig. ( 7) corresponds to the weak spin-orbit limit, whereitisfoundthatindisorderedferromagnets sx maygrosslyoverestimatetheGilbertdampingbecauseits inter-band contributiondoes not vanish even as SO tends to zero. As explained in Section III, this flaw may be re- paired by adding the leading order impurity vertex cor- rection. The torque-correlation formula is free from such problem because Kvanishes identically in absence of SO interaction. Thus the main practical advantage of Kis that it yields a physically sensible result without having to resort to vertex corrections. Continuing with Fig.( 7), at weak disorder the intra-band contributions dominate and therefore sxandKcoincide; even Λ agrees, because for intra-band transitions at weak spin-orbit interaction the vertex corrections are unimportant. Fig. ( 8) cor- responds to the strong spin-orbit case. In this case, at low disorder sxandKagree well with each other, but differ from the exact result because higher order vertex corrections alter the intra-band part substantially. For a similar reason, neither sxnorKagree with the exact Λ at higher disorder. Based on these model calculations, we do not believe that there are any objective grounds to prefer either the Ktorque-correlation or the sxspin-flip formula estimate of αGwhen spin-orbit interactions are strong and αGis dominated by inter-band relaxation. A precise estimation of αGunder these circumstances ap- pears to require that the character of disorder, incud- ing its spin-dependence, be accounted for reliably and that the vertex-correction Dyson equation be accurately solved. Carrying out this program remains a challenge both because of technical complications in performing the calculation for general band structures and because disorder may not be sufficiently well characterized. Analogous considerations apply for Figs. ( 9) and ( 10), which show results for the four-band model re- lated to (Ga,Mn)As. These figures show results similar to those obtained in the strong spin-orbit limit of the M2DEG (Fig. 8). Overall, our study indicates that thetorque-correlation formula captures the intra-band contributions accurately when the vertex corrections are unimportant, while it is less reliable for inter-band con- tributions unless the predominant inter-band transitions connect states that are close in energy. The torque- correlation formula has the practical advantage that it correctly gives a zero spin relaxation rate when there is no spin-orbit coupling in the band structure and spin- independent disorder. The damping it captures derives entirely from spin-orbit coupling in the bands. It there- fore incorrectly predicts, for example, that the damp- ing rate vanishes when spin-orbit coupling is absent in0.00 0.10 0.20 0.30 0.40 0.50 1/(εFτ)0.000.100.200.300.400.50αGp=0.4nm−3 (εF=380 meV) ; x=0.08 ; uz=0 sx Λ K FIG.9:GaMnAs : Comparison ofGilbertdampingpredicted using spin-flip and torque matrix element formulas, as well a s the exact vertex corrected result. pis the hole concentration that determines the Fermi energy EFandxis the Mn frac- tion. Due to the strong intrinsic SO, this figure shows simila r features as Fig.( 8). 0.00 0.10 0.20 0.30 0.40 1/(εFτ)0.000.050.100.150.20αGp=0.8nm−3 (εF=605 meV) ; x=0.04 ; uz=0 sx Λ K FIG. 10: GaMnAs : Comparison of Gilbert damping pre- dicted using spin-flip and torque matrix element formulas, a s well as the exact vertex corrected result. In relation to Fig . ( 9) the effective spin-orbit interaction is stronger, due to a largerpand a smaller x. the bands and the disorder potential is spin-dependent. Nevertheless, assuming that the dominant disorder is normally spin-independent, the K-formula may have a pragmatic edge over the sx-formula in weakly spin-orbit coupled systems. In strongly spin-orbit coupled systems there appears to be little advantage of one formula over the other. We recommend that inter-band and intra- band contributions be evaluated separately when αGis evaluated using the torque-correlation formula. For the intra-band contribution the sxandKlife-time formulas are identical. The model calculations reported here sug-10 gest that vertex corrections to the intra-band contribu- tion do not normally have an overwhelming importance. We conclude that αGcan be evaluated relatively reliably when the intra-band contribution dominates. When the inter-band contribution dominates it is important to as- sess whether or not the dominant contributions are com- ing from bands that are nearby in momentum space, or equivalently whether or not the matrix elements which contribute originate from pairs of bands that are ener- getically spaced by much less than the exchange spin- splitting at the same wavevector. If the dominant con- tributions are from nearby bands, the damping estimate should have the same reliability as the intra-band contri- bution. If not, we conclude that the αGestimate should be regarded with caution. To summarize, this article describes an evaluation of Gilbert damping for two simple models, a two- dimensionalelectron-gasferromagnetmodelwith Rashba spin-orbit interactions and a four-band model which pro- vides an approximate description of (III, Mn)V of fer- romagnetic semiconductors. Our results are exact in the sense that they combine time-dependent mean field theory6with an impurity ladder-sum to all orders, hence giving us leverage to make the following statements.First, previously neglected higher order vertex correc- tions become quantitatively significant when the intrin- sic spin-orbit interaction is larger than the exchange splitting. Second, strong intrinsic spin-orbit interaction leads to the the supremacy of intra-band contributions in (Ga,Mn)As, with the corresponding monotonic decay of the Gilbert damping as a function ofdisorder. Third, the spin-torque formalism used in ab-initio calculations of the Gilbert damping is quantitatively reliable as long as the intra-band contributions dominate andthe exchange field is weaker than the spin-orbit splitting; if these con- ditions are not met, the use of the spin-torque matrix element in a life-time approximation formula offers no significant improvement overthe originalspin-flip matrix element. Acknowledgments The authors thank Keith Gilmore and Mark Stiles for helpful discussions and feedback. This work was sup- ported by the Welch Foundation and by the National Science Foundation under grant DMR-0606489. 1Foranintroductoryreviewsee D.C. RalphandM.D.Stiles, J. Magn. Mag. Mater. 320, 1190 (2008). 2J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag- netic Structures III: Fundamentals of Nanomagnetism (Springer-Verlag, New York, 2005). 3V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769 (1972). 4V. Kambersky, Czech J. Phys. B 26, 1366 (1976); V. Kam- bersky, Czech J. Phys. B 34, 1111 (1984). 5K. Gilmore, Y.U.IdzerdaandM.D. Stiles, Phys.Rev.Lett. 99, 27204 (2007); V. Kambersky, Phys. Rev. B 76, 134416 (2007). 6Ion Garate and A.H. MacDonald, arXiv:0808.1373. 7O. Gunnarsson, J. Phys. F 6, 587 (1976). 8For reviews see T. Jungwirth et al., Rev. Mod. Phys. 78, 809 (2006); A.H. MacDonald, P. Schiffer and N. Samarth, Nature Materials 4, 195 (2005). 9These simplified models sometimes have the advantage that their parameters can be adjusted phenomenologically to fit experiments, compensating for inevitable inaccura- cies inab initio electronic structure calculations. This ad- vantage makes p−dmodels of (III,Mn)V ferromagnets particularly useful. s−dmodels of transition elements are less realistic from the start because they do not account for the minority-spin hybridized s−dbands which are present at the Fermi energy. 10This is not the most general type of disorder for quasi- particles with spin >1/2, but it will be sufficient for the purpose of this work. 11H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan 75, 113706 (2006). 12We assume that the spins of magnetic impurities are frozenalong the staticpart of the exchange field. In reality, the direction of the impurity spins is a dynamical variable that is influenced by the magnetization precession. 13G.D. Mahan, Many-Particle Physics (3rd Ed.), Physics of Solids and Liquids Series (2000) 14A possible exception is the ferromagnetic 2DEG recently discovered in GaAs/AlGaAs heterostructures with Mn δ- doping; see A. Bove et. al, arXiv:0802.3871v3. 15J.J. Sakurai, Modern Quantum Mechanics , Addison- Wesley (1994). 16E.M. Hankiewicz, G. Vignale and Y. Tserkovnyak, Phys. Rev. B 75, 174434 (2007). In their case the inter-band splitting in the Green’s function is Ω, while in our case it is 2b. In addition, we neglect interactions between band quasiparticles. 17C(1) and C(∞) differ by as much as 25%; the disparity between B(1) andB(∞) may be even larger. 18The disorder dependence in αGoriginates not only from the vertex part, but from the Green’s functions as well. It is useful to recall thatR GσG−σ∝1/(b+isg(σ)γ) andR GσGσ∝1/γ. 19P. Yu, M. Cardona, Fundamentals of Semiconductors (3rd Ed.), Springer (2005). 20Notwithstanding that the four-band model is a SO → ∞ limit of the more general six-band model, we shall tune the effective spin-orbit strength via p(hole concentration) and γ3. 21Strictly speaking, it is |sx α,β|2≃ |Kα,β|2what is needed, rather than sx α,β≃Kα,β. The former condition is less de- manding, and can occasionally be satisfied when Eα−Eβ is of the order of the exchange splitting.

1505.04087v2.Reliable_Damping_of_Free_Surface_Waves_in_Numerical_Simulations.pdf

Reliable Damping of Free Surface Waves in Numerical Simulations Robinson Peri c1, Moustafa Abdel-Maksoud Hamburg University of Technology (TUHH), Institute for Fluid Dynamics and Ship Theory (M8), Hamburg, Germany Abstract This paper generalizes existing approaches for free-surface wave damping via momentum sinks for ow simulations based on the Navier-Stokes equations. It is shown in 2D ow simulations that, to obtain reliable wave damping, the coecients in the damping functions must be adjusted to the wave pa- rameters. A scaling law for selecting these damping coecients is presented, which enables similarity of the damping in model- and full-scale. The in- uence of the thickness of the damping layer, the wave steepness, the mesh neness and the choice of the damping coecients are examined. An ecient approach for estimating the optimal damping setup is presented. Results of 3D ship resistance computations show that the scaling laws apply to such simulations as well, so the damping coecients should be adjusted for every simulation to ensure convergence of the solution in both model and full scale. Finally, practical recommendations for the setup of reliable damping in ow simulations with regular and irregular free surface waves are given. Keywords: Damping of free surface waves, absorbing layer, volume of uid (VOF) method, damping coecient, scaling law 1. Introduction In free surface ow simulations based on the Navier-Stokes equations, it is often required to model an innite domain, which basically means min- imizing undesired wave re ections at wave-maker and domain boundaries, 1Corresponding author. Tel. +49 40 42878 6031. E-mail adress: robinson.peric@tuhh.de Preprint submitted to Ship Technology Research May 22, 2017arXiv:1505.04087v2 [physics.flu-dyn] 19 May 2017while choosing the solution domain as small as possible in order to lower the computational eort. Since such simulations are among the computa- tionally most eortful techniques to solve numerical ow problems, they are employed when simpler approaches (e.g. potential ow methods) cannot be used or when higher accuracy is required. As the schemes contain numer- ical errors (discretization, iteration, modelling (in some cases)), to achieve the required accuracy it is necessary to minimize avoidable uncertainties; of these, the performance of the wave damping approach is often among the most critical. The elimination of wave re ections at the domain boundaries is commonly achieved by (i)increasing the domain size (ii)beaches (e.g. Lal and Elangovan (2008)): a slope in the domain bottom leads to wave breaking and energy dissipation as in experiments (iii) grid damping (e.g. Kraskowski (2010), Peri c and Abdel-Maksoud (2015)): continuously increasing the cell size towards the corresponding domain boundary increases numerical discretization and iteration errors, thus damping the wave (this approach is also called numerical beach or grid extrusion) (iv) active wave absorption techniques (e.g. Cruz (2008); Higuera et al. (2013); Sch aer and Klopman (2000)): a boundary-based wave-maker generates waves which eliminate the incoming waves via destructive interference (v)solution-forcing orsolver-coupling (e.g. Ferrant et al. (2008), Guignard et al. (1999), Kim et al. (2012), Kim et al. (2013), W ockner-Kluwe (2013)): the ow is forced to a known solution in the vicinity of the boundary or the Navier-Stokes-based ow solver is coupled to another (e.g. potential ow based) solver (vi) damping layer approaches (e.g. Cao et al. (1993); Choi and Yoon (2009); Ha et al. (2011); Israeli and Orszag (1981); Park et al. (1999)): the damping layer (also called sponge layer, absorbing layer, damping zone, porous media layer) is a zone set up next to the corresponding boundaries, in which momentum sinks are included in the governing equations to damp the waves propagating through the zone 2Apart from the above methods, further approaches have been developed for other governing equations, like potential ow with boundary element method or Boussinesq-type equations (see e.g. Grilli and Horrillo (1997) and ref- erences therein). However, many of these approaches have not yet been transferred to Navier-Stokes-type equations. From the above mentioned ap- proaches, (i) is the least feasible due to its (in many cases enormous) inherent increase in computational eort. With (ii) it is dicult to minimize re ec- tions to less then 10% of the incoming wave, which is a problem in experi- ments as well. Approaches (iv) and (v) have recently attracted much interest and produced good results; however, since they are often used to simultane- ously generate and damp waves, it is likely that their damping performance varies depending on the kind of waves they are currently generating. At the time of writing, (iii) and (vi) are the most widely-used wave damping ap- proaches in Navier-Stokes-type equation ow solvers, implemented in most commercial and research codes. Although both approaches have been used with success, the disadvantage of (iii) is that its performance depends on grid, time step, temporal/spatial discretization schemes, etc., which for suf- ciently ne discretization is not the case for (vi), as the results in this work suggest. Thus (iii) is less predictable than (vi) regarding the damping qual- ity. This work focuses exclusively on damping layer approaches (vi) based on momentum source terms, since a detailed investigation and comparison of all previously mentioned approaches was not possible in the scope of this study. This paper discusses momentum source terms based on linear or quadratic damping functions as dened in Sect. 3. The amount and character of the damping is controlled by coecients in the damping functions and the thickness of the damping layer. Such approaches are widely used and already implemented in several open source as well as commercial computational uid dynamics (CFD) solvers. Section 3 shows how the existing implementations can be generalized. Two of the most widely used implementations, the ones from CD-adapco STAR-CCM+, based on Choi and Yoon (2009), and ANSYS Fluent, based on Park et al. (1999), are discussed in Sect. 4. Various other implementations of linear or quadratic damping exist, see e.g. Cao et al. (1993) and Ha et al. (2011). The nomenclature from Sect. 4 is used throughout the work. For all simulations in this study, the damping function from Choi and Yoon (2009) is used, since it is rather generic and can be set up to act similar to the other approaches mentioned. Thus the results are easily applicable to all 3damping approaches used in other CFD codes which can be generelized as described in Sect. 3. Although widely used, much about the damping func- tions remains unknown at the time of writing. It has been observed that the coecients in the damping functions, and thus the damping performance, are case-dependent. In the above mentioned codes, these damping coecients can be modied by the user. However, no guidelines seem to exist for this. Thus in practice, either the default settings or values from experience are used as coecients in the damping functions. If during or after the simula- tion it is observed that the damping does not work satisfactorily, then the damping coecients are modied by trial and error and the simulation is restarted. This procedure is repeated until acceptable damping is obtained, a process which requires human interaction and can cost considerable addi- tional time and computational eort. Especially in the light of the increased automation of CFD computations, it would be preferable to be sure about the damping quality before the simulation. Similar to wave damping in experi- ments, where according to Lloyd (1989) seldom data on beach performance is published, also with CFD simulations, the performance of the wave damping is often not suciently accounted for. Thus with most CFD publications, it is dicult to judge on the damping quality even if the damping setup is given. The aim of the present work is to clarify how the damping functions work, on which factors the damping quality depends and how reliable wave damping can be set up case-independently, so that no more ne-tuning of the damping coecients is necessary. In Sect. 3, the generalized forms of linear and quadratic damping func- tions are given. In the following Sect. 4, the damping approach used in this work is described. Moreover, it is shown exemplarily for two widely used damping functions how these can be generalized to the equations given in Sect. 3 and how results from one implementation can be transferred to another. Starting from the analogy of the damped harmonic oscillator, Sect. 5 attempts to show similarities to damping phenomena outside the hydrody- namics eld and constructs scaling laws for wave damping based on this information. These scaling laws are fully formulated and veried in Sects. 13 and 14. Furthermore, recommendations for selecting the damping coe- cients are given. Sections 8 to 12 investigate via 2D ow simulations how the damping qual- ity and behavior is in uenced by the choice of the coecients in the damping 4functions, the thickness of the damping layer, the computational grid and the wave steepness. As the investigations so far have been mainly concerned with regular, monochromatic waves, recommendations for the damping for irregular waves are given in Sect. 15 and illustrated with simulation results. Since wave damping is also widely used to speed up convergence for ex- ample in ship resistance computations, it is veried in Sect. 16 via 3D ow simulations that the presented scaling laws hold for such cases as well. 2. Governing Equations and Solution Method The governing equations for the simulations are the Navier-Stokes equa- tions, which consist of the equation for mass conservation and the three equations for momentum conservation: d dtZ VdV+Z Sv
ndS= 0; (1) d dtZ VuidV+Z Suiv
ndS= Z S(ijijpii)
ndS+Z VgiidV+Z VqidV : (2) HereVis the control volume (CV) bounded by the closed surface S, vis the velocity vector of the uid with the Cartesian components ui,nis the unit vector normal to Sand pointing outwards, tis time,pis the pressure, and are uid density and dynamic viscosity, ijare the components of the viscous stress tensor, ijis the unit vector in direction xj,gcomprises the body forces andqiis an optional momentum source term. For the present simulations, the only body force considered was the gravitational acceleration, i.e. g= (0;0;9:81m s2)T. Only incompressible Newtonian uids are considered in this study. Thus ijtakes the form ij=@ui @xj+@uj @xi : No turbulence modeling was applied to the above equations since, unless waves break, the ow inside them can be considered practically laminar and all structures of interest can be resolved with acceptable computational ef- fort. For all simulations, the software STAR-CCM+ 8 :02:008 was used. The 5volume of uid (VOF) method implemented in the STAR-CCM+ software is used to account for the two phases (air and water). Further details on the method can be found in Muzaferija and Peri c (1999). The governing equa- tions are applied to each cell and discretized according to the Finite Volume Method (FVM). All integrals are approximated by the midpoint rule. The interpolation of variables from cell center to face center and the numerical dierentiation are performed using linear shape functions, leading to ap- proximations of second order. The integration in time is based on assumed quadratic variation of variables in time, which is also a second-order approx- imation. Each algebraic equation contains the unknown value from the cell center and the centers of all neighboring cells with which it shares common faces. The resulting coupled equation system is then linearized and solved by the iterative STAR-CCM+ implicit unsteady segregated solver, using an algebraic multigrid method with Gauss-Seidl relaxation scheme, V-cycles for pressure and volume fraction of water, and exible cycles for velocity cal- culation. For each time step, one iteration consists of solving the governing equations for the velocity components, the pressure-correction equation (us- ing the SIMPLE method for collocated grids to obtain the pressure values and to correct the velocities) and the transport equation for the volume frac- tion of water. For further information on the discretization of and solvers for the governing equations, the reader is referred to Ferziger and Peri c (2002) or the STAR-CCM+ software manual. 3. General Formulation for Linear and Quadratic Damping Linear wave damping is obtained by inserting the momentum source term qd;lin iforqiin Eq. (2): qd;lin i=Ci;linui; (3) with coecient Ci;lin, which usually depends on the spatial location. Ci;linreg- ulates the strength of the damping and is used to provide a smooth blending- in of the damping, by increasing the amount of damping from weak damping where the waves enter the damping layer to strong damping at the end of the damping layer. This is to prevent undesired re ections at the entrance to the damping layer as shown in Sects. 8 and 9. Commonly, the blending-in is realized in an exponential or quadratic fashion. Quadratic wave damping takes the form qd;quad i =Ci;quadjuijui; (4) 6with coecient Ci;quad, which regulates strength and blending-in of the damp- ing. In contrast to Eq. (3), uid particles with higher velocities experience disproportionately high damping. In most CFD codes, Ci;linandCi;quadcan be adjusted by the user. Usually, but not necessarily, the momentum source terms are only applied to the equation for the vertical velocity component. 4. Common Implementations of Linear and Quadratic Wave Damp- ing A widely used approach is the one in Choi and Yoon (2009), which is e.g. implemented in the commercial software code STAR-CCM+ by CD-adapco. It features a combination of linear and quadratic damping, which allows the use of either one or a combination of both approaches. Taking zas the vertical direction (i.e. perpenticular to the free surface) and was the vertical velocity component, then the following source term appears for qiin Eq. (2): qd z=(f1+f2jwj)e1 e11w ; (5) =xxsd xedxsdn : (6) Herexstands for the wave propagation direction with xsdbeing the start andxedthe endx-coordinate of the damping layer, thus the thickness of the damping layer xd=jxedxsdj.f1is the damping constant for the linear part andf2is the damping constant for the quadratic part of the damping term. The default values are f1= 10:0 s1andf2= 10:0 m1according to the STAR-CCM+ manual (release 8 :02:008). The terme1 e11blends in the damping term, i.e. it is zero at xsd, and it equals one at xed.describes vianthe character of the blending functions, i.e. for n= 1 the blending is nearly linear. When increasing n, the blending becomes smoother at the entrance to the damping layer and at the same time more abrupt at xed, see Fig. 1. 7Figure 1: The functione(x)1 e11evaluated for n= 1,n= 2 andn= 4 over the dimensionless distancexxsd xedxsd; a wave enters the damping layer atxxsd xedxsd= 0 and is damped during propagation towardsxxsd xedxsd= 1, where the damping layer ends Another widely used approach was presented by Park et al. (1999), which is implemented in ANSYS Fluent.: qd z=(0:5f3jwj) 1zzfs zbzfs w ; (7) with damping constant f3,as given in Eq. (6) with n= 2, vertical coordi- natezandz-coordinates of the domain bottom zband the free water surface zfs. The default value for f3is 10:0 m1. This approach can be generalized to a quadratic damping function accord- ing to Eq. (4). It corresponds to the quadratic part of Eq. (5), except for a quadratic instead of exponential blending in x-direction and an added verti- cal fade-in. The latter term is a linear fade between domain bottom, where no damping is applied, to free surface level, where full damping is applied; for practical deep water conditions of several wavelengths water depth, the in uence of the z-fading term on the applied damping becomes small, since most wave energy is concentrated in the vicinity of the free surface. Thus for such cases, this damping approach can be modeled using Eqs. (5) and (6) 8with accordingly adjusted coecients. In a similar manner, also the other damping layer approaches from Sect. 1 (vi) can be modeled. Therefore in the following, only the approach by Choi and Yoon (2009) will be considered to study wave damping. Thus the ndings in this work can easily be transferred to other damping approaches. 5. Dependence of Damping Coecients For gravity waves, all scaling laws should be consistent with Froude scal- ing. Thus to obtain similar wave damping for waves of dierent scale, the blending part of Ci;linandCi;quadfrom Eqs. (3) and (4) must be geomet- rically similar. Therefore, the thickness xdof the damping layer must be directly proportional to the wavelength . For the remaining part of Ci;lin andCi;quad, which regulates the magnitude of the damping, the scaling laws can be obtained by dimensional analysis. Due to its dimension of [ s1],Ci;lin is directly proportional to the wave frequency !, whereasCi;quadhas the di- mension [m1] and is thus directly proportional to 1(or!2in deep water). Thus to achieve similar damping when changing scale and/or wave, it is nec- essary to: 9Scaling Laws 1. Set the damping thickness xdso that it is geometrically similar to the damping layer thickness xd;refof the reference case (i.e. scale xdwith the percentual change in wavelength) xd=xd;ref
 ref; (8) 2. For linear damping , scalef1with the change of wave frequency ! f1=f1;ref
! !ref; (9) with wave frequency !refof the reference case. This holds for all damp- ing formulations that can be generalized by Eq. (3). 3. For quadratic damping , scalef2with the change of wavelength to the power of minus one f2=f2;ref
ref ; (10) with wavelength refof the reference case. This holds for all damping formulations that can be generalized by Eq. (4). 6. Numerical Simulation Setup Figure 2 shows the solution domain, a 2D deep water wave tank with lengthLx= 6and height Lz= 4:5, given in relation to wavelength . It is lled with water to a depth of d= 4, the rest of the tank is lled with air. The origin of the coordinate system is set in the bottom left front corner of the tank in Fig. 2. The top boundary (i.e. z=Lz) is a pressure outlet boundary, i.e. the pressure there is set constant and equal to atmospheric pressure. This corresponds to an open water tank, where air can ow in and out through the tank top. The x=Lxboundary is a pressure outlet, where volume fraction and hydrostatic pressure are prescribed for an undisturbed free surface. The waves are generated by prescribing velocities and volume fraction of a 5th-order Stokes waves according to Fenton (1985) at thex= 0 boundary. The wave damping layer extends over a distance of xd= 2in boundary-normal direction from the x=Lxboundary. Thus the 10waves are created at x= 0, propagate in x-direction, enter the damping layer atx=Lxxd, are subjected to damping until x=Lxis reached, where the remaining waves are re ected and, while further subjected to damping, propagate back to x=Lxxd. If the damping was set up correctly, then the re ected waves are either completely damped or their height is decreased so much, that their in uence on the simulation results can be neglected. Otherwise they will be evident in the simulation results as additional error, which can be substantial as will be shown in the following sections. Figure 2: 2D wave tank lled with water (light gray) and air (white). The damping layer is shown in dark grey Local mesh renement is used, so that the grid is nest in the vicinity of the free surface, where the waves are discretized by roughly 100 cells per wavelength and 16 cells per wave height, as seen in Fig. 3. The temporal discretization involved more than 500 time steps
tper wave period T, thus in all investigated cases the Courant number C= (ui
t)=
xiremains well below 0:5 for every cell with size
xiand corresponding uid velocity ui. Waves were generated for over 12 wave periods with 8 iterations per time step and under-relaxation of 0 :4 for pressure and 0 :9 for all other variables. The discretization is based on the grid convergence study conducted by Peri c (2013). 11Figure 3: Computational grid for the whole domain (right), and a close up of the dis- cretization around free surface level (left) The numerical setup is the same for all simulations in this study, only that the scale, wave and damping parameters are varied. Variations of the setup are mentioned where they occur. 7. Assessing the Damping Performance for Regular, Monochro- matic Waves When wave re ections occur within a damping zone, the waves will prop- agate back into the solution domain with a diminished height. Due to the resulting superposition of the partly re ected wave and the original wave, the original wave will seem to grow and shrink in a time-periodic fashion. For regular, monochromatic waves, this occurs uniformly in all parts of the domain where the two wave systems are superposed as seen later e.g. in Sect. 11 from Fig. 9. In the present work, this phenomenon is called a partial standing wave , since the more the damping deviates from the optimal value, the more does the phenomenon resemble a standing wave, until for the bounding cases (i.e. zero damping or innitely large damping) there is 100% re ection, and a standing wave occurs. Assessing the amount of re ections for regular monochromatic waves is an intricate issue. Since only a single wave period occurs, a wave spectrum 12cannot be constructed. As seen from Fig. 9, the wave height envelope changes over space. However, this cannot be predicted since it is not known in advance at which point in the damping layer the re ection mainly occurs. Thus from the recordings of a single wave probe, the height of the re ected wave cannot be determined. However, the surface elevation in close vicinity to the boundary to which the damping zone is attached shows how much of the incident wave reaches the domain boundary despite the damping, since a maximum amplication occurs at the domain boundary when the wave is re ected. Thus in this study, rst the free surface elevation is recorded at x= 5:75, i.e. next to the domain boundary. The average wave height at this location is obtained and plottet in relation to the average wave height of the undamped wave, to show how much the wave height is reduced after propagating from the entrance to the outer boundary of the damping layer. As shown in Sects. 8 and 9, this is a good criterion for insucient to optimal wave damping, where wave re ections occur mainly at the outer boundary of the damping layer, atx=Lx; however, this criterion will not detect re ections if the damping is too strong, since then the waves will be re ected at the entrance to the damping layer. Therefore, secondly a re ection coecient CRis computed as proposed by Ursell et al. (1960). CRcorresponds to the ratio of the heights of the incident wave to the re ected wave. It can be written as CR= (HmaxHmin)=(Hmax+Hmin); (11) whereHmaxis the maximum and Hminthe minimum value of the wave height envelope. It holds 0 CR1, so thatCR= 1 for perfect wave re ection andCR= 0 for no wave re ection. As explained above, the wave height for a partial standing wave is not constant, but oscillates around a mean value. This occurs uniformly in all domain parts where the partial standing wave has fully developed. Thus HmaxandHminare obtained in the following fashion in this work: The free surface elevation in the whole tank is recorded at 40 evenly timed instances per wave period starting at 10 periods simulation time for 2 wave periods. For each recording j, the average wave height Hjis calculated for the inter- val 2x4, which is adjacent to the damping zone but not subject to wave damping and suciently away from the inlet; furthermore, the partial standing wave at this location is fully developed during this time interval, 13while wave re-re ections at the wave-maker have not yet developed signi- cantly. Therefore, the maximum Hmaxand minimum Hminof all Hjvalues can be taken as HmaxandHmin, respectively. This approach detects all wave re ections that propagate back into the solution domain. The accuracy of the scheme can be increased if the surface elevation in the tank is recorded in smaller time intervals, however this also increases the computational eort signicantly. Furthermore, the above procedure requires undisturbed regular monochromatic waves. Yet in many applications, uid-structure interactions create ow disturbances, so the above scheme cannot be applied directly in the simulations, but an additional 2D simulation of undisturbed wave prop- agation and damping for otherwise the same setup is required to obtain CR. As long as this process is not fully automized, running and post-processing the 2D simulation requires additional human eort. Overall, this procedure is rather eortful, which explains why CRis rarely computed in practice. With the two presented approaches, it is possible to distinguish between those wave re ections, which occur mainly at the entrance to the damping zone, and those re ections which occur mostly at the boundary to which the damping zone is attached. Furthermore, the in uence of wave re ections on the solution can be quantied. 8. Variation of Damping Coecient for Linear Damping To investigate which range for the linear damping coecient according to Eq. (5) produces satisfactory wave damping, waves with wavelength = 4 m and height 0 :16 m are investigated. Only linear damping is considered, sof2= 0. Simulations are performed for damping coecient f1between f12[0:625 s1;1000 s1]. The recorded surface elevations near the end of the damping layer in Fig. 4 show that the damping is strongly in uenced by the choice off1, while the wave phase and period remain nearly unchanged. 14Figure 4: Surface elevation scaled by height Hof the undamped wave over time scaled by the wave period T; recorded at x= 5:75for simulations with f12[0:625 s1;80 s1] and f2= 0 Figure 5 shows how a variation of f1aects the re ection coecient CR, which describes the amount of re ected waves that are present in the solution domain outside of the damping zone, and the mean measured wave height Hmeanrecorded in close vicinity to the boundary to which the damping layer is attached. For smaller damping coecients ( f1.20 s1), wave re ections occur mainly at the domain boundary to which the damping zone is attached. Here, the both show the same trend, with CR<H mean=(2H) since the re ected waves loose further height until they leave the damping zone atx= 4. Forf1<1:25 s1, the in uence of the damping is so small that a partial standing wave (as described in Sect. 7) higher than the initial wave appears even inside the damping layer. For stronger damping ( f1>20 s1), wave re ection occurs mainly at the entrance to the damping zone. Thus Hmean continually decreases when increasing f1, since less of the incoming wave can pass through the damping zone, so the water surface will be virtually at near the domain boundary. This is also visible later in Figs. 13 from Sect. 13. The aforementioned increase in the amount of wave re ections detectable in the solution domain can be seen in the curve for CR. This shows that, for a given fade-in function and damping layer thickness, 15there is an optimum for f1so that the wave re ections propagating back into the solution domain are minimized. The best damping was achieved for f1= 10 s1, in which case the eects of wave re ections on the wave height within the solution domain are less than 0 :7%. Figure 5: Mean wave height Hmean recorded at x= 5:75scaled by twice the height Hof the undamped wave and re ection coecient CRover damping coecient f1, whilef2= 0 This provides the following conclusions: Hmean is not a suitable indicator for damping quality if the damping is stronger than optimal, however it is useful to characterize where the re ections originate from: if CRshows that noticeable re ections are present within the solution domain, while at the same timeHmeanis negligibly small, then the re ections cannot occur at the domain boundary, but must occur closer to the entrance to the damping layer. Signicant re ections ( CR>2%) in form of partial standing waves appear for roughlyf1<5 s1andf1>80 s1. The height of the partial standing wave increases the more f1deviates from the regime where satisfactory damping is observed; however, the increase is slower if the damping is stronger than optimal instead of weaker. 9. Variation of Damping Coecient for Quadratic Damping To investigate which range for the quadratic damping coecient accord- ing to Eq. (5) produces satisfactory wave damping, waves with wavelength 16= 4 m and height 0 :16 m are investigated. Only quadratic damping is con- sidered, so f1= 0. Simulations are performed for damping coecient f2 betweenf22[0:625 m1;10240 m1]. The recorded surface elevations near the end of the damping layer in Fig. 6 show that the damping is strongly in uenced by the choice of f2, while the wave phase and period remain nearly unchanged. Figure 6: Surface elevation scaled by height Hof the undamped wave over time scaled by the wave period T; recorded at x= 5:75for simulations with f22[0:625 m1;10240 m1] andf1= 0 Figure 7 shows how a variation of f2aects the re ection coecient CR and the mean wave height Hmeanrecorded in close vicinity to the boundary to which the damping layer is attached. The results show the same trends as the ones in Sect. 8. For the given fade-in function and damping layer thickness, there is an optimum for f2 so that the wave re ections propagating back into the solution domain are minimized. The best damping was achieved for f2= 160 m1, in which case the eects of wave re ections on the wave height within the solution domain are less than 0 :7%. The eects of wave re ections increase the further f2 deviates from the optimum. For smaller f2values, the waves are re ected mainly at the domain boundary, for larger f2values the re ection occurs mainly at the entrance to the damping zone. Signicant re ections ( CR> 2%) in form of partial standing waves appear for roughly f2<80 m1and f2>640 m1. 17Figure 7: Mean wave height Hmean recorded at x= 5:75scaled by twice the height Hof the undamped wave and re ection coecient CRover damping coecient f2, whilef1= 0 Compared to the linear damping functions from the previous section, the use of quadratic damping functions does not oer a signicant improvement in damping quality. With an optimal setup, both approaches provide roughly the same damping quality if the setup is optimized. However, the range of wave frequencies that are damped satisfactorily is narrower for quadratic damping. 10. In uence of Computational Mesh on Achieved Damping The simulations from Sect. 8 are rerun with same setup except for a grid coarsened by factor 2. Thus whereas the ne mesh simulations discretize the wave with 100 cells per wavelength and 16 cells per wave height, the coarse mesh simulations have 50 cells per wavelength and 8 cells per wave height. All coarse grid re ection coecients dier from their corresponding ne grid re ection coecient by CR;coarse =CR;ne0:8%. This is also visible in Fig. 8. Thus for sucient resolution, the damping eectiveness can be considered grid-independent. This is expected since the damping-related terms in Eqs. (2) to (4) do not depend on cell volume. The used grids in this study are thus adequate. 18Figure 8: Re ection coecient CRover damping coecient f1for coarse and ne mesh simulations, with f2= 0 11. In uence of the Thickness of the Damping Layer The simulation for = 4:0 m,H= 0:16 m andf1= 10 s1from Sect. 8 is repeated for xd= 0;0:25;0:5;0:75;1:0;1:25;1:5;2:0;2:5with otherwise the same setup. The evolution of the free surface elevation in the tank over time in Fig. 9 shows that xdhas a strong in uence on the achieved damping. Setting xd= 0deactivates the damping and produces at rst a nearly perfect standing wave, which then degenerates due to the in u- ence of the pressure outlet boundary, since prescribing hydrostatic pressure establishes an oscillatory in-/out ow of water through this boundary which disturbs the standing wave. For xd= 0:5, a strong partial standing wave oc- curs, and for xd= 1:0only slight re ections are still observable CR1:6%. For largerxd, the in uence of wave re ections continues to decrease. This is evident from the plot of CRfor the simulations shown in Fig. 10. 19Figure 9: Free surface elevation over x-location in tank shown for 40 equally spaced time instances over one period starting at t= 16 s; from top to bottom xd= 0;0:5;1:0;1:5;2:0;2:5; the damping zone is depicted as shaded gray 20Figure 10: Re ection coecient CRover damping zone thickness xd Subsequently, the simulations from Sect. 8 have been rerun with the damping thickness set to xd= 1. Comparing the resulting curves for CR overf1shows that increasing xdnot only improves the damping quality; it also widens the range of damping coecients for which satisfactory damping is obtained; thus the wave damping then becomes less sensitive to !the more xdincreases. However, this also increases the computational eort. Figure 11: Re ection coecient CRover damping coecient f1forxd= 1andxd= 2, whilef2= 0 21The results show that, if the damping coecients are set up close to the optimum and it is desired that CR<2%, thenxd= 1suces. This knowl- edge is useful, since by reducing xdthe computational domain can be kept smaller and thus the computational eort can be reduced. However, if better damping is desired or when complex ow phenomena are considered, espe- cially when irregular waves or wave re ections from bodies are present, then the damping layer thickness should be increased to damp all wave compo- nents successfully. The present study suggests that a damping layer thickness of 1:5xd2can be recommended. 12. In uence of Wave Steepness on Achieved Damping The simulations in this section are based on those from Sect. 8, i.e. = 4:0 m,H= 0:16 m and varying f1in the range [0 :625 s1;1000 s1]. The simulations were rerun with the same setup except for two modications: The wave height was changed to H= 0:4 m, resulting in a steepness of H= = 0:1 instead of the previous H= = 0:04. Furthermore, the grid was adjusted to maintain the same number of cells per wave height as well as per wavelength, so that both results are comparable. As can be seen from Figure 12, the in uence of the increased wave steep- ness is comparatively small, except for the cases with signicantly smaller than optimum damping ( f12:5 s1). For the rest of the range, i.e. 5 f1 1000 s1, the dierence in re ection coecients is only CR(H= = 0:1) = CR(H= = 0:04)1:7%. Therefore, although the damping performs slightly better for waves of smaller steepness, the in uence of wave steepness can be assumed negligible for most practical cases. If stronger wave steepness is con- sidered and less uncertainty is required, then the thickness of the damping layer can be further increased to decrease CR. 22Figure 12: Re ection coecient CRover damping coecient f1for waves of same period but diering steepness H= = 0:04 andH= = 0:1, whilef2= 0 13. The Scaling Law for Linear Damping In order to verify the assumed scaling law for linear damping from Sect. 5, the simulation setup with the best damping performance ( f1= 10:0 s1) from Sect. 8 was scaled geometrically and kinematically so that the generated waves are completely similar, for wavelengths = 0:04 m;4 m;400 m and thus corresponding heights of H= 0:0016 m;0:16 m;16 m. This corresponds to a realistic scaling, since geometrically scaling by 1 : 100 is common in both experimental and computational model- and full scale investigations. As necessary requirement to obtain similar damping (i.e. similarity of CRand surface elevation), the damping length xdis scaled directly proportional to the wavelength, according to Eq. (8). At rst, the simulations are run with no scaling of f1, so thatf1= 10 s1in all cases. Figure 13 shows the free surface in the tank after the simulations. In the vicinity of the inlet (0 <x=< 1) the eects of wave re ections have not yet fully established, thus the dierences between the curves are small. This shows that the wave-maker generated similar waves in the three simulations. As the plot shows an arbitrarily selected time instant, the eects of wave re ections cannot be judged from it, since the partial standing waves were not captured at their maximum amplication. Therefore, for = 400 m no noticeable dierence to the reference case = 4 m is seen outside the damping zone (0 < x= < 4), whereas obvious re ections are visible for 23= 0:04 m. However regarding the surface elevation within the damping zone (4<x=< 6), the three curves dier considerably, and thus the wave damping also does not act in a similar fashion. A close look at the surface elevation in the damping zones shows that wave re ections will mostly occur at domain boundary x== 6 for= 0:04 m, in contrast to = 400 m, where they will mostly occur close to the entrance to the damping zone at x== 4. Figure 13: Surface elevation in the whole domain after 12:6 periods; for similar waves with wavelengths = 0:04 m;4 m;400 m; no scaling of f1, thusf1= 10 s1andf2= 0 for all cases Subsequently, based on the ndings in Sect. 5, the simulations are rerun withf1scaled with wave frequency !according to Eq. (9), based on reference caseref= 4 m. It is apparent from Fig. 14, that the surface elevations are similar everywhere in the tank and that the proposed scaling law is correct. 24Figure 14: Surface elevation in the whole domain after 12:6 periods; for similar waves with wavelengths = 0:04 m;4 m;400 m; scaling off1according to Eq. (9) , thusf1= 10 s1;3:16 s1;1 s1andf2= 0 Table 1 compares the re ection coecients CRfor both unscaled and correctly scaled f1. Whenf1is held constant, an up- or down-scaling of the wave with factor 100 will lead to a signicant increase in re ections. Therefore without adjusting f1, it is not possible to obtain similar damping in model- and full scale. If insteadf1is scaled according to Eqs. (8) and (9), CRstays nearly the same; slight uctuations of CRare due to the scheme for obtaining CR. Therefore, the damping quality can be reliably reproduced when the scaling law is applied. This is even evident in the CRvalues for xed f1: For the scaled waves considered, f1is either roughly 10 times larger ( = 400 m) or smaller ( = 0:04 m) than the optimum value; compared with the 10 times larger or smaller than optimal f1values from Fig. 5, the CRvalues coincide. 25Table 1: Re ection coecient CRfor unscaled and correctly scaled simulations CR(xedf1)CR(scaledf1) 0:04 m 39:8% 1 :1% 4 m 0:7% 0 :7% 400 m 2:4% 0 :6% Practical Recommendation When using linear damping according to Eqs. (5) and (6), the following damping setup can be recommended based on the results from Sects. 8 to 15: f1= 1! ; (12) with 1=, wave frequency !,f2= 0,xd= 2andn= 2. 14. The Scaling Law for Quadratic Damping In order to verify the assumed scaling law for quadratic damping from Sect. 5, the simulation setup with the best damping performance ( f2= 160:0 m1) from Sect. 9 was scaled geometrically and kinematically so that the generated waves are completely similar, for wavelengths = 0:04 m;4 m;400 m and thus corresponding heights of H= 0:0016 m;0:16 m;16 m. As in the pre- vious section, the damping length xdis scaled directly proportional to the wavelength, according to Eq. (8). At rst the simulations are run with no scaling of f2, so thatf2= 160 m1 in all cases. Subsequently, based on the ndings in Sect. 5, the simulations are rerun with f2scaled with wave frequency !squared according to Eq. (10), based on reference case = 4 m. It is apparent from Fig. 15, that the surface elevations are similar everywhere in the tank and that the proposed scaling law is correct. 26Figure 15: Surface elevation in the whole domain after 12:6 periods; for similar waves with wavelengths = 0:04 m;4 m;400 m; scaling off2according to Eq. (10) , thusf2= 160 m1;16 m1;1:6 m1andf1= 0 Table 2 compares the re ection coecients CRfor both unscaled and correctly scaled f2. Whenf2is held constant, an up- or down-scaling of the wave with factor 100 will lead to a signicant increase in re ections. If insteadf2is scaled according to Eqs. (8) and (9), CRstays nearly the same; slight uctuations of CRare due to the scheme for obtaining CR. This shows that the damping quality can be reliably reproduced when the scaling law is applied. Table 2: Re ection coecient CRfor unscaled and correctly scaled simulations CR(xedf2)CR(scaledf2) 0:04 m 65:9% 1 :1% 4 m 0:6% 0 :6% 400 m 17:7% 0 :5% The present results show that if the damping parameters are not ad- justed when performing model- and full-scale simulations, then unsatisfac- tory damping can be expected. 27Compared to linear wave damping, quadratic damping has a narrower range of wave frequencies for which satisfactory damping is obtained. There- fore, since the damping performance is more sensitive to changes in wave frequency for quadratic (or higher order) damping functions, and since with optimum set up the damping performance is the same as with linear damp- ing, it is recommended to use linear damping functions. For this reasons, only linear damping was examined in the following Sects. 15 and 16. Practical Recommendation When using quadratic damping according to Eqs. (5) and (6), the following damping setup can be recommended based on the results from Sect. 9: f2= 21; (13) with 2= 2
102, wavelength ,f1= 0,xd= 2andn= 2. 15. Damping of Irregular Free-Surface Waves The previous results indicate that, to simulate irregular waves, the damp- ing setup should be based on the wave component with the longest wave- length. If the damping setup is optimal for the longest wave component, then on the one hand the damping coecient for all shorter wave components will be smaller then optimal, which has a negative eect on the damping perfor- mance for the corresponding wave component as seen in Sect. 8. On the other hand, the damping layer thickness relative to the wavelength will be larger for these wave components, which has a positive eect on the damping performance as shown in Sect. 11. From the previous results, the latter eect is expected be the prevailing in uence on the damping quality. Furthermore, the shorter wave components are discretized with less cells per wavelength, so the discretization errors are stronger for these wave components, which provides a faster wave energy dissipation for these components which is also benecial in this respect. Exemplarily, linear damping of three superposed wave components with same steepnessHi i= 0:03 and wavelengths diering by 400% is demonstrated in the following. At the wave-maker, the following surface elevation is pre- scribed (t) =3X i=1Hi 2cos(!i(t+ 1:19 s) +i); (14) 28with wave height Hi, wave frequencies !i, phase shifts iand timet. The actual parameters are shown in Table 3. Table 3: Parameters of the wave components ii(m)Hi(m)!i(rad s)i(rad) 1 4:0 0:12 3:926 0:0 2 2:0 0:06 5:5510:9176 3 1:0 0:03 7:851 3:543 Velocity and volume fraction corresponding to Eq. (14) are taken from linear wave theory and are applied at the inlet boundary x= 0 as linear superposition. The domain length is Lx= 31. The grid dimensions in free surface zone are
x= 0:0195 m and
z= 0:0024 m with a time step of
t= 0:001 s. The damping setup is based on the wave component with the longest wavelength according to Eq. (12): The damping layer thickness is xd= 21 and the damping coecients are f1= 20 s1andf2= 0. With this setup, all wave components were damped satisfactorily without noticeable re ections. As can be seen from Fig. 16, the waves enter the damping layer at x= 4 m and after propagating a distance of 2 1into the damping layer, the wave height is reduced by two orders of magnitude. Thus linear damping seems well suited for the damping of irregular waves. Although the approach from Sect. 5 to determine CRcannot be applied in this case, the results from Sect. 8 show that for all wave components re ections will mostly occur at the domain boundary, and not at the entrance to the damping layer. Thus in this case, measuring the surface elevation in close vicinity to the corresponding domain boundary is enough to evaluate the damping performance. 29Figure 16: Free surface elevation in the solution domain after 20 s; top: whole domain; bottom: close up of the last part of the damping layer However, for realistic wave spectra the above approach is not feasible, as these consist of wave components over a broad range of wavelengths. Thus selecting the component with the longest wavelength for the damping zone setup would not be practical, since this would require a very large damping zone and increase the total amount of grid cells and computational eort substantially (possibly one or more orders of magnitude if the mesh size is not changed). For practical purposes, an ecient approach is outlined in the following. According to Lloyd (1989), most realistic sea spectra are narrow banded. This means that most wave energy is concentrated in a narrow band of wave frequencies around a peak frequency. Although larger wavelengths occur as well, these carry a comparatively small amount of energy. Thus we propose to set up the wave damping based on the peak wave frequency. As seen in Sect. 11 the range of frequencies that are damped can be modied with the 30damping zone thickness, so the broader-banded the spectrum is, the larger hasxdto be. Exemplarily, this approach is investigated for the widely used JONSWAP spectrum, which was originally developed by Hasselmann et al. (1973). Ir- regular waves are prescribed at the inlet boundary, with parameters peak wave period Tp= 1:6 s, signicant wave height Hs= 0:12 m, peak shape pa- rameter = 3:3, spectral width parameter = 0:07 for!!pand= 0:09 for!>! p. The wave spectrum was discretized into 100 components. The choice of xd= 2peak, with wavelength peakcorresponding to the peak wave frequency, seems adequate for such cases. No visible disturbance eects were noted in the simulation. The wave heights over time are shown recorded close to the wave-maker in Fig. 17 and recorded close to the bound- ary to which the damping zone is attached in Fig. 18. This shows that the average wave height is reduced by roughly two orders of magnitude. Figure 17: Surface elevation over time recorded directly before the wave-maker 31Figure 18: Surface elevation over time recorded in close vicinity to the boundary, to which the damping layer is attached From the surface elevation in the tank in Fig. 19 no undesired wave re ections can be noticed. Although a more detailed study of the error of this approximation regarding dierent parameters of the spectrum was not possible in the scope of this study, the proposed approach seems to work reasonably well for practical purposes. Further research in this respect is recommended to reduce uncertainties regarding the re ections. Figure 19: Surface elevation in the whole domain at t15:6 s 16. Application of Scaling Law to Ship Resistance Prediction This section compares results from model and full scale resistance compu- tations in 3D of the Kriso Container Ship (KCS) at Froude number 0 :26. The 32simulations in this section are based on the simulations reported in detail in Enger et al. (2010). For detailed information on the setup and discretiza- tion, the reader referred to Enger et al. (2010). In the following, only a brief overview of the setup and dierences to the original simulations are given for the sake of brevity. The present model scale simulation diers only in the damping setup (and slight modications of the used grid) from the ne grid simulation in Enger et al. (2010). The KCS is xed in its oating position at zero speed. To simulate the hull being towed at speed U, this velocity is applied at the inlet domain boundaries and the hydrostatic pressure of the undisturbed water surface is applied at the outlet boundary behind the ship. The domain is initialized with a at water surface and ow velocity U for all cells. As time accuracy is not in the focus here, rst-order implicit Euler scheme is used for time integration. Apart from the use of the k- turbulence model by Launder and Spalding (1974), the computational setup is similar to the one used in the rest of this work. The computational grid consists of roughly 3 million cells. For comparability, we compare only the pressure components of the drag and vertical forces, which are obtained by integrating the x-component for the drag and z-component for the vertical component of the pressure forces over the ship hull. The simulation starts at 0 s and is stopped at tmax= 90 s simulation time. The Kelvin wake of the ship can be decomposed into a transversal and a divergent wave component. The wave damping setup is based on the ship-evoked transversal wave (wave- lengtht3:1 m), the phase velocity of which equals the service speed U. Wave damping according to Eqs. (5) and (6) has been applied to inlet, side and outlet boundaries with parameters xd= 2:3t,f1= 22:5 s1,f2= 0, andn= 2. This setup provided satisfactory convergence of drag and vertical forces in model scale. The wake pattern for the nished simulation is shown in Fig. 20 and the results are in agreement with the ndings from Enger et al. (2010). The obtained resistance coecient CT;sim= 3:533
103compares well with the experimental data ( CT;exp= 3:557
103, 0:68% dierence to CT;sim) by Kim et al. (2001) and to the simulation results by Enger et al. (2001) (CT;Enger = 3:561
103, 0:11% dierence to CT;exp). 33Figure 20: Wave prole for model scale ship with f1= 22:5 s1att= 90 s Additionally, full scale simulations are performed with Froude similarity. The scaled velocity and ship dimensions are shown in Table 4. The grid is similar to the one for the model scale simulation except scaled with factor 31:6. Assuming similar damping can be obtained with the presented scaling laws,xdwas scaled according to Eq. 8 by factor 31 :6 as well. Otherwise the setup corresponds to the one from the model scale simulation. 34Table 4: KCS parameters scale waterline length L(m) service speed U(m=s) model 7:357 2 :196 full 232:5 12 :347 Figure 21 shows drag and vertical forces over time when both model and full scale simulations are run with the same value for damping coecient f1. In contrast to the model scale forces, the full scale forces oscillate in a complicated fashion. Therefore without a proper scaling of f1, no converged solution can be obtained for the full scale case. 35Figure 21: Drag (top image) and vertical (bottom image) pressure forces on ship over time for model (red) and full scale (grey); no scaling of f1, thusf1= 22:5 s1is the same in both simulations Finally, the full scale simulation is rerun with f1scaled according to Eq. 9 to obtain similar damping as in the model scale simulation with f1= 22:5 s1. The correctly scaled value for the full scale simulation is thus f1;full=f1;model
!full=!model = 22:5 s1
1=p 31:64 s1. The resulting drag and vertical forces in Fig. 22 show that indeed similar damping is obtained, since in both cases the forces converge in a qualitatively similar fashion. Note that a perfect match of the curves in Fig. 22 is not expected, since Froude-similarity is given, but not Reynolds-similarity. Thus although the pressure components of the forces on the ship will converge to the same values (if scaled by L3), the way they converge (amplitude and frequency of the oscillation) may not be 36exactly similar, since this depends on the solution of all equations. However, the tendency of the convergence will be qualitatively similar as shown in the plots. Figure 22: Drag (top image) and vertical (bottom image) pressure forces on ship over time for model and full scale; the damping setup for the full scale simulation is obtained by scaling the model scale setup according to Eqs. 8 and 9 17. Discussion and Conclusion In order to obtain reliable wave damping with damping layer approaches, the damping coecients must be adjusted according to the wave parameters, as shown in Sects. 8, 9, 13 and 14. It is described in Sect. 7 that the procedure to quantify the damping quality is quite eortful, and thus it 37is seldom carried out in practice. This underlines the importance of the present ndings for practical applications, since unless the damping quality can be reliably set to ensure that the in uence of undesired wave re ections are small enough to be neglected, a large uncertainty will remain in the simulation results. The optimum values for damping coecients f1andf2 can be assumed not to depend on computational grid, wave steepness and thicknessxdof the damping layer as shown in Sects. 10, 11 and 12. As shown in Sect. 11, the damping layer thickness has the strongest in uence on the damping quality. If it increases, the range of waves that will be damped satisfactorily broadens and the re ection coecient for the optimum setup shrinks; unfortunately, the computational eort increases at the same time as well, thus optimizing the damping setup is important. In contrast, Sect. 12 shows that the wave steepness has a smaller eect on the damping, with the tendency towards better damping for smaller wave steepness. For suciently ne discretizations, Sect. 10 shows that the damping can be considered not aected by the grid. A practical approach for ecient damping of irregular waves has been presented in Sect. 15. The scaling laws in Sect. 5 and recommendations given in Sects. 13 and 14 provide a reliable way to set up optimum wave damping for any regular wave. Moreover, similarity of the wave damping can be guaranteed in model- and full-scale simulations as shown in Sects. 13, 14 and 16. The ndings can easily be applied to any implementation of wave damping which accords to Sect. 3. References [1] Cao, Y., Beck, R. F. and Schultz, W. W. 1993. An absorbing beach for numerical simulations of nonlinear waves in a wave tank, Proc. 8th Intl. Workshop Water Waves and Floating Bodies , 17-20. [2] Choi, J. and Yoon, S. B. 2009. Numerical simulations using momentum source wave-maker applied to RANS equation model, Coastal Engineer- ing, 56, (10), 1043-1060. [3] Cruz, J. 2008. Ocean wave energy, Springer Series in Green Energy and Technology , UK, 147-159. [4] Enger, S., Peri c, M. and Peri c, R. 2010. Simulation of Flow Around KCS- Hull, Gothenburg 2010: A Workshop on CFD in Ship Hydrodynamics, Gothenburg. 38[5] Fenton, J. D. 1985. A fth-order Stokes theory for steady waves, J. Waterway, Port, Coastal and Ocean Eng. , 111, (2), 216-234. [6] Ferrant, P., Gentaz, L., Monroy, C., Luquet, R., Ducrozet, G., Alessan- drini, B., Jacquin, E., Drouet, A. 2008. Recent advances towards the viscous ow simulation of ships manoeuvring in waves, Proc. 23rd Int. Workshop on Water Waves and Floating Bodies , Jeju, Korea. [7] Ferziger, J. and Peri c, M. 2002. Computational Methods for Fluid Dy- namics, Springer. [8] Grilli, S. T., Horrillo, J. 1997. Numerical generation and absorption of fully nonlinear periodic waves. J. Eng. Mech. , 123, (10), 1060-1069. [9] Guignard, S., Grilli, S. T., Marcer, R., Rey, V. 1999. Computation of shoaling and breaking waves in nearshore areas by the coupling of BEM and VOF methods, Proc. ISOPE1999 , Brest, France. [10] Ha, T., Lee, J.W. and Cho, Y.S. 2011. Internal wave-maker for Navier- Stokes equations in a three-dimensional numerical model, J. Coastal Research , SI 64, 511-515. [11] Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruse- man, P., Meerburg, A., M uller, P., Olbers, D. J., Richter, K., Sell, W. and Walden, H. 1973. Measurements of wind-wave growth and swell de- cay during the Joint North Sea Wave Project (JONSWAP), Deutsche Hydrographische Zeitschrift , (8), Reihe A. [12] Higuera, P., Lara, J. L., Losada, I. J. 2013. Realistic wave generation and active wave absorption for Navier-Stokes models Application to OpenFOAM r,Coastal Eng. , 71, 102-118. [13] Jha, D. K. 2005. Simple Harmonic Motion and Wave Theory, Discovery Publishing House. [14] Kim, J., O'Sullivan, J., Read, A. 2012. Ringing analysis of a vertical cylinder by Euler overlay method, Proc. OMAE2012 , Rio de Janeiro, Brazil. 39[15] Kim, J., Tan, J. H. C., Magee, A., Wu, G., Paulson, S., Davies, B. 2013. Analysis of ringing ringing response of a gravity based structure in extreme sea states, Proc. OMAE2013 , Nantes, France. [16] Kim, W. J., Van, S. H., Kim, D. H. 2001. Measurement of ows around modern commercial ship models, Experiments in Fluids , 31, Springer- Verlag, 567-578. [17] Kraskowski, M. 2010. Simulating hull dynamics in waves using a RANSE code, Ship technology research , 57, 2, 120-127. [18] Lal, A., Elangovan, M. 2008. CFD simulation and validation of ap type wave-maker, WASET , 46, 76-82. [19] Launder, B.E., and Spalding, D.B. 1974. The numerical computation of turbulent ows, Comput. Meth. Appl. Mech. Eng. , 3, 269-289. [20] Lloyd, A. R. J. M. 1989. Seakeeping: Ship Behaviour in Rough Weather, Ellis Horwood Limited. [21] Muzaferija, S. and Peri c, M. 1999. Computation of free surface ows us- ing interface-tracking and interface-capturing methods, Nonlinear Water Wave Interaction, Chap. 2, 59-100, WIT Press, Southampton. [22] Park, J. C., Kim, M. H. and Miyata, H. 1999. Fully non-linear free- surface simulations by a 3D viscous numerical wave tank, International Journal for Numerical Methods in Fluids , 29, 685-703. [23] Peri c, R. 2013. Internal generation of free surface waves and application to bodies in cross sea, MSc Thesis, Schriftenreihe Schibau, Hamburg University of Technology, Hamburg, Germany. [24] Peri c, R., Abdel-Maksoud, M., 2015. Assessment of uncertainty due to wave re ections in experiments via numerical ow simulations, Proc. ISOPE2015 , Hawaii, USA. [25] Sch aer, H.A., Klopman, G. 2000. Review of multidirectional active wave absorption methods, Journal of Waterway, Port, Coastal, and Ocean Engineering , 88-97. 40[26] Ursell, F., Dean, R. G. and Yu, Y. S. 1960. Forced small-amplitude water waves: a comparison of theory and experiment, Journal of Fluid Mechanics , 7, (01), 33-52. [27] W ockner-Kluwe, K. 2013. Evaluation of the unsteady propeller perfor- mance behind ships in waves, PhD thesis at Hamburg University of Technology, Schriftenreihe Schibau , 667, Hamburg. 41

1602.07325v1.Experimental_Investigation_of_Temperature_Dependent_Gilbert_Damping_in_Permalloy_Thin_Films.pdf

1 Experimental Investigation of Temperature-Dependent Gilbert Damping in Permalloy Thin Films Yuelei Zhao1,2†, Qi Song1,2†, See-Hun Yang3, Tang Su1,2, Wei Yuan1,2, Stuart S. P. Parkin3,4, Jing Shi5*, and Wei Han1,2* 1International Center for Quantum Materials, Peking University, Beijing, 100871, P. R. China 2Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China 3IBM Almaden Research Center, San Jose, California 95120, USA 4Max Planck Institute for Microstructu re Physics, 06120 Halle (Saale), Germany 5Department of Physics and Astronomy, Univers ity of California, Riverside, California 92521, USA †These authors contributed equally to the work *Correspondence to be addressed to: jing.shi @ucr.edu (J.S.) and weihan@pku.edu.cn (W.H.) Abstract The Gilbert damping of ferromagnetic materials is arguably the most important but least understood phenomenological parameter that dictates real-time magnetization dynamics. Understanding the physical origin of the Gilbert damping is highly relevant to developing future fast switching spintronics devices such as magnetic sensors and magnetic random access memory. Here, we report an experimental stud y of temperature-dependent Gilbert damping in permalloy (Py) thin films of varying thicknesses by ferromagnetic resonance. From the thickness dependence, two independent cont ributions to the Gilbert damping are identified, namely bulk damping and surface damping. Of particular inte rest, bulk damping decreases monotonically as the temperature decreases, while surface da mping shows an enhancement peak at the 2 temperature of ~50 K. These results provide an important insight to the physical origin of the Gilbert damping in ultr athin magnetic films. Introduction It is well known that the magnetization dynamics is described by the Landau-Lifshitz-Gilbert equation with a phenomenological parameter called the Gilbert damping ( α),1,2: eff SdM dMMH Mdt M dtαγ=− × + × (1) where M is the magnetization vector, γis the gyromagne tic ratio, and SM M= is the saturation magnetization. Despite intense theore tical and experimental efforts3-15, the microscopic origin of the damping in ferromagnetic (FM) metallic ma terials is still not well understood. Using FM metals as an example, vanadium doping decreases the Gilbert damping of Fe3 while many other rare-earth metals doping increase s the damping of permalloy (Py)4-6,16. Theoretically, several models have been developed to explain some key characteristics. For example, spin-orbit coupling is proposed to be the intrinsic or igin for homogenous time-varying magnetization9. The s-d exchange scattering model assumes that damp ing results from scattering of the conducting spin polarized electrons with the magnetization10. Besides, there is the Fermi surface breathing model taking account of the spin scattering with the lattice defects ba sed on the Fermi golden rule11,12. Furthermore, other damping mechanisms in clude electron-electron scattering, electron- impurity scattering13 and spin pumping into the adjacent nonmagnetic layers14, as well as the two magnon scattering model, which refers to that pa irs of magnon are scatte red by defects, and the ferromagnetic resonance (FMR) mode moves into short wavelength spin waves, leading to a 3 dephasing contribution to the linewidth15. In magnetic nanostructu res, the magnetization dynamics is dictated by the Gilbert damping of the FM materials which can be simulated by micromagnetics given the boundaries and dimens ions of the nanostructures. Therefore, understanding the Gilbert damping in FM materials is particularly important for characterizing and controlling ultrafast responses in magnetic nanostructures that ar e highly relevant to spintronic applications such as magne tic sensors and magnetic random access memory17. In this letter, we report an expe rimental investigation of the G ilbert damping in Py thin films via variable temperature FMR in a modified multi-functional insert of physical property measurement system with a coplanar waveguide (see methods for details). We choose Py thin films since it is an interesting FM metallic material for spintronics due to its high permeability, nearly zero magnetostriction, low coercivity, a nd very large anisotropi c magnetoresistance. In our study, Py thin films are gr own on top of ~25 nm SiO 2/Si substrates with a thickness ( d) range of 3-50 nm by magnetron sputtering (see methods for details). A capping layer of TaN or Al 2O3 is used to prevent oxidation of the Py during m easurement. Interestingly, we observe that the Gilbert damping of the thin Py films ( d <= 10 nm) shows an enhanced peak at ~ 50 K, while thicker films ( d >= 20 nm) decreases monotonically as the temperature decreases. The distinct low-temperature behavior in the Gilbert dampi ng in different thickness regimes indicates a pronounced surface contribution in the thin limit. In fact, from the linear relationship of the Gilbert damping as a function of the 1/ d, we identify two contribu tions, namely bulk damping and surface damping. Interestingl y, these two contributions show very different temperature dependent behaviors, in whic h the bulk damping decreases m onotonically as the temperature decreases, while the surface damping indicates an enhancement peak at ~ 50 K. We also notice that the effective magnetization sh ows an increase at the same temperature of ~50 K for 3 and 5 4 nm Py films. These observations could be all related to the magnetization reorientation on the Py surface at a certain temperatur e. Our results are important for theoretical investigation of the physical origins of Gilbert damping and also us eful for the purpose of designing fast switching spintronics devices. Results and Discussion Figure 1a shows five representative curves of the forward amplitude of the complex transmission coefficients (S 21) vs. in plane magnetic field meas ured on the 30 nm Py film with TaN capping at the frequencies of 4, 6, 8, 10 an d 12 GHz and at 300 K after renormalization by subtracting a constant background. These experiment al results could be fitted using the Lorentz equation18: 2 21 0 22() () ( )resHSSHH HΔ∝Δ+ − (2) where S0 is the constant describing the coefficient for the transmitted microwave power, H is the external magnetic field, Hres is the magnetic field under the resonance condition, and ΔH is the half linewidth. The extracted ΔH vs. the excitation frequency ( f) is summarized in Figures 1b and 1c for the temperature of 300 K and 5 K respect ively. The Gilbert damp ing could be obtained from the linearly fitted curves (red lin es), based on the following equation: 02()H fHπαγΔ= + Δ (3) in which γ is the geomagnetic ratio and ΔH0 is related to the inhom ogeneous properties of the Py films. The Gilbert damping at 300 K and 5 K is calculated to be 0.0064 ± 0.0001 and 0.0055 ± 0.0001 respectively. 5 The temperature dependence of the Gilbert damp ing for 3-50 nm Py films with TaN capping layer is summarized in Figure 2a. As d decreases, the Gilbert damping increases, indicative of the increasing importance of the film surfaces. Interestingly, fo r thicker Py films (e.g. 30 nm), the damping decreases monotonically as the temper ature decreases, which is expected for bulk materials due to suppressed sca ttering at low temperature. As d decreases down to 10 nm, an enhanced peak of the damping is obser ved at the temperature of ~ 50 K. As d decreases further, the peak of the damping becomes more pronounce d. For the 3 nm Py film, the damping shows a slight decrease first from 0.0126 ± 0.0001 at 3 00 K to 0.0121 ± 0.0001 at 175 K, and a giant enhancement up to 0.0142 ± 0.0001 at 50 K, and then a sharp decrease back down to 0.0114 ± 0.0003 at 5 K. The Gilbert damping as a function of the Py th icknesses at each temperature is also studied. Figure 2b shows the thickness dependence of the Py damping at 300 K. As d increases, the Gilbert damping decreases, which indicates a surface/interface enhanced damping for thin Py films19. To separate the damping due to the bul k and the surface/interface contribution, the damping is plotted as a function of 1/ d, as shown in Figure 2c, and it follows this equation as suggested by theories19-21. 1()BSdαα α=+ (4) in which the Bα and Sα represent the bulk and surface da mping, respectively. From these linearly fitted curves, we are able to separate the bulk damping term and the surface damping term out. In Figure 2b, the best fitted parameters for Bα and Sα are 0.0055 ± 0.0003 and 0.020 ± 0.002 nm. To be noted, there are two insulating mate rials adjacent to the Py films in our studies. 6 This is very different from previous studies on Py/Pt bilayer systems, where the spin pumping into Pt leads to an enhanced magnetic dampi ng in Py. Hence, the enhanced damping in our studies is very unlikely resulti ng from spin pumping into SiO 2 or TaN. To our knowledge, this surface damping could be related to interfacial spin f lip scattering at the interface between Py and the insulating layers, which ha s been included in a generalized spin-pumping theory reported recently21. The temperature dependence of the bulk damp ing and the surface damping are summarized in Figures 3a and 3b. The bulk damping of Py is ~0.0055 at 300 K. As the temperature decreases, it shows a monotonic decrea se and is down to ~0.0049 at 5 K. Th ese values are consistent with theoretical first principle calculations21-23 and the experimental valu es (0.004-0.008) reported for Py films with d ≥ 30 nm24-27. The temperature dependence of the bulk damping could be attributed to the magnetization rela xation due to the spin-lattice scattering in the Py films, which decreases as the temperature decreases. Of particular interest, the surface damping sh ows a completely different characteristic, indicating a totally different mechanism from th e bulk damping. A strong enhancement peak is observed at ~ 50 K for the surface damping. Could this enhancement of this surface/interface damping be due to the strong spin-orbit coupli ng in atomic Ta of Ta N capping layer? To investigate this, we measure the damping of the 5 nm and 30 nm Py films with Al 2O3 capping layer, which is expected to exhibit much lo wer spin-orbit coupling compared to TaN. The temperature dependence of the Py damping is su mmarized in Figures 4a and 4b. Interestingly, the similar enhancement of the damping at ~ 50 K is observed for 5 nm Py film with either Al 2O3 capping layer or TaN layer, whic h excludes that the origin of the feature of the enhanced 7 damping at ~50 K results from th e strong spin-orbit coupling in TaN layer. These results also indicate that the mechanism of this feature is most likely related to the common properties of Py with TaN and Al 2O3 capping layers, such as the crysta lline grain boundary and roughness of the Py films, etc. One possible mechanism for the observed peak of the damping at ~50 K could be related to a thermally induced spin reorientation transition on the Py surface at that temperature. For example, it has been show n that the spin reorientation of Py in magnetic tunnel junction structure happens due to the competition of different magne tic anisotropies, which c ould give rise to the peak of the FMR linewidth around the temperature of ~60 K28. Furthermore, we measure the effective magnetization ( Meff) as a function of temperature. Meff is obtained from the resonance frequencies ( fres) vs. the external magnetic field via the Kittel formula29: 12() [ ( 4 ) ]2res res res efffH H Mγππ=+ (4) in which Hres is the magnetic field at the resonance condition, and Meff is the effective magnetization which contains the saturation ma gnetization and other anisotropy contributions. As shown in Figures 5a and 5b, the 4π*M eff for 30 nm Py films w ith TaN capping layer are obtained to be ~10.4 and ~10.9 kG at 300 K and 5 K respectively. The temperature dependences of the 4π*M eff for 3nm, 5 nm, and 30 nm Py films are s hown in Figures 6a-6c. Around ~50 K, an anomaly in the effective magnetization for thin Py films (3 and 5 nm) is observed. Since we do not expect any steep change in Py’s saturation magnetization at this temperature, the anomaly in 4π*M eff should be caused by an anisot ropy change which coul d be related to a sp in reorientation. However, to fully understand the underlying mechan isms of the peak of the surface damping at ~ 50 K, further theoretical and e xperimental studies are needed. 8 Conclusion In summary, the thickness and temperature dependences of the Gilbert damping in Py thin films are investigated, from which the contributio n due to the bulk damping and surface damping are clearly identified. Of particular interest, the bulk damping decreases monotonically as the temperature decreases, while the surface damping develops an enhancement peak at ~ 50 K, which could be related to a thermally induced spin reorientation for the surface magnetization of the Py thin films. This model is also consistent with the observation of an enhancement of the effective magnetization below ~50 K. Our expe rimental results will contribute to the understanding of the intrinsic and ex trinsic mechanisms of the Gilber t damping in FM thin films. Methods Materials growth. The Py thin films are deposited on ~25 nm SiO 2/Si substrates at room temperature in 3×10- 3 Torr argon in a magnetron sputtering sy stem with a base pressure of ~ 1×10-8 Torr. The growth rate of the Py is ~ 1 Å/s. To prevent ex situ oxidation of the Py film during the measurement, a ~ 20 Å TaN or Al 2O3 capping layer is grown in situ environment. The TaN layer is grown by reactive sputtering of a Ta target in an argon-nitrogen gas mixture (ratio: 90/10). For Al 2O3 capping layer, a thin Al (3 Å) layer is deposited first, and the Al 2O3 is deposited by reactive spu ttering of an Al target in an ar gon-oxygen gas mixture (ratio: 93/7). FMR measurement. The FMR is measured using the vector network analyzer (VNA, Agilent E5071C) connected with a coplanar wave guide30 in the variable temperature insert of a Quantum Design Physical Properties Measuremen t System (PPMS) in the temperature range from 300 to 2 K. The Py sample is cut to be 1 × 0.4 cm and attached to the coplanar wave guide 9 with insulating silicon paste. For each temper ature from 300 K to 2 K, the forward complex transmission coefficients (S 21) for the frequencies between 1 - 15 GHz are recorded as a function of the magnetic field sweeping from ~2500 Oe to 0 Oe. Contributions J.S. and W.H. proposed and supervised the studies. Y.Z. and Q.S. performed the FMR measurement and analyzed the data. T.S. and W.Y. helped the measurement. S.H.Y. and S.S.P.P. grew the films. Y.Z., J.S. and W.H. wrote the manuscript. All authors commented on the manuscript and contributed to its final version. Acknowledgements We acknowledge the fruitful discussions with Ryuichi Shindou, Ke Xia, Ziqiang Qiu, Qian Niu, Xincheng Xie and Ji Feng and the support of National Basic Research Programs of China (973 Grants 2013CB921903, 2014CB920902 and 2015 CB921104). Wei Han also acknowledges the support by the 1000 Talents Program for Young Scientists of China. Competing financial interests The authors declare no compe ting financial interests. References: 1 Landau, L. & Lifshitz, E. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8, 153 (1935). 2 Gilbert, T. L. A phenomenological theory of damping in ferromagnetic materials. Magnetics, IEEE Transactions on 40, 3443-3449, doi:10.1109/TMAG.2004.836740 (2004). 10 3 Scheck, C., Cheng, L., Barsukov, I., Frait, Z. & Bailey, W. E. Low Relaxation Rate in Epitaxial Vanadium-Doped Ultrathin Iron Films. Phys. Rev. Lett. 98, 117601 (2007). 4 Woltersdorf, G., Kiessling, M., Meyer, G., Th iele, J. U. & Back, C. H. Damping by Slow Relaxing Rare Earth Impurities in Ni 80Fe20 Phys. Rev. Lett. 102, 257602 (2009). 5 Radu, I., Woltersdorf, G., Kiessling, M., Meln ikov, A., Bovensiepen, U., Thiele, J. U. & Back, C. H. 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Ferromagnetic reso nance linewidth in metallic thin films: Comparison of measurement methods. J. Appl. Phys. 99, 093909 (2006). 12 Figure Captions Figure 1. Measurement of Gilbert damping in Py thin films via ferromagnetic resonance (Py thickness = 30 nm). a, Ferromagnetic resonance spectra of the absorption for 30 nm Py thin films with TaN capping layer at gigahertz frequencies of 4, 6, 8, 10 and 12 GHz at 300 K after normalization by background subtraction. b, c, The half linewidths as a function of the resonance frequencies at 300 K and 5 K respectively. The red solid lines indicate the fitted lines based on equation (3), where the Gilbert damp ing constants could be obtained. Figure 2. Temperature dependence of the Gilber t damping of Py thin films with TaN capping. a, The temperature dependence of the Gilbert damping fo r 3, 5, 10, 15, 20, 30, and 50 nm Py films. b, The Gilbert damping as a function of the Py thickness, d, measured at 300 K. c, The Gilbert damping as a function of 1/ d measured at 300 K. The linear fitting corresponds to equation (4), in which the slope and the intercep t are related to the surf ace contribution and bulk contribution to the total Gilber t damping. Error bars correspond to one standard deviation. Figure 3. Bulk and surface damping of Py thin films with TaN capping layer. a, b, The temperature dependence of the bulk damping an d surface damping, respectively. The inset table summarizes the experimental values reported in early studies. Error bars correspond to one standard deviation. Figure 4. Comparison of the Gilbert damping of Py films with different capping layers. a, b, Temperature dependence of the Gilbert dampi ng of Py thin films with TaN capping layer 13 (blue) and Al 2O3 capping layer (green) for 5 nm Py a nd 30 nm Py, respectively. Error bars correspond to one standard deviation. Figure 5. Measurement of effective magnetizat ion in Py thin films via ferromagnetic resonance (Py thickness = 30 nm). a, b, The resonance frequencies vs. the resonance magnetic field at 300 K and 5 K, respectively. The fitted li nes (red curves) are obtained using the Kittel formula. Figure 6. Effective magnetization of Py fi lms as a function of the temperature. a, b, c, Temperature dependence of the effective magnetizati on of Py thin films of a thickness of 3 nm, 5 nm and 30 nm Py respectively. In b, c, the blue/green symbols correspond to the Py with TaN/Al 2O3 capping layer. 0 500 1000 1500 2000 -0.3 -0.2 -0.1 0.0 0.1 4 6 8 10 12 S 21 (dB) H (Oe) T=300 K f (GHz) 0 4 8 12 16 0 10 20 30 H (Oe) f (GHz) T=300 K 0 4 8 12 16 0 10 20 30 H (Oe) f (GHz) T=5 K b c a Figure 10 50 100 150 200 250 300 0.006 0.008 0.010 0.012 0.014 d (nm) 3 15 5 20 10 30 50 a Temperature (K) 0.0 0.1 0.2 0.3 0.004 0.006 0.008 0.010 0.012 0.014 a 1/ d (nm -1 ) 0 10 20 30 0.006 0.008 0.010 0.012 0.014 d (nm) a a b c Figure 20 50 100 150 200 250 300 0.0040 0.0045 0.0050 0.0055 0.0060 Theory Ref. 21, 22 Ref. 23 Temperature (K) a B a a Exp. 0.006 Ref. 24 0.004 - 0.008 Ref. 25 0.007 Ref. 26 0.0067 Ref. 27 0 50 100 150 200 250 300 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 Temperature (K) a S (nm) b Figure 30 50 100 150 200 250 300 0.004 0.006 0.008 0.010 5 nm Py/TaN 5 nm Py/Al 2 O 3 Temperature (K) a 0 50 100 150 200 250 300 0.004 0.005 0.006 0.007 30 nm Py/TaN 30 nm Py/Al 2 O 3 Temperature (K) a a b Figure 4a b 0 500 1000 1500 2000 0 4 8 12 16 f (GHz) H (Oe) T=300 K 0 500 1000 1500 2000 0 4 8 12 16 f (GHz) H (Oe) T=5 K Figure 58.6 8.8 9.0 9.2 9.4 9.6 4 M eff (kG) 5 nm Py/TaN 5 nm Py/Al 2 O 3 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 4 M eff (kG) 3 nm Py/TaN 0 50 100 150 10.6 10.7 10.8 10.9 11.0 30 nm Py/TaN 30 nm Py/Al 2 O 3 4 M eff (kG) a b c Temperature (K) Figure 6

1704.01559v1.Relativistic_theory_of_magnetic_inertia_in_ultrafast_spin_dynamics.pdf

Relativistic theory of magnetic inertia in ultrafast spin dynamics Ritwik Mondal,Marco Berritta, Ashis K. Nandy, and Peter M. Oppeneer Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden (Dated: November 12, 2018) The influence of possible magnetic inertia effects has recently drawn attention in ultrafast mag- netization dynamics and switching. Here we derive rigorously a description of inertia in the Landau-Lifshitz-Gilbert equation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy- Wouthuysen transformation up to the order of 1=c4gives the intrinsic inertia of a pure system through the 2ndorder time-derivative of magnetization in the dynamical equation of motion. Thus, the inertial damping Iis a higher order spin-orbit coupling effect, 1=c4, as compared to the Gilbert damping that is of order 1=c2. Inertia is therefore expected to play a role only on ultra- short timescales (sub-picoseconds). We also show that the Gilbert damping and inertial damping are related to one another through the imaginary and real parts of the magnetic susceptibility tensor respectively. PACS numbers: 71.15.Rf, 75.78.-n, 75.40.Gb I. INTRODUCTION The foundation of contemporary magnetization dy- namics is the Landau-Lifshitz-Gilbert (LLG) equation which describes the precession of spin moment and a transverse damping of it, while keeping the modulus of magnetization vector fixed [1–3]. The LLG equation of motion was originally derived phenomenologically and the damping of spin motion has been attributed to rela- tivistic effects such as the spin-orbit interaction [1, 4–6]. In recent years there has been a flood of proposals for the fundamental microscopic mechanism behind the Gilbert damping: the breathing Fermi surface model of Kamber- ský, where the damping is due to magnetization preces- sion and the effect of spin-orbit interaction at the Fermi surface [4], the extension of the breathing Fermi surface model to the torque-torque correlation model [5, 7], scat- tering theory description [8], effective field theories [9], linear response formalism within relativistic electronic structure theory [10], and the Dirac Hamiltonian theory formulation [11]. For practical reasons it was needed to extend the orig- inal LLG equation to include several other mechanisms [12, 13]. To describe e.g. current induced spin-transfer torques, the effects of spin currents have been taken into account [14–16], as well as spin-orbit torques [17], and the effect of spin diffusion [18]. A different kind of spin relaxation due to the exchange field has been intro- duced by Bar’yakhtar et al.[19]. In the Landau-Lifshitz- Bar’yakhtar equation spin dissipations originate from the spatial dispersion of exchange effects through the second order space derivative of the effective field [20, 21]. A further recent work predicts the existence of extension terms that contain spatial as well temporal derivatives of the local magnetization [22]. Another term, not discussed in the above investiga- tions, is the magnetic inertial damping that has recently ritwik.mondal@physics.uu.sedrawnattention[23–25]. Originally, magneticinertiawas discussed following the discovery of earth’s magnetism [26]. Within the LLG framework, inertia is introduced as an additional term [24, 27–29] leading to a modified LLG equation, @M @t= MHe+M @M @t+I@2M @2t ;(1) where is the Gilbert damping constant [1–3], the gy- romagnetic ratio, Hethe effective magnetic field, and I is the inertia of the magnetization dynamics, similar to the mass in Newton’s equation. This type of motion has the same classical analogue as the nutation of a spinning symmetric top. The potential importance of inertia is il- lustrated in Fig. 1. While Gilbert damping slowly aligns the precessing magnetization to the effective magnetic field, inertial dynamics causes a trembling or nutation of the magnetization vector [24, 30, 31]. Nutation could consequently pull the magnetization toward the equa- tor and cause its switching to the antiparallel direction [32, 33], whilst depending crucially on the strength of the magnetic inertia. The parameter Ithat character- izes the nutation motion is in the most general case a tensor and has been associated with the magnetic suscep- tibility [29, 31, 33]. Along a different line of reasoning, Fähnle et al.extended the breathing Fermi surface model to include the effect of magnetic inertia [27, 34]. The technological importance of nutation dynamics is thus its potential to steer magnetization switching in memory devices [23–25, 32] and also in skyrmionic spin textures [35]. Magnetization dynamics involving inertial dynam- ics has been investigated recently and it was suggested that its dynamics belongs to smaller time-scales i.e., the femtosecond regime [24]. However, the origin of inertial damping from a fundamental framework is still missing, and, moreover, although it is possible to vary the size of the inertia in spin-dynamics simulations, it is unknown what the typical size of the inertial damping is. Naturally the question arises whether it is possible to derive the extended LLG equation including iner-arXiv:1704.01559v1 [cond-mat.other] 20 Mar 20172 MHeff Precession Nutation Figure 1. (Color online) Schematic illustration of magnetiza- tion dynamics. The precessional motion of Maround Heis depicted by the blue solid-dashed curve and the nutation is shown by the red curve. tia while starting from the fully relativistic Dirac equa- tion. Hickey and Moodera [36] started from a Dirac Hamiltonian and obtained an intrinsic Gilbert damping term which originated from spin-orbit coupling. How- ever they started from only a part of the spin-orbit cou- pling Hamiltonian which was anti-hermitian [37, 38]. A recent derivation based on Dirac Hamiltonian theory for- mulation [11] showed that the Gilbert damping depends strongly on both interband and intraband transitions (consistent with Ref. [39]) as well as the magnetic sus- ceptibility response function, m. This derivation used the relativistic expansion to the lowest order 1=c2of the hermitian Dirac-Kohn-Sham (DKS) Hamiltonian includ- ing the effect of exchange field [40]. In this article we follow an approach similar to that of Ref. [11] but we consider higher order expansion terms of the DKS Hamiltonian up to the order of 1=c4. This is shown to lead to the intrinsic inertia term in the modi- fied LLG equation and demonstrates that it stems from a higher-order spin-orbit coupling term. A relativistic origin of the spin nutation angle, caused by Rashba-like spin-orbit coupling, was previously concluded, too, in the context of semiconductor nanostructures [41, 42]. In the following, we derive in Sec. II the relativistic correction terms to the extended Pauli Hamiltonian up to the order of 1=c4, which includes the spin-orbit inter- action and an additional term. Then the corresponding magnetization dynamics is computed from the obtained spin Hamiltonian in Sec. III, which is shown to contain the Gilbert damping and the magnetic inertial damping. Finally, we discuss the size of the magnetic inertia in re- lation to other earlier studies.II. RELATIVISTIC HAMILTONIAN FORMULATION We start our derivation with a fully relativistic par- ticle, a Dirac particle [43] inside a material and in the presence of an external field, for which we write the DKS Hamiltonian: H=c
(peA) + ( 1)mc2+V 1 =O+ ( 1)mc2+E; (2) whereVis the effective crystal potential created by the ion-ion, ion-electron and electron-electron interactions, A(r;t)is the magnetic vector potential from the external field,cis the speed of light, mis particle’s mass and 1 is the 44unit matrix. andare the Dirac matrices that have the form = 0 0 ;  = 10 01 ; whereis the Pauli spin matrix vector and 1is22unit matrix. TheDiracequationisthenwrittenas i~@ (r;t) @t= H (r;t)for a Dirac bi-spinor . The quantityO=c
(peA)defines the off-diagonal, or odd terms in the matrix formalism and E=V 1are the diagonal, i.e., even terms. The latter have to be multiplied by a 22block diagonal unit matrix in order to bring them in a matrix form. To obtain the nonrelativistic Hamiltonian and the relativistic corrections one can write down the Dirac bi- spinor in double two component form as (r;t) = (r;t) (r;t) ; and substitute those into the Dirac equation. The up- per two components represent the particle and the lower two components represent the anti-particle. However the question of separating the particle’s and anti-particle’s wave functions is not clear for any given momentum. As the part
pis off-diagonal in the matrix formalism, it retains the odd components and thus links the particle- antiparticle wave function. One way to eliminate the an- tiparticle’s wave function is by an exact transformation [44] which gives terms that require a further expansion in powers of 1=c2. Another way is to search for a represen- tation where the odd terms become smaller and smaller and one can ignore those with respect to the even terms and retain only the latter [45]. The Foldy-Wouthuysen (FW) transformation [46, 47] was the very successful at- tempt to find such a representation. It is an unitary transformation obtained by suitably choosing the FW operator, UFW=i 2mc2O: (3) The minus sign in front of the operator is because and Oanti-commute with each other. The transformation of thewavefunctionadoptstheform 0(r;t) =eiUFW (r;t)3 such that the probability density remains the same, j j2=j 0j2. The time-dependent FW transformation can be expressed as [45, 48] HFW=eiUFW Hi~@ @t eiUFW+i~@ @t:(4) The first term can be expanded in a series as eiUFWHeiUFW=H+i[UFW;H] +i2 2![UFW;[UFW;H]] +::::+in n![UFW;[UFW;:::[UFW;H]:::]] +::: :(5) The time dependency enters through the second term of Eq. (4) and for a time-independent transformation one works with@UFW @t= 0. It is instructive to note that the aim of the whole procedure is to make the odd termssmaller and one can notice that as it goes higher and higher in the expansion, the corresponding coefficients decrease of the order 1=c2due to the choice of the unitary operator. After a first transformation, the new Hamilto- nian will contain new even terms, E0, as well as new odd terms,O0of1=c2or higher. The latter terms can be used to perform a next transformation having the new unitary operator as U0 FW=i 2mc2O0. After a second transfor- mation the new Hamiltonian, H0 FWis achieved that has the odd terms of the order 1=c4or higher. The trans- formation is a repetitive process and it continues until the separation of positive and negative energy states are guaranteed. After a fourth transformation we derive the new trans- formed Hamiltonian with all the even terms that are cor- rect up to the order of1 m3c6as [48–50] H000 FW= ( 1)mc2+O2 2mc2O4 8m3c6 +E1 8m2c4h O;[O;E] +i~_Oi + 16m3c6fO;[[O;E];E]g+ 8m3c6n O;h i~_O;Eio + 16m3c6n O;(i~)2Oo : (6) Note that [A;B]defines the commutator, while fA;Bgrepresents the anti-commutator for any two operators Aand B. A similar Foldy-Wouthuysen transformation Hamiltonian up to an order of 1=m3c6was derived by Hinschberger and Hervieux in their recent work [51], however there are some differences, for example, the first and second terms in the second line of Eq. (6) were not given. Once we have the transformed Hamiltonian as a function of odd and even terms, the final form is achieved by substituting the correct form of odd terms Oand calculating term by term. Evaluating all the terms separately, we derive the Hamiltonian for only the positive energy solutions i.e. the upper components of the Dirac bi-spinor as a 22matrix formalism [40, 51, 52]: H000 FW=(peA)2 2m+Ve~ 2m
B(peA)4 8m3c2e~2 8m2c2r
Etot+e~ 8m3c2n (peA)2;
Bo e~ 8m2c2
[Etot(peA)(peA)Etot] e~2 16m3c4f(peA);@tEtotgie~2 16m3c4
[@tEtot(peA) + (peA)@tEtot]; (7) where@t@=@tdefines the first-order time derivative. Thehigherorderterms( 1=c6ormore)willinvolvesimilar formulations and more and more time derivatives of the magnetic and electric fields will appear that stem from the time derivative of the odd operator O[48, 51]. The fields in the last Hamiltonian (7) are defined as B=rA, the external magnetic field, Etot=Eint+ Eextare the electric fields where Eint=1 erVis the internal field that exists even without any perturbation andEext=@A @tis the external field (only the temporal part is retained here because of the Coulomb gauge). The spin Hamiltonian The aim of this work is to formulate the magnetiza- tion dynamics on the basis of this Hamiltonian. Thus,we split the Hamiltonian into spin-independent and spin- dependentpartsandconsiderfromnowonelectrons. The spin Hamiltonian is straightforwardly given as HS(t) =e mS
B+e 4m3c2n (peA)2;S
Bo e 4m2c2S
[Etot(peA)(peA)Etot] ie~ 8m3c4S
[@tEtot(peA) + (peA)@tEtot]; (8) where the spin operator S= (~=2)has been used. Let us briefly explain the physical meaning behind each term that appears inHS(t). The first term defines the Zee- man coupling of the electron’s spin with the externally applied magnetic field. The second term defines an indi- rect coupling of light to the Zeeman interaction of spin and the optical B-field, which can be be shown to have4 the form of a relativistic Zeeman-like term. The third term implies a general form of the spin-orbit coupling that is gauge invariant [53], and it includes the effect of the electric field from an internal as well as an external field. Thelasttermisthenewtermofrelevanceherethat has only been considered once in the literature by Hin- schberger et al.[51]. Note that, although the last term in Eq. (8) contains the total electric field, only the time- derivative of the external field plays a role here, because the time derivative of internal field is zero as the ionic potential is time independent. In general if one assumes a plane-wave solution of the electric field in Maxwell’s equation asE=E0ei!t, the last term can be written as e~! 8m3c4S
(Ep)and thus adopts the form of a higher- order spin-orbit coupling for a general E-field. The spin-dependent part can be easily rewritten in a shorter format using the identities: A(peA)(peA)A= 2A(peA) +i~rA (9) A(peA) + (peA)A=i~rA(10) for any operator A. This allows us to write the spin Hamiltonian as HS=e mS
B+e 2m3c2S
B p22eA
p+3e2 2A2 e 2m2c2S
 Etot(peA) +ie~ 4m2c2S
@tB +e~2 8m3c4S
@ttB: (11) Here, the Maxwell’s equations have been used to derive the final form that the spatial derivative of the electric field will generate a time derivative of a magnetic field such that rEext=@B @t, whilst the curl of a internal field results in zero as the curl of a gradient function is always zero. The final spin Hamiltonian (11) bears much importance for the strong laser field-matter interaction as it takes into account all the field-spin coupling terms. It is thus the appropriate fundamental Hamiltonian to understand the effects of those interactions on the mag- netization dynamics described in the next section. III. MAGNETIZATION DYNAMICS In general, magnetization is given by the magnetic mo- ment per unit volume in a magnetic solid. The magnetic momentisgivenby gBhSi,wheregistheLandég-factor andBis the unit of Bohr magneton. The magnetization is then written M(r;t) =X jgB hSji; (12) where is the suitably chosen volume element, the sum jgoes over all electrons in the volume element, and h::i is the expectation value. To derive the dynamics, wetake the time derivative in both the sides of Eq. (12) and, withintheadiabaticapproximation, wearriveatthe equation of motion for the magnetization as [36, 54, 55] @M @t=X jgB 1 i~h Sj;HS(t) i:(13) Now the task looks simple, one needs to substitute the spin Hamiltonian (11) and calculate the commutators in order to find the equation of motion. Note that the dy- namics only considers the local dynamics as we have not taken into account the time derivative of particle density operator (for details, see [11]). Incorporating the latter would give rise the local as well as non-local processes (i.e., spin currents) within the same footing. The first term in the spin Hamiltonian produces the dynamics as @M(1) @t= MB; (14) with =gjej=2mdefines the gyromagnetic ratio and the Landé g-factor g2for spins, the electronic charge e < 0. Using the linear relationship of magnetization with the magnetic field B=0(H+M), the latter is replaced in Eq. (14) to get the usual form in the Landau- Lifshitz equations, 0MH, where 0=0 is the effective gyromagnetic ratio. This gives the Larmor pre- cessionofmagnetizationaroundaneffectivefield H. The effective field will always have a contribution from a ex- change field and the relativistic corrections to it, which has not been explicitly taken into account in this article, as they are not in the focus here. For detailed calcula- tions yet including the exchange field see Ref. [11]. The second term in the spin Hamiltonian Eq. (11) will result in a relativistic correction to the magnetization precession. Within an uniform field approximation (A= Br=2), the corresponding dynamics will take the form @M(2) @t= 2m2c2MBD p2eB
L+3e2 8(Br)2E ; (15) withL=rpthe orbital angular momentum. The presence of =2m2c2implies that the contribution of this dynamics to the precession is relatively small, while the leading precession dynamics is given by Eq. (14). For sake of completeness we note that a relativistic correc- tion to the precession term of similar order 1=m2c2was obtained previously for the exchange field [11]. The next term in the Hamiltonian is a bit tricky to handle as the third term in Eq. (11) is not hermitian, not even the fourth term which is anti-hermitian. However together they form a hermitian Hamiltonian [11, 37, 38]. Therefore one has to work together with those terms and cannot only perform the dynamics with an individual term. In an earlier work [11] we have shown that taking anuniformmagneticfieldalongwiththegauge A=Br 2 will preserve the hermiticity. The essence of the uniform field lies in the assumption that the skin depth of the5 electromagnetic field is longer than the thickness of the thin-film samples used in experiments. The dynamical equation of spin motion with the second and third terms thus thus be written in a compact form for harmonic ap- plied fields as [11] @M(3;4) @t=M A
@M @t ; (16) with the intrinsic Gilbert damping parameter Athat is a tensor defined by Aij= 0 4mc2X n;kh hripk+pkriihrnpn+pnrniiki  1+1 m kj:(17) Heremis the magnetic susceptibility tensor of rank 2 (a33matrix) and 1is the 33unit matrix. Note that for diagonal terms i.e., i=kthe contributions from the expectation values of rkpicancel each other. The damp- ing tensor can be decomposed to have contributions from an isotropic Heisenberg-like, anisotropic Ising-like and Dzyaloshinskii-Moriya-like tensors. The anti-symmetric Dzyaloshinskii-Moriya contribution has been shown to lead to a chiral damping of the form M(D@M=@t) [11]. Experimental observations of chiral damping have been reported recently [56]. The other cross term having the formEAin Eq. (11) is related to the angular mo- mentum of the electromagnetic field and thus provides a torque on the spin that has been at the heart of an- gular magneto-electric coupling [53]. A possible effect in spin dynamics including the light’s angular momentum has been investigated in the strong field regime and it has been shown that one has to include this cross term in the dynamics in order to explain the qualitative and quantitative strong field dynamics [57]. For the last term in the spin Hamiltonian (11) it is rather easy to formulate the spin dynamics because it is evidently hermitian. Working out the commutator with the spins gives a contribution to the dynamics as @M(5) @t=M@2B @t2; (18) with the constant = ~2 8m2c4. Let us work explicitly with the second-order time derivative of the magnetic induction by the relation B= 0(H+M), using a chain rule for the derivative: @2B @t2=@ @t@B @t =0@ @t@H @t+@M @t =0@2H @t2+@2M @t2 : (19) This is a generalized equation for the time-derivative of the magnetic induction which can be used even for non- harmonic fields. The magnetization dynamics is then given by @M(5) @t=0M@2H @t2+@2M @t2 :(20)Thus the extended LLG equation of motion will have these two additional terms: (1) a field-derivative torque and (2) magnetization-derivative torque, and they ap- pear with their 2ndorder time derivative. It deserves to be noted that, in a previous theory we also obtained a similar term–a field-derivative torque in 1storder-time derivative appearing in the generalized Gilbert damping. Specifically, the extended LLG equation for a general time-dependent field H(t)becomes @M @t= 0MH+Mh A
@H @t+@M @ti +0M@2H @t2+@2M @t2 ; (21) where AisamodifiedGilbertdampingtensor(fordetails, see [11]). However for harmonic fields, the response of the ferro- magnetic materials is measured through the differential susceptibility, m=@M=@H, because there exists a net magnetization even in the absence of any applied field. With this, the time derivative of the harmonic magnetic field can be further written as: @2H @t2=@ @t@H @M@M @t =@ @t 1 m
@M @t =@1 m @t
@M @t+1 m
@2M @t2: (22) In general the magnetic susceptibility is a spin-spin re- sponse function that is wave-vector and frequency depen- dent. Thus, Eq. (18) assumes the form with the first and second order time derivatives as @M(5) @t=M K
@M @t+I
@2M @t2 ;(23) where the parameters Iij=0 1+1 m
ijandKij= 0@t(1 m)ijare tensors. The dynamics of the second term is that of the magnetic inertia that operates on shorter time scales [25]. Having all the required dynamical terms, finally the full magnetization dynamics can be written by joining together all the individual parts. Thus the full magneti- zation dynamics becomes, for harmonic fields, @M @t=M 0H+
@M @t+I
@2M @t2 :(24) Note that the Gilbert damping parameter has two con- tributions, one from the susceptibility itself, Aij, which is of order 1=c2and an other from the time derivative of it,Kijof order 1=c4. Thus, ij=Aij+Kij. However we will focus on the first one only as it will obviously be the dominant contribution, i.e., ijAij. Even though we consider only the Gilbert damping term of order 1=c2in the discussions, we shall explicitly analyze the other term of the order 1=c4. For an ac susceptibility i.e., 1 m/ei!t we find thatKij/0@t(1 m)ij/i0!1 m, which suggests again that the Gilbert damping parameter of6 the order 1=c4will be given by the imaginary part of the susceptibility,Kij/0!Im 1 m
. The last equation (24) is the central result of this work, as it establishes a rigorous expression for the in- trinsic magnetic inertia. Magnetization dynamics in- cluding inertia has been discussed in few earlier articles [24, 30, 31, 58]. The very last term in Eq. (24) has been associated previously with the inertia magnetization dy- namics [32, 59, 60]. As mentioned, it implies a magne- tization nutation i.e., a changing of the precession angle as time progresses. Without the inertia term we obtain the well-known LLG equation of motion that has already been used extensively in magnetization dynamics simu- lations (see, e.g., [61–65]). IV. DISCUSSIONS Magnetic inertia was discussed first in relation to the earth’s magnetism [26]. From a dimensional analysis, the magnetic inertia of a uniformly magnetized sphere undergoing uniform acceleration was estimated to be of the order of 1=c2[26], which is consistent with the here- obtained relativistic nature of magnetic inertia. Our derivation based on the fundamental Dirac-Kohn- Sham Hamiltonian provides explicit expressions for both the Gilbert and inertial dampings. Thus, a comparison can be made between the Gilbert damping parameter and the magnetic inertia parameter of a pure system. As noticed above, both the parameters are given by the magnetization susceptibility tensor, however it should be noted that the quantiy hrpiis imaginary itself, because [11], hrpi=i~ 2mX n;n0;kf(Enk)f(En0k) EnkEn0kp nn0p n0n:(25) Thus the Gilbert damping parameter should be given by the imaginary part of the susceptibility tensor [36, 66]. On the other hand the magnetic inertia tensor must be given by the real part of the susceptibility [31]. This is in agreement with a recent article where the authors also found the same dependence of real and imaginary parts of susceptibility to the nutation and Gilbert damping re- spectively [33]. In our calculation, the Gilbert damping and inertia parameters adopt the following forms respec-tively, ij=i 0 4mc2X n;k[hripk+pkriihrnpn+pnrniik] Im 1 m
kj =0 ~ 4mc2X n;khripk+pkriihrnpn+pnrniik i~ Im 1 m
kj =X n;khripk+pkriihrnpn+pnrniik i~ Im 1 m
kj;(26) Iij=0 ~2 8m2c4h 1+Re 1 m
kji =~ 2mc2h 1+Re 1 m
kji ; (27) with0 ~ 4mc2. Note that the change of sign from damp- ingtensortotheinertiatensorthatisalsoconsistentwith Ref. [33], and also a factor of 2 present in inertia. How- ever, most importantly, the inertia tensor is ~=mc2times smallerthan the damping tensor as is revealed in our calculations. Considering atomic units we can evaluate 0 4c20:00066 413728:8109; ~ 2mc2 2c28:8109 213722:341013: This implies that the intrinsic inertial damping is typi- cally 4104times smaller than the Gilbert damping and it is not an independently variable parameter. Also, be- cause of its smallness magnetic inertial dynamics will be more significant on shorter timescales [24]. A further analysis of the two parameters can be made. OnecanusetheKramers-Kronigtransformationtorelate the real and imaginary parts of a susceptibility tensor with one another. This suggests a relation between the two parameters that has been found by Fähnle et al.[34], namelyI=,whereisarelaxationtime. Weobtain here a similar relation, I/ , where =~=mc2has time dimension. Even though the Gilbert damping is c2times larger than the inertial damping, the relative strength of the two parameters also depends on the real and imaginary parts of the susceptibility tensor. In special cases, when the real part of the susceptibility is much higher than the imaginary part, their strength could be comparable to each other. We note in this context that there exist materials where the real part of the susceptibility is 102 103times larger than the imaginary part. Finally, we emphasize that our derivation provides the intrinsic inertial damping of a pure, isolated system. For the Gilbert damping it is already well known that en- vironmental effects, such as interfaces or grain bound- aries, impurities, film thickness, and even interactions of7 the spins with quasi-particles, for example, phonons, can modify the extrinsic damping (see, e.g., [67–69]). Simi- larly, it can be expected that the inertial damping will become modified through environmental influences. An example of environmental effects that can lead to mag- netic inertia have been considered previously, for the case of a local spin moment surrounded by conduction elec- trons, whose spins couple to the local spin moment and affect its dynamics [31, 32]. V. CONCLUSIONS In conclusion, we have rigorously derived the magne- tization dynamics from the fundamental Dirac Hamilto- nian and have provided a solid theoretical framework for, and established the origin of, magnetic inertia in pure systems. We have derived expressions for the Gilbert damping and the magnetic inertial damping on the same footing and have shown that both of them have a rela- tivistic origin. The Gilbert damping stems from a gen- eralized spin-orbit interaction involving external fields, whiletheinertialdampingisduetohigher-order(in 1=c2) spin-orbit contributions in the external fields. Both have been shown to be tensorial quantities. For general time dependent external fields, a field-derivative torque with a 1storder time derivative appears in the Gilbert-type damping, and a 2ndorder time-derivative field torque ap- pears in the inertial damping. In the case of harmonic external fields, the expressions of the magnetic inertia and the Gilbert damping scalewith the real part and the imaginary part, respectively, of the magnetic susceptibility tensor, and they are op- posite in sign. Alike the Gilbert damping, the magnetic inertia tensor is also temperature dependent through the magnetic response function and also magnetic moment dependent. Importantly, we find that the intrinsic iner- tial damping is much smaller than the Gilbert damping, which corroborates the fact that magnetic inertia was neglected in the early work on magnetization dynamics [1–3, 19]. 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1012.5473v1.Screw_pitch_effect_and_velocity_oscillation_of_domain_wall_in_ferromagnetic_nanowire_driven_by_spin_polarized_current.pdf

arXiv:1012.5473v1 [cond-mat.other] 25 Dec 2010Screw-pitch effect and velocity oscillation of domain-wall in ferromagnetic nanowire driven by spin-polarized current Zai-Dong Li1,2,3, Qiu-Yan Li1, X. R. Wang3, W. M. Liu4, J. Q. Liang5, and Guangsheng Fu2 1Department of Applied Physics, Hebei University of Technol ogy, Tianjin 300401, China 2School of Information Engineering, Hebei University of Tec hnology, Tianjin, 300401, China 3Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China. 4Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China 5Institute of Theoretical Physics and Department of Physics , Shanxi University, Taiyuan 030006, China We investigate the dynamics of domain wall in ferromagnetic nanowire with spin-transfer torque. The critical current condition is obtained analytically. B elow the critical current, we get the static domain wall solution which shows that the spin-polarized cu rrent can’t drive domain wall moving continuously. In this case, the spin-transfer torque plays both the anti-precession and anti-damping roles, which counteracts not only the spin-precession driv en by the effective field but also Gilbert damping to the moment. Above the critical value, the dynamic s of domain wall exhibits the novel screw-pitch effect characterized by the temporal oscillati on of domain wall velocity and width, respectively. Both the theoretical analysis and numerical simulation demonstrate that this novel phenomenon arise from the conjunctive action of Gilbert-da mping and spin-transfer torque. We also find that the roles of spin-transfer torque are entirely contrary for the cases of below and above the critical current. PACS numbers: 75.75.+a, 75.60.Ch, 75.40.Gb Keywords: Screw-Pitch effect, Velocity Oscillation of Doma in-Wall, Spin-Polarized Current. A magnetic domain wall (DW) is a spatially local- ized configuration of magnetization in ferromagnet, in which the direction of magnetic moments inverses gradu- ally. Whenaspin-polarizedelectriccurrentflowsthrough DW, the spin-polarization of conduction electrons can transfer spin momentum to the local magnetization, thereby applying spin-transfer torque which can manip- ulate the magnetic DW without the applied magnetic field. This spin-transfer effect was theoretically proposed by Slonczewski [1] and Berger [2], and subsequently veri- fied experimentally [3]. As a theoretical model the mod- ified Landau-Lifshitz-Gilbert (LLG) equation [4–6] with spin-transfertorquewasderivedto describesuchcurrent- induced magnetization dynamics in a fully polarized fer- romagnet. With these novel forms of spin torque many interesting phenomena have been studied, such as spin wave excitation [4, 7, 8] and instability [4, 9], magneti- zation switching and reversal [10–13], and magnetic soli- tons [14, 15]. For the smooth DW, this spin torque can displace DW opposite to the current direction which has beenconfirmedexperimentallyinmagneticthinfilmsand magnetic wires [16–20]. With the remarkable experimental success measur- ing the motion of DW under the influence of current pulse,considerableprogresshasbeenmadetounderstand the current-induced DW motion in magnetic nanowire [5, 6, 17–20]. These studies haveimproved the pioneering work of current-driven DW motion by Berger [21]. Al- though both the theory and the quasi-static experiments have indicated that the spin-polarized current can cause DW motion, the current-driven DW dynamics is not well understood. The dynamics of magnetization described by the LLG equation admits the static solutions for DWmotion. In the presence of spin torque and the exter- nal magnetic field, it is difficult to derive the dynamic solutions. A circumvented approach is Walker solution analysis [22] for the moving DW in response to a steady magnetic field smaller than some critical value. How- ever, this approximation applying to DW motion driven by the electric current is unclear, and its reliability has to be verified theoretically and numerically. Inthispaper,wereportanalyticallythecriticalcurrent condition for anisotropic ferromagnetic nanowire driven only by spin-transfer torque. Below the critical cur- rent, the ferromagnetic nanowire admits only the final static DW solution which implies that the spin-polarized current can’t drive DW moving continuously. When the spin-polarized current exceeds the critical value, the dynamics of DW exhibits the novel Screw-pitch effect with the periodic temporal oscillation of DW velocity and width. A detail theoretical analysis and numerical sim- ulation demonstrate that this novel phenomenon arises from the natural conjunction action of Gilbert-damping and spin-transfer torque. We also observe that the spin- transfer torque plays the entirely opposite roles in the above two cases. At last, our theoretical prediction can be confirmed by the numerical simulation in terms of RKMK method [23]. We consider an infinite long uniaxial anisotropic fer- romagnetic nanowire, where the electronic current flows along the long length of the wire defined as xdirection whichis alsothe easyaxisofanisotropyferromagnet. For convenience the magnetization is assumed to be nonuni- form only in the direction of current. Since the magneti- zation varies slowly in space, it is reasonable to take the adiabatic limit. Then the dynamics of the localized mag-2 netization can be described by the modified LLG equa- tion with spin-transfer torque ∂M ∂t=−γM×Heff+α MsM×∂M ∂t+bJ∂M ∂x,(1) whereM≡M(x,t) is the localized magnetization, γ is the gyromagnetic ratio, αis the damping parameter, andHeffrepresents the effective magnetic field. The last term of Eq. (1) denotes the spin-transfer torque, wherebJ=PjeµB/(eMs),Pis the spin polarization of the current, jeis the electric current density and flows along the xdirection, µBis the Bohr magneton, eis the magnitude of electron charge, and Msis the sat- uration magnetization. For the uniaxial ferromagnetic nanowire the effective field can be written as Heff=/parenleftbig 2A/M2 s/parenrightbig ∂2M/∂x2+HxMx/Msex−4πMzez, whereAis the exchange constant, Hxis the anisotropy field, and ei, i=x,y,z,is the unit vector, respectively. Introducing the normalized magnetization, i.e., m=M/Ms, Eq. (1) can be simplified as the dimensionless form α1∂m ∂t=−m×heff−αm×(m×heff) +αb1m×∂m ∂x+b1∂m ∂x, (2) whereα1=/parenleftbig 1+α2/parenrightbig andb1=bJt0/l0. The time tand space coordinate xhave been rescaled by the characteristic time t0= 1/(16πγMs) and length l0=/radicalbig A/(8πM2s), respectively. The dimensionless effective field becomes heff=∂2m/∂x2+C1mxex−C2mzez, with C1=Hx/(16πMs) andC2= 0.25. In the following, we seek for the exact DW solutions of Eq. (2), and then study the dynamics of magnetization driven by spin-transfer torque. To this purpose we make the ansatz mx= tanhΘ 1,my=sinφ coshΘ 1,mz=cosφ coshΘ 1,(3) where Θ 1=k1x−ω1t, with the temporal and spatial independent parameters φ,k1, andω1to be determined, respectively. Substituting Eq. (3) into Eq. (2) we have k2 1=C1+C2cos2φ, (4) −ω1/parenleftbig 1+α2/parenrightbig =b1k1+C2sinφcosφ,(5) αb1k1cosφ=α/parenleftbig C1−k2 1/parenrightbig sinφ, (6) αb1k1sinφ=−αC2sin2φcosφ. (7) From the above equations we can get three cases of DW solutions for Eq. (2). Firstly, in the absence of damping Eqs. (4) to (7) admit the solution k1=±/radicalbig C1+C2cos2φ,ω1=−b1k1−C2 2sin2φ,(8) with the arbitrary angle φ. This solution show that the spin-transfer torque contributes a dimensionless veloc- ity−b1only without damping. The velocity of DW isformed by two parts, i.e., v=−(C2sin2φ)/(2k1)−b1, which can be affected by adjusting the angle φand the spin-transfer torque. Secondly, in the absence of spin torque, we have the solution of Eqs. (4) to (7) as ω1= 0, φ=±π/2,k1=±√C1, i.e., the static DW solution. In terms of RKMK method [23] we perform direct numeri- cal simulation for Eq. (2) with various initial condition, and all numerical results show that the damping drives the change of φwhich in turn affects the DW velocity and width defined by 1 /|k1|. At last φ=±π/2,ω1= 0, i.e., the DW loses moving, and the DW width attains its maximum value√C1, which confirms the Walker’s analysis [22] that the damping prevents DW from mov- ing without the external magnetic field or spin-transfer torque. However, as shown later, the presence of damp- ing is prerequisite for the novel Screw-pitch property of DW driven by spin-transfer torque. At last, we consider the case of the presence of damping and spin-transfer torque. Solving Eqs. (4) to (7) we have k1=±1 2(B1−/radicalbig B2),ω1= 0,sin2φ=−2b1k1 C2,(9) whereB1= 2C1+C2−b2 1,B2=/parenleftbig C2−b2 1/parenrightbig2−4C1b2 1. It is clear that Eq. (9) implies the critical spin- polarized current condition, namely bJ≤(/radicalbig C1+C2−/radicalbig C1)l0/t0, which is determined by the character velocity l0/t0, the anisotropic parameter C1, and the demagnetiza- tion parameter C2. Below the critical current, i.e., b2 1≤(√C1+C2−√C1)2, the DW width falls into the range that 1 //radicalbig C2 1+C1C2≤1/|k1| ≤1/C1. From Eq. (9) we get four solutions of φ, i.e.,φ=±π/2 + 1/2arcsin(2 b1k1/C2) fork1>0 andφ=±π/2− 1/2arcsin(2 b1|k1|/C2) fork1<0. In fact, the signs “+” and “ −” in Eq. (9) denotes kink and anti-kink so- lution, respectively, and the corresponding solution in Eq. (3) represents the static tail-to-tail or head-to-head N´ eel DW, respectively. This result shows that below the critical current, the final equilibrium DW solution must be realized by the condition that m×heff=b1∂m/∂x. It clearly demonstrates that the spin-transfer torque has two interesting effects. One is that the term b1∂m/∂x in Eq. (2) plays the anti-precession role counteracting the precession driven by the effective field heff. How- ever, the third term in the right hand of Eq. (2), namely αb1m×∂m/∂x, has the anti-damping effect counteract- ing the damping term −αm×(m×heff). It is to say that below the critical value, the spin-polarized current can’t drive DW moving continuously without the applied external magnetic field. When the spin-polarized current exceed the critical value, the dynamics of DW possesses two novel prop- erties as shown in the following section. Above the critical current, the precession term −m×heffcan’t be counteracted by spin-transfer torque, and the static DW3 solution of Eq. (2) doesn’t exist. Because the mag- netization magnitude is constant, i.e., m2= 1, so we havem·∂m/∂x= 0 which shows that the direction of ∂m/∂xis always perpendicular to the direction of m, or ∂m/∂x= 0. It is well known that a magnetic DW sep- arates two opposite domains by minimizing the energy. In the magnetic DW the direction of magnetic moments gradually changes, i.e., ∂m/∂x/negationslash= 0,so the direction of ∂m/∂xshould adopt the former case. Out of region of DW the normalized magnetization will site at the easy axis, i.e., mx= 1(or−1), in which ∂m/∂x= 0. Withtheaboveconsiderationwemakeadetailanalysis for Eq. (2). As a characteristic view we mainly consider the DW center, defined by mx= 0. The magnetic mo- ment must be in the y-zplane, while the direction of ∂m/∂xshould lay in x-axis (+x-axis for k1>0, and −x-axis for k1<0). In order to satisfy Eq. (2) the mag- netic moment in DW center should include both the pre- cession around the effective spin-torque field αb1∂m/∂x and the tendency along the direction of ∂m/∂xcontinu- ouslyfromthelasttwotermsintherighthandofEq. (2). The formerprecessionmotion implies that the parameter φwill rotate around x-axis continuously, while the lat- ter tendency forces the DW center moving toward to the opposite direction of the current, i.e., −x-axis direction, confirming the experiment [16–20] in magnetic thin films and magnetic wires. Combining the above two effects we find that this rotating and moving phenomenon is very similar to Screw-pitch effect. The continuous rotation of magnetic moment in DW center, i.e., the periodic change ofφ, can result in the periodic oscillation of DW veloc- ity and width from Eq. (8) under the action of the first two terms in the right hand of Eq. (2). It is interesting to emphasize that when the current exceeds the critical value, the term αb1m×∂m/∂xplays the role to induce the precession, while the term b1∂m/∂xhas the effect of damping, which is even entirely contrary to the case below the critical current as mentioned before. Combin- ing the above discussion we conclude that the motion of magnetic moment in the DW center will not stop, except it falls into the easy axis, i.e., out of the range of DW. In fact, all the magnetic moments in DW can be analyzed in detail with the above similar procedure. Now it is clear for the dynamics of DW driven only by spin-transfer torque. Coming back to Eq. (2) we can see that this novel Screw-pitch effect with the periodicoscillation of DW velocity and width occurs even at the conjunct action of the damping and spin-transfer torque. To confirm our theoretical prediction we perform direct numerical simulation for Eq. (2) with an arbitrary initial condition by means of RKMK method [23] with the cur- rent exceeding the critical value. In figure 1(a) to 1(c) we plot the time-evolution of the normalized magnetiza- tionm, while the displacement of DW center is shown in figure 1(d). The result in figure 1 confirms entirely our theoretical analysis above. The evolution of cos φand the DW velocity and width are shown in figure 2. From figure 2 we can see that the periodic change of cos φleads to the periodic temporal oscillation of DW velocity and width. From Eq. (8) and the third term of Eq. (2) we can infer that cos φpossesses of the uneven change as shown in figure 2(a), i.e., the time corresponding to 0< φ+nπ≤π/2 is shorter than that corresponding to π/2< φ+nπ≤π,n= 1,2..., in each period, and the DW velocity has the same character. It leads to the DW displacement firstly increases rapidly, and then slowly as shown in figure 1(d). This phenomenon clarifies clearly the presence of Screw-pitch effect . The DW velocity os- cillation driven by the external magnetic field has been observed experimentally [24]. Our theoretical prediction for the range of DW velocity oscillation driven by the above critical current could be observed experimentally. In summary, the dynamics of DW in ferromagnetic nanowire driven only by spin-transfer torque is theoret- ically investigated. We obtain an analytical critical cur- rent condition, below which the spin-polarized current can’t drive DW moving continuously and the final DW solution is static. An external magnetic field should be applied in order to drive DW motion. We also find that the spin-transfer torque counteracts both the precession drivenbytheeffectivefieldandtheGilbertdampingterm different from the common understanding. When the spin current exceeds the critical value, the conjunctive action of Gilbert-damping and spin-transfer torque leads naturally the novel screw-pitch effect characterized by the temporal oscillation of DW velocity and width. This work was supported by the Hundred Innovation Talents Supporting Project of Hebei Province of China, the NSF of China under grants Nos 10874038, 10775091, 90406017, and 60525417, NKBRSFC under grant No 2006CB921400, and RGC/CERG grant No 603007. [1] Slonczewski J C, 1996 J. Magn. Magn. Mater. 159 L1 [2] Berger L, 1996 Phys. Rev. B 54 9353 [3] Katine J A, Albert F J, Buhrman R A Myers E B, and Ralph D C 2000 Phys. Rev. Lett. 84 3149 [4] Bazaliy Y B, Jones B A, and Zhang Shou-Cheng, 1998 Phys. Rev. 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Berakdar J, 2006 Phys. Rev. B 74 054403 [21] Berger L, 1978 J. Appl. Phys. 49 2156 Berger L, 1984 J. Appl. Phys. 55 1954 Berger L, 1992 J. Appl. Phys. 71 2721 Salhi E and Berger L, 1993 J. Appl. Phys. 73 6405 [22] Thiele A A, 1973 Phys. Rev. B 7 391 Schryer N L, and Walker L R, 1974 J. Appl. Phys. 45 5406 [23] Munthe-Kaas H, 1995 BIT. 35 572 Engo K, 2000 BIT. 40 41 [24] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine J L, 2005 Nat. Mater. 4 741 Yang J, Nistor C, Beach G S D, and Erskine J L, 2008 Phys. Rev. B 77 014413 Figure Captions Fig. 1. The dynamics of DW above the critical current. (a)-(c) Evolution of the normalized magneti- zationm. (d) The displacement of DW driven only by spin-transfer torque. The parameters are α= 0.2, C1= 0.05,C2= 0.25,bJ= 0.6, and the initial angle φ= 0.01π. Fig. 2. (a) The evolution of cos φand the periodic oscillation of DW velocity. (b) The periodic temporal oscillation of DW width. The parameters are same as in figure 1.0 20 40 60 80 100 120 140 160 180 200-1-0.8-0.6-0.4-0.200.20.40.60.81 TimeEvolution of cos( I) and velocity of DW center(a)Velocity of DW center cos(I) of DW center0 20 40 60 80 100 120 140 160 180 20012345 TimeEvolution of DW width(b)

2308.03353v1.__textit_In_situ___electric_field_control_of_ferromagnetic_resonance_in_the_low_loss_organic_based_ferrimagnet_V_TCNE____x_sim_2__.pdf

1 In situ electric-field control of ferromagnetic resonance in the low- loss organic-based ferrimagnet V[TCNE] x∼2 Seth W. Kurfman1, Andrew Franson1, Piyush Shah2, Yueguang Shi3, Hil Fung Harry Cheung4, Katherine E. Nygren5, Mitchell Swyt5, Kristen S. Buchanan5, Gregory D. Fuchs4, Michael E. Flatté3,6, Gopalan Srinivasan2, Michael Page2, and Ezekiel Johnston-Halperin†1 1Department of Physics, The Ohio State University 2Materials and Manufacturing Directorate, Air Force Research Laboratory 3Department of Physics and Astronomy, University of Iowa 4Department of Physics, Cornell University 5Department of Physics, Colorado State University 6Department of Applied Physics, Eindhoven University of Technology †Corresponding author email: johnston-halperin.1@osu.edu We demonstrate indirect electric-field control of ferromagnetic resonance (FMR) in devices that integrate the low-loss, molecule-based, room-temperature ferrimagnet vanadium tetracyanoethylene (V[TCNE] x∼2) mechanically coupled to PMN-PT piezoelectric transducers. Upon straining the V[TCNE] x films, the FMR frequency is tuned by more than 6 times the resonant linewidth with no change in Gilbert damping for samples with α = 6.5 × 10−5. We show this tuning effect is due to a strain-dependent magnetic anisotropy in the films and find the magnetoelastic coefficient | λs| ∼ (1 − 4.4) ppm, backed by theoretical predictions from DFT calculations and magnetoelastic theory. Noting the rapidly expanding application space for strain-tuned FMR, we define a new metric for magnetostrictive materials, magnetostrictive agility, given by the ratio of the magnetoelastic coefficient to the FMR linewidth. This agility allows for a direct comparison between magnetostrictive materials in terms of their comparative efficacy for magnetoelectric applications requiring ultra-low loss magnetic resonance modulated by strain. With this metric, we show V[TCNE] x is competitive with other magnetostrictive materials including YIG and Terfenol- D. This combination of ultra-narrow linewidth and magnetostriction in a system that can be directly integrated into functional devices without requiring heterogeneous integration in a thin- film geometry promises unprecedented functionality for electric-field tuned microwave devices ranging from low-power, compact filters and circulators to emerging applications in quantum information science and technology. Keywords : magnetostriction, magnonics, molecule-based magnets 2 Introduction The use of electric fields for control of magnetism has been a long-term goal of magnetoelectronics in its many manifestations ranging from metal and semiconductor spintronics [1, 2], to microwave electronics [3, 4], to emerging applications in quantum information [5]. This interest arises from the potential for clear improvements in scaling, high-speed control, and multifunctional integration. However, while the promise of this approach is well established its realization has proven challenging due to the strong materials constraints imposed by the existing library of magnetic materials [3, 4]. The most common approach to achieving this local control is through linking piezoelectricity with magnetostriction to achieve electric- field control of magnetic anisotropy, either intrinsically through inherent coupling in multiferroic materials or extrinsically through piezoelectric/magnetic heterostructures [3]. Ideally, magnetic materials chosen for such applications should exhibit large magnetostriction, low magnetic damping and narrow linewidth (high- Q) ferromagnetic resonance (FMR), and robust mechanical stability upon strain cycling [4]. While the use of multiferroic materials promises relative simplicity in device design, they typically suffer from poor magnetic properties and minimal tunability. The alternative approach of employing heterostructures of magnetic thin films and piezoelectric substrates effectively creates a synthetic multiferroic exploiting the converse magnetoelectric effect (CME) and in principle allows independent optimization of piezoelectric and magnetic properties [6, 7, 8, 9, 10, 11]. Further, for applications that rely on magnetic resonance ( e.g., microwave electronics and magnon-based quantum information systems), the traditional metrics of magnetostriction, λs, or the CME coefficient, A, do not capture the critical parameters governing damping and loss in this regime ( e.g., linewidth or Gilbert damping coefficient). To date, materials with large magnetoelastic constants and CME coefficients ( e.g., Terfenol-D) suffer from high damping, broad magnetic resonance features, and are particularly fragile and brittle [4, 12]. Ferrites such as yttrium iron garnet (YIG), on the other hand, are attractive due to their low- loss magnetic resonance properties but typically exhibit minimal to moderate magnetoelastic coefficients. Further, these low-loss ferrites require high growth temperatures (800-900 ◦C) and lattice-matched substrates to produce high-quality material, which makes integrating these materials on-chip while maintaining low-loss properties challenging [13, 14, 15], and limits their applicability for magnetic microelectronic integrated circuits (MMIC). Accordingly, alternative low-loss, magnetostrictive materials with facile integration capabilities are desired for applications in electrically-controlled devices. Recently, a complementary material to YIG, vanadium tetracyanoethylene (V[TCNE] x, x ~ 2), has gained significant interest from the spintronics and quantum information science and engineering (QISE) communities due to its ultra-low damping under magnetic resonance and benign deposition characteristics [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. 3 Here we present the first systematic experimental study of the magnetostrictive properties of V[TCNE] x. Composite heterostructures of V[TCNE] x films and piezoelectric substrates demonstrate shifts in the FMR frequency by 35 – 45.5 MHz, or more than 6 linewidths upon application of compressive strains up to 𝜀= −2.4×10ିସ . Further, a systematic analysis shows that the Gilbert damping, α, and inhomogeneous broadening linewidth, 0, are insensitive to strain in this regime and robust to repeated cycling. Density- functional theory (DFT) calculations provide insight into the elastic and magnetoelastic properties of V[TCNE] x and predict a magnetoelastic coefficient 𝜆ሚଵ = −2.52 ppm. Experimental measures of the effective magnetoelastic constant, λs, determined by combining optical measurements of the distortion with the corresponding FMR frequency shifts, yield values of λs = −(1 − 4 .4) ppm, which is in good agreement with these DFT predictions. Finally, we define a new figure of merit, magnetostrictive agility, ζ , as the ratio of the magnitude of the magnetoelastic coefficient to the FMR linewidth ζ = |λs|/Γ that more closely aligns with the performance requirements for emerging applications of magnetostrictive materials. These results establish a foundation for utilizing strain or acoustic (phononic) excitations for highly efficient strain- modulated magnetoelectronic devices based on V[TCNE] x and other next-generation organic-molecule- based magnetically-ordered materials for coherent information processing and straintronic applications [28]. Results V[TCNE] x is a room-temperature, organic-molecule-based ferrimagnet ( Tc ∼ 600 K) that exhibits superb low-damping properties ( α = (3.98 ± 0.22) × 10−5) and high-quality factor FMR ( Q = fR/Γ > 3,000) [17, 18, 19, 20, 21, 22, 23, 24, 25]. V[TCNE] x thin films are deposited via chemical vapor deposition (CVD) at relatively low temperature and high pressure (50 ◦C and 35 mTorr, respectively) and is largely insensitive to substrate lattice constant or surface termination [17, 20]. Further, V[TCNE] x can be patterned via e-beam lithography techniques without increase in its damping [19]. The highly coherent and ultra-low loss magnonic properties of V[TCNE] x have driven interest in applications in microwave electronics [26, 29] and magnon-based quantum information science and engineering (QISE) [21, 30, 31]. These benign deposition conditions, combined with patterning that does not degrade performance, highlight the versatility of V[TCNE] x for facile on-chip integration with pre-patterned microwave circuits and devices [26, 27, 29, 32, 33, 34]. These excellent magnetic properties are even more surprising given that V[TCNE] x lacks long- range structural order [25]. Early studies indicated that V[TCNE] x films do not exhibit magnetic anisotropy beyond shape effects due to this lack of long-range ordering [19, 20]. However, recent FMR studies on V[TCNE] x nanowires, microstructures, and thin films [19, 20, 21], coupled with combined DFT and electron energy loss spectroscopy (EELS) of the crystal structure [25], suggest there is a residual nematic 4 ordering of the c-axis of the V[TCNE] x unit cell, giving rise to an averaged crystal field anisotropy that is sensitive to structural and thermally induced strain. However, dynamic measurements of these crystal fields and their dependence on strain are currently lacking, preventing a quantitative analysis of the magnetostrictive properties of this material. For this work, PMN-PT/epoxy/V[TCNE] x/glass heterostructure devices are fabricated such that upon electrically biasing the PMN-PT[001] substrate, the piezoelectric effect produces a lateral in-plane strain in the V[TCNE] x thin film, schematically shown in Fig. 1a. PMN-PT is selected for its strong piezoelectric effects and high strain coefficients ( d31 ∼ −(500 to 1 ,000) pm/V) to maximize the strain in the devices [3], and the epoxy encapsulation layer is selected to allow for device operation under ambient conditions [22]. This device structure allows investigation of the magnetoelastic properties of V[TCNE] x via standard FMR characterization and analysis. In the main text of this work, measurements on three devices denoted Samples 1 - 3 are presented. Sample 1 is measured via broadband FMR (BFMR) techniques. Sample 2 is studied via X-band (∼9.8 GHz) cavity FMR techniques. Sample 3 is used to directly measure and calibrate strain in the devices via optical techniques. Additional devices characterized via BFMR techniques are presented in the Supplemental Information and their characteristics summarized in Table 1. The BFMR response of Sample 1 is described in Fig. 1(b), where individual scans (inset) are fit to a Lorentzian lineshape to extract the resonance frequency as a function of applied field, HR vs fR. This data can be modeled by considering the V[TCNE] x thin film as an infinite sheet with attendant shape anisotropy and with a uniaxial crystal field anisotropy oriented in the out-of-plane direction (as described above). Accordingly, the Kittel equation for ferromagnetic resonance reduces to [19, 20, 21] (1) where fR = ω/2π is the FMR resonance frequency, γ is the gyromagnetic ratio, HR is the applied magnetic field at resonance, Heff = 4πMeff =4πMs−H⊥ is the effective magnetization of the V[TCNE] x film with saturation magnetization 4 πMs and uniaxial strain-dependent anisotropy field H⊥, and θ describes the orientation of the external field as defined in Fig. 1(a). This equation is valid for films where HR ≫ 4πMeff, and is appropriate here as the effective magnetization for V[TCNE] x is typically ∼100 G [19] while the resonance field is typically between 3500 – 3650 G at X-band frequencies (9.86 GHz) for all magnetic field orientations. As-grown films exhibit no in-plane anisotropy, consistent with the literature [19, 20, 21], and so ϕ-dependences are neglected. When the external magnetic field is held out-of-plane ( θ = 0◦), Eq. (1) reduces to (2) 5 Further information about the magnetic damping in thin films can be revealed by comparing the FMR full-width-half-max (FWHM) frequency linewidth, Γ, to the FMR resonance frequency, fR, via (for 4𝜋𝑀≪𝐻ோ and Γ ≪𝑓ோ [19]) Γ = 2αfR + Γ0 (3) where α is the (dimensionless) Gilbert damping constant and Γ 0 is the inhomogeneous broadening. It should be noted this form for the Gilbert damping utilizing the frequency-swept linewidth is appropriate directly for out-of-plane magnetized thin films due to symmetry conditions resulting in the linear relationship between 𝐻ோ and 𝑓ோ [19]. Accordingly, Eqs. (1 - 3) show that performing FMR at various frequencies, fields, and magnetization orientations with and without applied strains in V[TCNE] x thin films should provide information regarding the magnetoelastic properties of V[TCNE] x. Fitting the data from Sample 1 to Eq. (2) reveals an effective magnetization 4 πMeff = 106.2 G and gyromagnetic ratio | γ|/2π = 2.756 MHz/Oe, consistent with literature [17, 18, 19, 20, 21, 22]. The Lorentzian fits of the FMR response also reveal the linewidth Γ as a function of the resonant frequency, seen in Fig. 1(c), where fitting to Eq. (3) yields α = 1.02 ± 0.52 × 10−4 and Γ0 = 8.48 ± 1.22 MHz in Sample 1. These damping characteristics are also consistent with literature values for V[TCNE] x [19, 27], and show that the devices incorporate high-quality magnetic films exhibiting superb low-damping properties [32, 33, 34]. Moving beyond measurements of the as-grown strain-free sample, the FMR response of the device is now measured while straining the V[TCNE] x film (Fig. 1(d)). Comparing the FMR response of Sample 1 with no applied strain ( EB = 0 kV/cm) and maximum-applied strain ( EB = 13.3 kV/cm) yields a shift in the resonance frequency of 45.5 MHz at a resonance frequency fR = 9.8 GHz, corresponding to a CME coefficient A = 3.38 MHz cm/kV (1.23 Oe cm/kV). It is worth noting that while this absolute shift in frequency, and consequent value for CME, is modest when compared to other magnetostrictive materials [4], it represents a shift of over 6 magnetic resonance linewidths due to the ultra-low damping and narrow FMR linewidths of the V[TCNE] x thin film. This ability to shift cleanly on and off resonance with an applied electric field is central to the functionality of many dynamically tuned MMIC devices, motivating a more in depth and systematic investigation of this phenomenon. The magnetostriction in this composite device is explored by biasing the piezoelectric transducer between 0 kV/cm and 13.3 kV/cm, and the shift in the resonance frequency (for 𝜃= 0∘) tracks the linear strain produced by the transducer [35], as seen in Fig. 2(a). For maximally strained films, fitting to Eq. (2) now reveals 4 πMeff = 122.9 G, a difference of +16 .7 G between EB = 0 kV/cm and EB = 13.3 kV/cm (14% change). Panels (b-d) of Fig. 2 show the FMR linewidth Γ, inhomogeneous broadening Γ 0, and Gilbert damping α, for Sample 1 as a function of applied electric field (strain). These parameters do not vary over the entire tuning range and are robust to repeated cycling ( >300 cycles - see Supplemental Information). 6 This stability indicates that the shift in resonance frequency is due to a true magnetoelastic effect under linear deformation rather than some fatigue induced structure or morphology change in the film, and further demonstrates the potential for device applications. Finally, it is noteworthy that the linewidths and damping coefficients observed in these proof of principle devices are much narrower than typical magnetostrictive materials, but are roughly twice the value observed in optimized bare V[TCNE] x films (Γ is typically ~3 MHz [17, 19, 25]). This suggests that the tuning ratio of 6 times the linewidth may be further extended to more than 10 times the linewidth in fully optimized devices [19, 25]. To confirm that these shifts in the resonance position are due to strain-dependent crystal-field anisotropy in V[TCNE] x as prior studies suggest [20, 21], angular-dependent measurements on unstrained and maximally strained films are performed. Sample 2 is mounted in an X-band (∼9.8 GHz) microwave cavity so that the structure can be rotated to vary the polar angle, θ. In-plane ( 𝜃= 90∘) and out-of-plane ( 𝜃= 0∘) FMR spectra are shown in Supplemental Fig. S1, with FWHM linewidths of 2.17 Oe (5.97 MHz) and 2.70 Oe (7.45 MHz), respectively. By tracking the resonance field as a function of rotation and fitting to Eq. (1) the effective magnetization Heff = 4πMeff = 74.0 G is extracted for Sample 2, as seen in Fig. 3. The difference in 4πMeff between Samples 1 and 2 can be attributed to sample-to-sample variation and remains consistent with literature values [19]. Repeating the measurement with an applied bias of 13.3 kV/cm to the PMN-PT reveals an increase of 4 πMeff to 79.4 G, an increase of 5.4 Oe (8% change), which is like the change observed in Sample 1. This confirms that strain is modulating the magnetic anisotropy in V[TCNE] x through the crystal field term H⊥ where 4πMeff = 4πMs−H⊥. This strain-dependent crystal field H⊥ is consistent with and supports previous measurements of V[TCNE] x with both thermally and structurally induced strain [19, 20, 21]. An approximate upper bound to the strain in these devices can be simply calculated through the relation ε = d31EB = (d31VB)/t ∼ −(6 − 12) × 10−4 for typical PMN-PT d31 piezo coefficients [4, 35]. However, the addition of epoxy, V[TCNE] x, and the glass substrate affect the overall stiffness of the device, thereby the piezo coefficient changes from d31 of the bare piezo to an effective coefficient deff of the entire stack. This deff is directly measured by exploiting the color change of V[TCNE] x upon laser heating [23] to pattern fiducial marks on the samples and monitor their positions under strain using optical microscopy (see Supplemental Information). This approach yields an effective piezoelectric coefficient of deff ∼ −180 pm/V or strain of ε ∼ −2.4×10−4, reasonable for the PMN-PT heterostructures used here [4, 35]. Density functional theory calculations on the relaxed and strained V[TCNE] x unit cell provide further insight into the elastic and magnetoelastic properties of V[TCNE] x. These properties are calculated using the Vienna ab initio Simulation Package (VASP) (version 5.4.4) with a plane-wave basis, projector- augmented-wave pseudopotentials [36, 37, 38, 39], and hybrid functional treatment of Heyd-Scuseria- Ernzerhof (HSE06) [40, 41]. The experimentally verified [25] local structure of the V[TCNE] x unit cell is 7 found by arranging the central V atom and octahedrally-coordinated TCNE ligands according to experimental indications [42, 43, 44, 45, 46], and subsequently allowing the structure to relax by minimizing the energy. These DFT results previously produced detailed predictions of the structural ordering of V[TCNE] x, along with the optoelectronic and inter-atomic vibrational properties of V[TCNE] x verified directly by EELS [25] and Raman spectroscopy [23], respectively. This robust and verified model therefore promises reliable insight into the elastic and magnetoelastic properties of V[TCNE] x. The magnetoelastic energy density for a cubic lattice f = fel + fme = E/V is a combination of the elastic energy density (4) where Cij are the elements of the elasticity tensor and εij are the strains applied to the cubic lattice, and the magnetoelastic coupling energy density where Bi are the magnetoelastic coupling constants and αi where i ∈ {x,y,z} represent the cosines of the magnetization vector [47]. The elastic tensor C = Cij for V[TCNE] x is found by applying various strains to the unit cell and observing the change in the energy. The calculated Cij tensor results in a predicted Young’s modulus for V[TCNE] x YV = 59.92 GPa. By directly applying compressive and tensile in-plane strains to the DFT unit cell (i.e. in the equatorial TCNE ligand plane [25]), one may calculate the overall change in the total energy density, both parallel and perpendicular to the easy axis. The difference between these two, Δ𝐸, is the magnetic energy density change, which is proportional to the magnetoelastic coupling constant B1 [47] ∆E/V = −(ν2D + 1)B1ε|| (6) where ν2D is the 2-dimension in-plane Poisson ratio and ε|| is the applied in-plane (equatorial TCNE plane) epitaxial strain while allowing out-of-plane (apical TCNE direction) relaxation. The elastic and magnetoelastic coefficients are related via the magnetostriction constant λ100 = λs via . (7) As a result, the calculated changes of the magnetoelastic energy density with strain provide direct predictions of the elasticity tensor ( Cij) and magnetoelastic coefficients ( Bi) for V[TCNE] x. For polycrystalline samples of cubic materials, the overall (averaged) magnetoelastic coefficient λs also considers the off-axis contribution from λ111 such that λs = (2/5)λ100 + (3/5)λ111. However, the off-axis component is not considered here for two reasons: (i) the apical TCNE ligands are assumed to align along 8 the out-of-plane direction ( z-axis, θ = 0◦), and (ii) difficulties in calculating the magnetoelastic energy density changes upon applying a shear strain that provides the estimate of B2 needed to calculate λ111. The former argument is reasonable as previous experimental results indicate the magnetocrystalline anisotropy from strain is out-of-plane [21], consistent with the ligand crystal field splitting between the equatorial and apical TCNE ligands [25]. Further, the lack of in-plane ( ϕ-dependent) anisotropy suggests the distribution in the plane averages out to zero. Therefore, the magnetoelastic coefficient calculated here considers an average of the in-plane Cii components in determining λ100. That is, the DFT predicts a magnetoelastic coefficient for V[TCNE] x of (8) where 𝐶𝐼𝑃 = (1/2)(𝐶11 + 𝐶22) = 60.56 GPa and C12 = 37.84 GPa. Accordingly, utilizing the calculated value of B1 = 85.85 kPa (see Supplemental Information) predicts a theoretically calculated 𝜆ሚଵ = −2.52 ppm for V[TCNE] x magnetized along the apical TCNE ligand (i.e. θ = 0◦). Combining these ferromagnetic resonance, direct strain measurements, and DFT calculations provides the information necessary to determine the magnetoelastic properties of V[TCNE] x. Here, we follow the convention in the literature using the magnetoelastic free energy form from the applied stress σ = Y ε to the magnetostrictive material, Fme = (3/2)λsσ [2, 3, 4]. Accordingly, this free energy yields an expression for the strain-dependent perpendicular (out-of-plane) crystal field [3] (9) where λS is the magnetoelastic coefficient, Y is the Young’s modulus of the magnetic material, d31 is the piezoelectric coefficient of the (multiferroic) crystal, and EB is the electric field bias. Here, ε = d31EB is the strain in the magnetic layer obtained based on the assumption that the electrically induced strain is perfectly transferred to the magnetic film. For this study, the direct optical measurement of the strain in the V[TCNE] x films allows the modification of Eq. 9 by replacing d31EB by the measured ε = deffEB = −2.4 × 10−4 to account for the mechanical complexity of the multilayered device. The magnetoelastic coefficient of V[TCNE] x can then be calculated from Eq. 9 using the values of 4 πMS and H⊥ from FMR characterization, the direct measurement of ε from optical measurements, and the calculated value of YV = 59.92 GPa from DFT. Accordingly, inserting the corresponding values into Eq. 9 yields a magnetoelastic constant for V[TCNE] x of λs ∼ −1 ppm to λs ∼ − 4 ppm for the devices measured here. This range shows excellent agreement with the DFT calculations of the magnetoelastic coefficient 𝜆ሚଵ = −2.52 ppm from Eq. 8. This agreement provides additional support for the robustness of the DFT model developed in previous work [23, 25]. 9 Further, comparison with past studies of the temperature dependence of Heff [21] allows for the extraction of the thermal expansion coefficient of V[TCNE] x, 𝛼௧ = 11 ppm/K, at room temperature. Discussion The results above compare V[TCNE] x thin films to other candidate magnetostrictive materials using the established metrics of CME and λs. However, while these parameters are effective in capturing the impact of magnetoelastic tuning on the DC magnetic properties of magnetic thin films and magnetoelectric devices, they fail to capture the critical functionality for dynamic (AC) magnetoelectric applications: the ability to cleanly tune on and off magnetic resonance with an applied electric field. For example, Terfenol-D is considered a gold standard magnetostrictive material due to its record large magnetoelastic coefficient λs up to 2,000 and CME coefficients 𝐴 as large as 590 Oe cm/kV [3]. However, due to its broad ∼1 GHz FMR linewidths, large 4 πMeff > 9,000 G, high Gilbert damping α = 6×10ିଶ, and brittle mechanical nature, it is not practical for many applications in MMIC. As a result, we propose a new metric that appropriately quantifies the capability of magnetostrictive materials for applications in microwave magnonic systems [28, 33, 34] that takes into account both the magnetostrictive characteristics and the linewidth (loss) under magnetic resonance of a magnetically-ordered material. Accordingly, a magnetostrictive agility ζ is proposed here, which is the ratio of the magnetoelastic coefficient λs to the FMR linewidth (in MHz) ζ(fR) = |λs|/Γ. For the V[TCNE] x films studied here the magnetostrictive agility at X-band frequencies (9.8 GHz) is in the range ζ = {0.164 – 0.660}, comparable to YIG ζ = {0.139 − 0.455} and Terfenol-D ζ = {0.301 – 0.662} as shown in Table 1. Further, we note that the growth conditions under which high-quality V[TCNE] x films can be obtained make on-chip integration with microwave devices significantly more practical than for YIG, and that the narrow linewidth (low loss) is more attractive for applications such as filters and microwave multiplexers than Terfenol-D. Conclusion We have systematically explored indirect electric-field control of ferromagnetic resonance in the low- loss organic-based ferrimagnet V[TCNE] x in V[TCNE] x/PMN-PT heterostructures. These devices demonstrate the ability to shift the magnetic resonance frequency of V[TCNE] x by more than 6 linewidths upon application of compressive in-plane strains 𝜀 ~ 10ିସ . Further, we find there is no change in the magnetic damping of the films with strain and that the samples are robust to repeated cycling (> 300 cycles), demonstrating the potential for applications in MMIC without sacrificing the ultra-low damping of magnetic resonance in V[TCNE] x. The changes in the FMR characteristics along with direct optical 10 measurements of strain provide an experimentally determined range for the magnetoelastic coefficient, λS = −(1 − 4 .3) ppm, showing excellent agreement to DFT calculations of the elastic and magnetoelastic properties of V[TCNE] x. Finally, we present a discussion on the metrics used in the magnetostriction community wherein we point out the shortcomings on the commonly used metrics of the magnetostriction and CME coefficients. In this context, we propose a new metric, the magnetostrictive agility, ζ, for use of magnetoelastic materials for coherent magnonics applications. These results develop the framework necessary for extended studies into strain-modulated magnonics in V[TCNE] x. Additionally, these magnetoelastic properties in V[TCNE] x suggest that large phonon- magnon coupling in V[TCNE] x might be achieved, necessary and useful for applications in acoustically- driven FMR (ADFMR) or, in conjunction with high-Q phonons, for quantum information applications [28]. Other recent work has identified V[TCNE] x as a promising candidate for QISE applications utilizing superconducting resonators [31] and NV centers in diamond ranging from enhanced electric-field sensing [5] to coupling NV centers over micron length scales [30]. These findings lay a potential framework for investigating the utilization of V[TCNE] x in quantum systems based on magnons and phonons. Acknowledgments S. W. K. developed the project idea, and S. W. K., E. J.-H., P. S., and M. P. developed the project plan for experimental analysis. S. W. K. fabricated the V[TCNE] x heterostructure devices, performed FMR characterization and analysis, and wrote the manuscript. A. F. developed the analysis software used for fitting FMR linewidths and extracting parameter fits. P. S., G. S., and M. P. provided PMN-PT substrates. Y. S. performed and analyzed DFT calculations of the elasticity tensor and the magnetoelastic coefficients for V[TCNE] x. H. F. H. C. performed and analyzed optical measurements of strain in the devices. K. E. N. and M. S. performed BLS measurements on V[TCNE] x/Epoxy devices to extract elastic properties. All authors discussed the results and revised the manuscript. S. W. K., A. F., and E. J.-H. were supported by NSF DMR-1808704. P. S., G. S, and M. P. were supported by the Air Force Office of Scientific Research (AFOSR) Award No. FA955023RXCOR001. The research at Oakland University was supported by grants from the National Science Foundation (DMR-1808892, ECCS-1923732) and the Air Force Office of Scientific Research (AFOSR) Award No. FA9550-20-1-0114. Y. S. and M. E. F. were supported by NSF DMR-1808742. H. F. H. C. and G. D. F. were supported by the DOE Office of Science (Basic Energy Sciences) grant DE-SC0019250. K. E. N., M. S., and K. S. B. were supported by NSF-EFRI grant NSF EFMA-1741666. The authors thank and acknowledge Georg Schmidt, Hans Hübl, and Mathias Kläui for fruitful discussions. 11 References [1] E. Y. Vedmedenko, R. K. Kawakami, D. D. Sheka, P. Gambardella, A. Kirilyuk, A. Hirohata, C. Binek, O. Chubykalo-Fesenko, S. Sanvito, B. J. Kirby, J. Grollier, K. Everschor-Sitte, T. Kampfrath, C.-Y. You, and A. Berger. “The 2020 magnetism roadmap”. J. Phys. D: Appl. Phys , 53(453001), 2020. [2] C. Song, B. Cui, F. Li, X. Zhou, and F. 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The pressure inside the CVD reactor for all growths was 35 mmHg, and TCNE, V(CO) 6, and the substrates are held at 65◦C, 10◦C, and 50◦C, respectively. All substrates were cleaned via solvent chain (acetone, methanol, isopropanol, and deionized (DI) water (×2)) and dried with N 2, followed by a 10 minute UV/Ozone clean in a UVOCS T10x 10/OES to remove any residual organic contaminants. Nominally 400 nm V[TCNE] x films are deposited onto microscope cover glass substrates ( t = 100µm). These V[TCNE] x/glass substrates are then mechanically fixed to a PMN-PT transducer (4 mm×10 mm ×0.15 mm) with an OLED epoxy (Ossila E130) to create a PMN-PT/Epoxy/V[TCNE] x/Glass heterostructure. The epoxy here not only protects V[TCNE] x from oxidation [22], but also propagates lateral strain into V[TCNE] x film from the piezo transducer upon biasing. While the primary deformation in the piezo transducer is along the poling direction of the 16 PMN-PT ( z), the distortion of the PMN-PT in the thickness direction also produces a lateral in-plane strain in the PMN-PT through the Poisson effect (i.e. one must consider here the d31 piezo coefficient of PMN-PT). Therefore, the primary strain experienced by the V[TCNE] x film is in-plane. The PMN-PT electrodes are connected to a Keithley 2400 voltage source so that electric fields up to EB = VB/tPMN−PT = 13.3 kV/cm can be applied across the PMN-PT layer. Ferromagnetic Resonance Characterization Broadband FMR (BFMR) measurements on Sample 1 and Supplemental Devices A-C were taken using a commercial microstrip (Southwest Microwave B4003-8M-50) and Agilent N5222A vector network analyzer (VNA). The devices are mounted so that the magnetic field is normal to the V[TCNE] x film (𝜃 = 0∘ ). S21 measurements (P = −20 dBm) show the FMR peak upon matched magnetic field and frequency conditions in accordance with Eq. 2. A Keithley 2400 Sourcemeter is used to apply up to 200 V to the piezoelectric transducers – accordingly, the maximum-applied strain in the 150 μm PMN-PT corresponds to an electric field 𝐸 = 13.3 kV/cm as mentioned in the main text. All angular-dependent FMR measurements (Sample 2) were performed in a Bruker X-band (~9.6 GHz) EPR (Elexsys 500) spectrometer. The frequency of the microwave source is tuned to match the resonant frequency of the cavity before each scan to ensure optimal cavity tuning. All scans had a 0.03 G modulation field at 100 kHz modulation frequency and were performed at the lowest possible microwave power (0.2 μW) to prevent sample heating and non-linear effects distorting the FMR lineshape. The V[TCNE] x/PMN-PT devices are mounted on a sapphire wafer and loaded into glass tubes for FMR measurements such that the samples can be rotated in-plane (IP: 𝜃 = 90∘ ) to out-of-plane (OOP: 𝜃 = 0∘ ) for FMR measurements in 10 degree increments, where resonance occurs upon matched field and frequency conditions according to Eq. 1. Density Functional Theory Calculations The pseudopotentials used are default options from VASP’s official PAW potential set, with five valence electrons per vanadium, four per carbon and five per nitrogen [36, 37, 38, 39]. For the rest of the calculation we used 400 eV for the energy cutoff and a Γ centered 5x5x3 k-mesh sampling. From these results, the elastic tensor Cij for V[TCNE] x is calculated. Using the elastic tensor, the Young’s modulus for V[TCNE] x is averaged over the C11, C22, and C33 components to yield YV = 59.92 GPa. From the DFT calculations, the full elastic matrix from the Cij is given by (in units of GPa) 𝐶= ⎣⎢⎢⎢⎢⎡66.4437.847.96 37.8454.683.79 7.96 3.79 58.641.38−0.200.53 0.09−1.55−0.37 −0.69 0.76 0.31 1.38 0.09 −0.69 −0.20 −1.55 0.76 0.53 −0.37 0.3135.16 0.25 −0.95 0.25 6.65 −0.17 −0.95 −0.17 9.94 ⎦⎥⎥⎥⎥⎤ 17 Optical Measurements of Strain in V[TCNE] x V[TCNE] x films can be patterned via laser heating techniques, whereupon the material changes color when heated above its thermal degradation temperature ( ∼ 370 K) [16, 23]. To more appropriately calibrate strain in the V[TCNE] x films versus applied bias, we directly measure the deformation in the films by exploiting the color change of V[TCNE] x upon laser heating [23] and optical microscopy techniques. Fresh V[TCNE] x/PMN-PT devices are exposed to a focused laser spot to create ad hoc fiducial marks on the film in Sample 3 (Supplementary Fig. S5) in a 50 µm × 50 µm square. By measuring the distance between these laser-written structures with and without applied strain, we can precisely and directly measure the strain in the V[TCNE] x films upon electric bias thus allowing a more precise calculation of magnetoelastic coefficients. Using these methods, we apply a bias of 13.3 kV/cm on Sample 3 and find a strain ε ∼ 2.4×10−4 which is in reasonable agreement with estimated values of strain using the thickness of the PMN-PT (150 µm) and typical piezo coefficient d31 ∼ 500 − 1000 pm/V). 18 Figure 1: (a) Effective device schematic, coordinate system, and wiring diagram for V[TCNE] x/PMN-PT heterostructures. (b) Ferromagnetic resonance frequency fR vs external field Hext with the field held OOP ( θ = 0◦) measured via BFMR. The external field is held constant as the microwave frequency is swept. (Inset) Representative BFMR scan at f0 = 9.8 GHz, Hext = 3,660 Oe. (c) FMR linewidth Γ versus FMR frequency for OOP field ( θ = 0◦). A linear fit (red line) extracts the dimensionless Gilbert damping parameter α = 1.02 ± 0.52 × 10−4 and the inhomogeneous broadening Γ 0 = 8.48 ± 1.22 MHz. (d) BFMR scans for unstrained (0 kV/cm – black) and maximally strained (13.3 kV/cm – red). The shift in the FMR frequency is ∼45 MHz, a shift ∼4 linewidths. 19 Figure 2: V[TCNE] x damping analysis with applied strain: (a) Plot showing differential (shifted) FMR resonance position f R−f0 where f 0 is the resonance at 9.8 GHz, (b) FWHM linewidth Γ, (c) inhomogeneous broadening Γ 0, and (d) Gilbert damping α versus applied electric field bias E B. While the resonance position shifts by multiple linewidths, there is negligible effect in the linewidth or damping of the material. 20 Figure 3: Cavity X-band FMR measurements on V[TCNE] x/PMN-PT devices. Angular dependence of FMR resonant field H R at 𝑓ோ ∼ 9.6 GHz is measured without (black squares) and with (red circles) strain. In-plane and out-of-plane peak-to-peak (FWHM) linewidths are 1.25 (2.16) Oe and 1.56 (2.70) Oe, respectively. 21 Table 1: Extracted parameters from V[TCNE] x strain devices compared to YIG, Terfenol-D, and other magnetostrictive materials. Asterisk indicates the frequency-equivalent linewidth calculated from the field-swept FMR linewidth and accounts for the ellipticity of FMR precession for in-plane magnetized materials following the method in Ref. [56]. 22 Supplemental Information: In situ electric-field control of ferromagnetic resonance in the low-loss organic-based ferrimagnet V[TCNE] x∼2 Cavity X-band FMR of Sample 2 Figure S1: X-band (9.8 GHz) cavity FMR scans of Sample 2 for DC magnetic field in-plane ( 𝜃=90°) and out- of-plane ( 𝜃 = 0° ). Peak-to-peak (p2p) linewidths and resonance positions determined from a fit to a Lorentzian derivative, from which the full-width-half-max (FWHM) linewidth Γ is found by multiplying by √3. 23 V[TCNE] x Resonance Frequency with Piezo Switching Strain Figure S2: V[TCNE] x out-of-plane magnetized differential resonance frequency 𝛿𝑓ோ=9.8 𝐺𝐻𝑧−𝑓ோ as a function of applied bias voltage to 150 µm PMN-PT . The “butterfly” hysteresis arises from the hysteretic behavior of the piezo strain upon the polarization direction switching. The inset shows frequency-swept FMR spectra and fits at maximum and minimum frequency shift. 24 Figure S3: Differential resonance frequency ( 𝑓ோ,=9.8 GHz) in the same device from Fig. S2 switching between 𝑉= ±25 V (𝐸= ±1.67 kV/cm) as a function of the number of the number of switches between positive and negative applied strain. The FWHM linewidth of the V[TCNE] x remains effectively constant for over 300 positive/negative (tensile/compressive) strain applications, and the resonant frequency for the respective compressive and tensile additionally remains effectively constant. These conditions were selected based on the linewidth and resonance frequency tuning such that the resonance features do not overlap, thereby demonstrating a means to electrically-bias a device on and off resonance. 25 V[TCNE] x Density Functional Theory Calculations of Strain-Dependent Magnetoelastic Energy V[TCNE] x Optical Strain Characterization Figure S5: Positions of fiducial marks “burned” onto the V[TCNE] x film measured via optical techniques upon biasing a 150 µm PMN-PT piezo transducer. The extracted strain in x and y is averaged to 𝜀 = 2.4×10ିସ and is used to calculate the magnetoelastic coefficients 𝜆ௌ presented in the text. Figure S4: DFT-calculated magnetic energy difference of the V[TCNE] x unit cell upon manipulating the applied strain. The orange line is a tangential linear fit at 𝜀 = 0 to solve for 𝛥𝐸/𝑉 in the main text that provides the magnetoelastic coupling 𝐵ଵ. 26 Additional V[TCNE] x Device Strain Characterizations Figure S6: Supplemental devices measured via BFMR techniques ( 𝜃=0°). Supplemental device C varies from the others only by the thickness of the PMN-PT piezo ( 𝑡= 500 𝜇𝑚 ), so that the electric field across the device (hence the strain) is adjusted accordingly. 27 Additional Linewidth and Damping Analysis: Supplemental Device C ( 𝒕𝑷= 𝟓𝟎𝟎 𝝁𝒎) Figure S7: Gilbert analysis of Supplemental Device C as a function of applied electric-field bias. (a) Resonance frequency for an out-of-plane magnetization orientation and applied external field 𝐻ோ= 3,658.8 G (𝑓ோ,= 9.83 GHz). (b) FWHM linewidth corresponding to the resonance frequencies in panel (a). (c) Inhomogeneous broadening and (d) Gilbert damping parameters. Note there is negligible change in linewidth, inhomogeneous broadening, and Gilbert damping for positive and negative bias up to the piezo switching fields at ±3.2 kV/cm. The error bars in (a) and (b) are smaller than the markers used for the data points.

0908.0600v1.Persistence_effects_in_deterministic_diffusion.pdf

Persistence effects in deterministic diffusion Thomas Gilbert1,and David P. Sanders2, † 1Center for Nonlinear Phenomena and Complex Systems, Universit ´e Libre de Bruxelles, C. P . 231, Campus Plaine, B-1050 Brussels, Belgium 2Departamento de F ´ısica, Facultad de Ciencias, Universidad Nacional Aut´onoma de M ´exico, Ciudad Universitaria, 04510 M ´exico D.F ., Mexico In systems which exhibit deterministic diffusion, the gross parameter dependence of the diffusion coefficient can often be understood in terms of random walk models. Provided the decay of correlations is fast enough, one can ignore memory effects and approximate the diffusion coefficient according to dimensional arguments. By successively including the effects of one and two steps of memory on this approximation, we examine the effects of “persistence” on the diffusion coefficients of extended two-dimensional billiard tables and show how to properly account for these effects, using walks in which a particle undergoes jumps in different directions with probabilities that depend on where they came from. PACS numbers: 05.60.Cd, 05.45.-a, 05.10.-a, 02.50.-r I. INTRODUCTION Diffusion is a fundamental macroscopic phenomenon in physical systems, which, for instance, characterizes the spreading of tracer particles in a solvent. At a mescoscopic scale, it can be traced to the cumulative effect of many “ran- dom” displacements, as in Brownian motion [1]. At the under- lying microscopic scale, however, the dynamics of a system are deterministic. Deterministic diffusion concerns the study of microscopic models whose deterministic dynamics also ex- hibit diffusive behavior at a macroscopic scale [2, 3, 4]. A particularly appealing, physically motivated model which does exhibit this phenomenon is the periodic Lorentz gas [5]. Here, independent point particles in free motion un- dergo elastic collisions with fixed hard disks in a periodic ar- ray. The diffusive motion can then be considered to be a result of the chaotic nature of the microscopic dynamics, accord- ing to which nearby initial conditions separate exponentially fast due to the convex nature of the obstacles. Thus a cloud of (non-interacting) particles in this Lorentz gas spreads out over time in a way similar to that of solutions of the diffusion equation, [x(t)x(0)]2 4Dt; (1) where x(t)denotes the position of a tracer at time t, with ini- tial position x(0), and the mean squared displacement is com- puted as an average h
iover many realizations of this process. The diffusion coefficient, D, is a constant which depends on the geometrical parameters of the system, i.e., the underlying microscopic dynamics. The diffusion coefficient summarizes the macroscopic be- havior of the system while capturing the microscopic prop- erties of the dynamics that lead to it. A central question in deterministic diffusion is to understand how this dependence Electronic address: thomas.gilbert@ulb.ac.be †Electronic address: dps@fciencias.unam.mx; URL: http://sistemas. fciencias.unam.mx/ ~dsanderson the geometrical parameters comes about. This has been ad- dressed in particular by Machta and Zwanzig [6], who showed that in the limit where the obstacles are close together, the mo- tion reduces to a stochastic Bernoulli-type hopping process— arandom walk —between “traps”. By calculating the diffu- sion coefficient of this random walk, they were able to obtain a reasonable agreement with the numerically-measured value of the diffusion coefficient. The approach of Machta and Zwanzig was extended heuris- tically by Klages and Dellago [7], by including important physical effects not taken into account in the simple random- walk picture, namely a possibly non-isotropic probability of changing directions, and of crossing two traps at once. Klages and Korabel [8] then provided an alternate approach, in which they employed a Green-Kubo expansion of the diffusion co- efficient to obtain a series of increasingly accurate approx- imations, based on numerically-calculated multi-step transi- tion probabilities. In one particular Lorentz gas model, they showed that their results are in good agreement with this ex- pansion, see also [4]. Nonetheless, as we emphasize below, the physical motivation, and indeed the physical meaning, of this approach, are not clear. The purpose of this paper is to show that in fact the correct expansion beyond the Machta-Zwanzig approximation is to incorporate this type of correction in the framework of persis- tent random walks . In other words, to be consistent, memory effects of a given length, whether one or several steps, must be accounted for through their contribution at all orders in the Green-Kubo formula relating the diffusion coefficient to the velocity auto-correlations. This is physically strongly moti- vated, and provides the correct way of incorporating correla- tion effects, in principle, of any finite order. In the usual periodic Lorentz gas on a triangular lattice con- sidered in [8], the model is sufficiently isotropic that higher or- der contributions are small and can be safely neglected. How- ever, when the correlation effects are very strong, this is no longer the case. We introduce a billiard model with this prop- erty and show that, whereas a Green-Kubo–type expansion fails except very close to the Machta-Zwanzig limiting case, the approximation based on a first-order persistent random walk is in reasonable agreement with the data. By further con-arXiv:0908.0600v1 [nlin.CD] 5 Aug 20092 FIG. 1: Periodic Lorentz gas on a triangular lattice. A typical tra- jectory is shown, which starts at the upper right disk in the initial cell—marked by the highlighted triangle—and moves across the ta- ble, performing a diffusive motion. sidering memory effects up to two successive steps, we find that the agreement between the numerically-measured diffu- sion coefficient of the billiard table and that of the second- order persistent random walk extends to an appreciably larger range of parameters. II. PERIODIC LORENTZ GAS ON A TRIANGULAR LATTICE Consider the periodic Lorentz gas on a triangular lattice, shown in Fig. 1. The centers of three nearby disks are iden- tified as the vertices of equilateral triangles which, in our no- tation, will be taken to be of unit side length. Denoting by r the radius of the disks, we let d12rdenote the spacing between disks. When d=0, the triangles form closed traps from which the tracer particles cannot escape. If, however, 0<d1, then we expect the particles to remain inside the cell for a long time before they can escape to another cell. This argument can be made precise using ergodic theory [6]. Given a tracer of unit velocity, the mean trapping time tis given in terms of the ratio of the cell area to the lengths of the holes by [9] t=pp 3=4pr2=2 3d=pp 3=4p(1d)2=8 3d:(2) Assuming that this time is longer than the typical decay of correlations [18], Machta and Zwanzig [6] argued that the dif- fusion coefficient of the Lorentz gas can be approximated by a memory-less random walk (called the “short-memory approx- imation” in [10]), DMZ=`2 4t; (3)where, in our notation, the lattice spacing `=1=p 3. Taking further account of memory effects, Klages & Ko- rabel [8] noted that the Machta-Zwanzig approximation (3) is the zeroth order expansion of a series given by the Green- Kubo formula for the diffusion coefficient: D=DMZ 1+2¥ å k=1hv0
vki! ; (4) where vkis the jump vector (“velocity”) between traps on the kth step, andhv0
vkiareauto-correlations of the velocity at steps 0 and k. The approach taken in [8] was to truncate the expression (4) at a finite value of k, assuming that all higher-order cor- relations are 0. They showed that by numerically calculating the terms appearing in this equation, the results evaluated by this truncation converged to the numerically-obtained diffu- sion coefficient. However, this ad hoc truncation has no physical meaning: ifhv0
v1i6=0, it is not true that higher-order correlations hv0
vkivanish. Rather, assuming that the process has mem- ory of the previous step alone—which we will refer to as the single-step memory approximation—one must compute the correlationshv0
vkiin a consistent way, and evaluate the dif- fusion coefficient (4) by taking all the k’s into consideration. A. Single-step memory approximation The process for which the motion of a particle at step k+1 depends explicitly on the state at step kis known as a persis- tentorcorrelated random walk [11, 12, 13, 14]. The tech- nique for studying such walks is well developed, and consists of treating it as a random walk with internal states, which, in this case, describe the direction with which the walker arrives at a site. Considering a persistent random walk on a honeycomb lat- tice, we denote by Pbthe conditional probability to return in the direction opposite to the current one, Prto turn right and Plto turn left. In terms of these quantities, the velocity auto- correlation is found to be [15] hv0
vki=(1)k 2(" PbPr+Pl 2ip 3 2(PrPl)#k +" PbPr+Pl 2+ip 3 2(PrPl)#k) ; (5) where i=p1. In the case of a symmetric walk for which Pr=PlPs= (1Pb)=2, the diffusion coefficient (4) is D1SMA =DMZ3(1Pb) 1+3Pb: (6) In comparison, the first order approximation made in [8] is to write D(1) KK=1+2hv0
v1i=DMZ(23Pb); (7)3 which corresponds to the first-order approximation of Eq. (6) when expanding D1SMA about the isotropic process for which Pb=1=3. B. Two-step memory approximation For a persistent process with two-step memory approxima- tion, there are nine conditional transition probabilities, whichwe denote by Pbb,Pbr,Pbl,Prb,Prr,Prl,Plb,PlrandPll, where, for instance, Pbrdenotes the probability that the tracer first moves backwards and then turns right, and similarly for the other symbols. Although the corresponding Markov chain in- volves a 99 stochastic matrix, there are symmetries in the system which can be exploited to reduce the computations to 33 matrices involving the transition probabilities listed above. The computation of the velocity auto-correlation then yields [15] hv0
vki= (1)k3 2 1 1 12 640 @Pbb Plb Prb fPblfPllfPrl f2Pbrf2Plrf2Prr1 Ak10 @1 0 0 0f0 0 0 f21 A+0 @Pbb Plb Prb f2Pblf2Pllf2Prl fPbrfPlrfPrr1 Ak10 @1 0 0 0f20 0 0 f1 A3 750 @p1 p2 p31 A; (8) where f=exp(2ip=3)and(p1;p2;p3)are the first three components of the stationary distribution of the Markov chain, which, for a left-right symmetric process for which Prr=PllPss,Prb=PlbPsbandPrl=Plr=1PssPsb, read p1=Psb 33Pbb+3Psb;p2=p3=1Pbb 66Pbb+6Psb: (9) The corresponding diffusion coefficient (4) is D2SMA =DMZ3(1Pbb)(1+PbbPsb)(2Psb2Pss) (1Pbb+Psb)[Psb(7+Pbb8Pss)+2(1+Pbb)Pss4P2 sb]: (10) This compares to the second-order approximation following the truncation scheme in [8], D(2) KK=D(1) KK+2hv0
v2i; =DMZ55Pbb7Psb+9PbbPsb 22Pbb+2Psb: (11) We note that this expression is actually different from that given in [8], where the stationary distribution (9) was erro- neously assumed to be uniform. C. Numerical results The transition probabilities of the single- and two-step memory approximation random walks can be computed for the Lorentz gas by estimating the relative frequencies of the corresponding events and taking into consideration the left- right symmetry of these transitions. Plugging their values into Eqs. (6) and (10), we obtain the corresponding coefficients and compare them to the diffusion coefficient of the billiard calculated from direct numerical simulations of the billiard dynamics. These results are shown in Fig. 2, including the results of the truncations (7) and (11). In the limit d!0, we see that the Machta-Zwanzig ap- proximation (3) is recovered. Looking at a broader range of parameter values, whereas the single-step approximation yields a good estimate of the actual diffusion coefficient ofthe Lorentz gas only for values of d<103, the extent of the range of validity of the two-step approximation is much larger, d<0:1. 00.060.121.1.21.4 d -10-8-6-4-20.91.01.11.21.31.4 logHdL4Dtl-2 FIG. 2: Numerical computation of the diffusion coefficient of the pe- riodic Lorentz gas on a triangular lattice, here divided by the dimen- sional factor, Eq. (3). The lines correspond to the different approx- imate results discussed above: (long-dashed, black line) single-step memory approximation (6); (dot-dashed, red line) first-order trunca- tion (7); (solid, magenta line) two-step memory approximation (10); (dashed green) second-order truncation (11). As seen in the figure, the successive approximations (6) and (10) are slightly better than their respective counterparts (7) and (11). The differences between the results of Eqs. (6) and4 (7) on the one hand, and Eqs. (10) and (11) on the other hand are, however, quite small and not everywhere easy to appre- ciate. The reason is that the transition probabilities are nearly isotropic for all values of din the range of allowed values, viz.p 3=4<d<1=2 (the lower bound corresponds to the finite- horizon condition). This fact, which we illustrate in Fig. 3 for the transition probabilities of the single-step memory approxi- mation, explains the validity of the Klages–Korabel truncation scheme in this case. àààààààààààà àààààààààààààààààààæææææææææææææææææææææææææææææææ 0.000.020.040.060.080.100.120.260.280.300.320.340.360.380.40 dPb,Ps FIG. 3: (Color online) Numerical computation of the transition prob- abilities Pb(squares) and Ps(circles) of the single-step memory ap- proximation associated to the periodic Lorentz gas on a triangular lattice. Similar results were reported in [7]. III. BILLIARD TABLE ON A SQUARE LATTICE In order to better emphasize the validity of our approxima- tion scheme over simpler truncation methods, we turn to a class of billiard tables such as shown in Fig. 4, which alter- nates small and large disks on a square lattice. By inserting rigid horizontal and vertical barriers between the small disks, with gaps of size din their centers, we introduce a control pa- rameter of relevance to the dynamical properties of the model which is independent of the other system parameters. How- ever large the gaps, the trajectories in these billiards can be reconstructed from the trajectories of the same billiard with a single unit cell on the torus. The geometry of the lattice is such that the distance between neighboring disks is equal to the lattice spacing, which we take to be unity, `=1. Given the radii riof the large disks androof the small disks, the trapping time is t=p1p(r2 i+r2 o) 4d: (12) Given these two parameters, the Machta-Zwanzig approxima- tion is here again given by Eq. (3). Note that in the present model, DMZdepends linearly on d, contrary to the Lorentz gas discussed in the previous section, for which the area of the cells varied with d. FIG. 4: Periodic Lorentz table on a square lattice. A typical trajectory is shown, with the initial position marked by a thick dot. We take the inner and outer radii to be respectively ri=0:36`andro=0:15` and vary only the size of the gaps. A. Single-step memory approximation Consider a persistent random walk with single-step mem- ory on a square lattice. We denote by Pf,Pr,PbandPl, respec- tively, the conditional probabilities that the particle proceeds in the same direction, turns right, reverses direction, or turns left. The velocity auto-correlation is hv0
vki=1 2f[PfPbi(PrPl)]k+[PfPb+i(PrPl)]kg: (13) Plugging this into (4) and assuming a symmetric process such thatPr=PlPs= (1PfPb)=2, we obtain the diffusion coefficient D1SMA =DMZ1+PfPb 1Pf+Pb: (14) This expression compares to D(1) KK=1+2hv0
v1i=DMZ(1+2Pf2Pb); (15) which is the first-order approximation of Eq. (14) when ex- panding D1SMA about the isotropic process for which Pf= Pb=1=4. B. Two-step memory approximation Applying the same procedure to the persistent random walk with two-step memory, we have 16 conditional transition probabilities, in terms of which it is possible to write the ve- locity auto-correlation in the compact form [15]5 hv0
vki=2 1 1 1 12 6640 B@Pff Plf Pbf Prf iPfliPlliPbl iPrl PfbPlbPbbPrb iPfriPlriPbriPrr1 CAk10 B@1 0 0 0 0i0 0 0 01 0 0 0 0i1 CA +0 B@Pff Plf Pbf Prf iPfliPlliPbliPrl PfbPlbPbbPrb iPfriPlriPbr iPrr1 CAk10 B@1 0 0 0 0i0 0 0 01 0 0 0 0 i1 CA3 7750 B@p1 p2 p3 p41 CA; (16) where the pi’s are the first four components of the stationary distribution of this process, which, assuming a left-right symmetric process, have the expressions p1=PbfPsb+PsfPbbPsf 4(1Pff+PsbPffPsb+Pbf(Pfb+Psb)+Pbb(1+PffPsf)+Psf+PfbPsf); p2=1PbbPbfPfbPff+PbbPff 8(1Pff+PsbPffPsb+Pbf(Pfb+Psb)+Pbb(1+PffPsf)+Psf+PfbPsf); p3=PsbPffPsb+PfbPsf 4(1Pff+PsbPffPsb+Pbf(Pfb+Psb)+Pbb(1+PffPsf)+Psf+PfbPsf); p4=1PbbPbfPfbPff+PbbPff 8(1Pff+PsbPffPsb+Pbf(Pfb+Psb)+Pbb(1+PffPsf)+Psf+PfbPsf):(17) The diffusion coefficient of this process may then be ob- tained by substituting Eqs. (16)-(17) into Eq. (4) and summing over k. We will however not write down its lengthy explicit expression, referring the reader instead to [15]. Let us also notice that Eqs. (16)-(17) can be used to write down the corresponding second-order approximation in [8], here properly taking into account the stationary distribution, D(2) KK=1+2hv0
v1i+2hv0
v2i=D(1) KK+2hv0
v2i:(18) C. Numerical results The transition probabilities of the random walks with single- and two-step memory approximation can be computed numerically for the billiard table by estimating the relative fre- quencies of the corresponding events and taking into account the left-right symmetry of these transitions. as above. Plug- ging their values into Eqs. (14) and (16), we obtain the cor- responding coefficients and compare them to the numerically- computed diffusion coefficient of the billiard. These results are reported in Fig. 5, including the results of the truncations (15) and (18). In the limit d!0, we again see that the Machta-Zwanzig approximation (3) is recovered. Zooming into the lower range of the parameter d, we see that the single-step approximation yields a good estimate of the actual diffusion coefficient of the Lorentz gas only for values of d<2103. The extent of the range of validity of the two-step approximation, on the other hand, is again much larger, d<0:5. The other main observation is that the truncated estimates 00.20.40.600.51 d -6-5-4-3-2-100.00.20.40.60.81.0 logHdL4Dtl-2FIG. 5: Numerical computation of the diffusion coefficient of the bil- liard table on a square lattice as shown in Fig. 4, here divided by the dimensional factor, Eq. (3). The lines correspond to the different ap- proximate results discussed above: (long-dashed, black line) single- step memory approximation (14); (dot-dashed, red line) first-order truncation (15); (solid, magenta line) two-step memory approxima- tion, obtained from Eqs. (4) and (16); (dashed, green line) second- order truncation (18). (15) and (18) are inaccurate as soon as d>0:05. The rea- son can be traced to the anisotropy of the hopping processes. Figure 6 shows the transition rates of the single-step mem- ory random walk. Although the probability of a right or left turn remains close to 1 =4 throughout the parameter range, the backscattering probability starts growing linearly above 1=4 with small d’s and saturates near 1 =2 at around d=0:5. Correspondingly, the forward-scattering probability decreases6 and is close to zero at around d=0:5. èèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè ìììììììììììììììììììììììììììììììììì 0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6 dPf,Ps,Pb FIG. 6: (Color online) Numerical computation of the transition prob- abilities Pb(squares), Pf(circles) and Ps(diamonds) of the single-step memory approximation associated to the billiard table on a square lattice. These rates reflect the anisotropy of the process. Looking at the transition probabilities of the two-step mem- ory process, shown in Fig. 7, we notice the differences among these probabilities, for instance comparing Pbb,PfbandPsb, as well as between these probabilities and that of the single-step memory process, in our example Pb. These differences jus- tify the necessity of resorting to the two-step memory process over the single-step process. IV . CONCLUSIONS The diffusion coefficients of deterministic systems with rapid decay of correlations can be well approximated by that of correlated walks, where a walker’s transition probabilities are determined according to its motion over the last few steps. Billiards provide good, physically motivated, examples of such systems. The Machta-Zwanzig dimensional prediction [6], according to which the diffusion coefficient is approxi- mated by the ratio between the distance between traps squaredand the trapping time, provides a gross estimate of this quan- tity. However, the importance of memory effects in the de- terministic diffusion of tracer particles is apparent as soon as one moves away from the limit where the trapping times are infinite. Truncation schemes based on the Green-Kubo formula, such as considered in [4, 8], may provide accurate results for models with little anisotropy, but they are physically inconsis- tent: given a hopping process with finite memory effects, ve- locity auto-correlations of all orders yield non-vanishing con- tributions to the Green-Kubo formula. This is particularly clear where anisotropies come into play. Estimates of the diffusion coefficients based on persistent ran- dom walks, however, do provide accurate results where the truncation schemes breakdown. It will be interesting to find out how these results trans- pose to models where disorder is present, such as with tagged- particle diffusion in interacting particle systems. Dimensional predictions similar to the Machta-Zwanzig one also appeared recently in the context of models of heat conduction [16, 17]. Estimating the deviations of the heat conductivities of these models from dimensional predictions remains an open prob- lem. Acknowledgments The authors thank Felipe Barra, Mark Demers, Hern ´an Lar- ralde and Carlangelo Liverani 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 Bel- gian Federal Government under the Inter-university Attrac- tion 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 Brux- elles, where most of this work was carried out. TG acknowl- edges the hospitality of the Weizmann Institute of Science, where part of this work was completed. [1] S. Chandrasekar, Stochastic problems in physics and astronomy Rev. Mod. Phys. 15, 1(1943). [2] P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cam- bridge University Press, Cambridge, UK, 1998). [3] J. R. Dorfman, An Introduction to Chaos in Nonequilib- rium Statistical Mechanics (Cambridge University Press, Cam- bridge, UK, 1999). [4] R. Klages, Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics (World Scientific, Singa- pore, 2007). [5] L. A. Bunimovich and Ya. Sinai, Markov Partition for Dis- persed Billiard Comm. Math. Phys. 78, 247 (1980). Statistical properties of Lorentz gas with periodic configuration of scatter- ersComm. Math. Phys. 78, 479 (1980). [6] J. Machta and R Zwanzig, Diffusion in a periodic Lorentz gas Phys. Rev. Lett. 501959 (1983).[7] R. Klages and C. Dellago, Density-dependent diffusion in the periodic Lorentz gas J. Stat. Phys. 101145 (2000). [8] R. Klages and N. Korabel, Understanding deterministic diffu- sion by correlated random walks J. Phys A math. gen. 354823 (2002). [9] N. Chernov and R. Markarian, Chaotic billiards Math. Surveys and Monographs 127(AMS, Providence, RI, 2006). [10] R. Zwanzig, From classical dynamics to continuous time ran- dom walks J. Stat. Phys. 30255 (1983). [11] B. D. Hughes, Random Walks and Random Environments. Vol- ume 1: Random Walks (Clarendon Press, Oxford, 1995) [12] G. H. Weiss and R. J. Rubin Random Walks: Theory and Selected Applications Advances in Chemical Physics 52563 (1983) [13] J. W. Haus and K.W. Kehr, Diffusion in regular and disordered lattices Phys. Rep. 150, 263 (1987).7 èèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèììììììììììììììììììììììììììììììììì 0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6 dPff,Pfb,Pfs èèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèìììììììììììììììììììììììììììììììììì 0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6 dPbf,Pbb,Pbs èèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè ìììììììììììììììììììììììììììììììììì òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò 0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6 dPsf,Pss,Psb FIG. 7: (Color online) Numerical computation of the transition probabilities Pxb(squares), Pxf(circles) and Pxs(diamonds) of the two-step memory approximation associated to the billiard table on a square lattice, where x stands respectively for f (left), b (middle) and s (right). In the right figure, the triangles are the probabilities of turning right after turning left and, the other way around, turning left after turning right. The differences between these three figures and the single-step transition probabilities shown in Fig. 6 justify resorting to a two-step memory process. [14] G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994). [15] T. Gilbert and D. P. Sanders, Diffusion coefficients for multi- step persistent random walks on lattices unpublished. [16] P. Gaspard and T. Gilbert, Heat conduction and Fourier’s law by consecutive local mixing and thermalization Phys. Rev. Lett. 101020601 (2008). [17] T. Gilbert and R. Lefevere, Heat Conductivity from MolecularChaos Hypothesis in Locally Confined Billiard Systems Phys. Rev. Lett. 101200601 (2008). [18] The exponential decay of correlations has been proved in the periodic Lorentz gas, see N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls , in: Hard Ball Systems and the Lorentz Gas , ed. by D. Szasz, Encyclopaedia of Mathematical Sciences 101, 89 (Springer, 2000).

1309.7483v1.High_efficiency_GHz_frequency_doubling_without_power_threshold_in_thin_film_Ni81Fe19.pdf

arXiv:1309.7483v1 [cond-mat.mtrl-sci] 28 Sep 2013High-efficiency GHz frequency doubling without power thresh old in thin-film Ni 81Fe19 Cheng Cheng1and William E. Bailey1 Materials Science and Engineering Program, Department of A pplied Physics and Applied Mathematics, Columbia University, New York, NY 10027 We demonstrate efficient second-harmonic generation at moderat e input power for thin film Ni 81Fe19undergoing ferromagnetic resonance (FMR). Powers of the gene r- atedsecond-harmonicareshowntobequadraticininputpower, wit hanupconversion ratio three orders of magnitude higher than that demonstrated in ferrites1, defined as ∆P2ω/∆Pω∼4×10−5/W·Pω, where ∆ Pis the change in the transmitted rf power and Pis the input rf power. The second harmonic signal generated exhibit s a significantly lower linewidth than that predicted by low-power Gilbert damping, and is excited without threshold. Results are in good agreement with an analytic, approximate expansion of the Landau-Lifshitz-Gilbert (LLG) equa tion. 1Nonlinear effects in magnetizationdynamics, apart frombeing offun damental interest1–4, have provided important tools for microwave signal processing, es pecially in terms of fre- quency doubling and mixing5,6. Extensive experimental work exists on ferrites1,4,6, tradi- tionally used in low-loss devices due to their insulating nature and narr ow ferromagnetic resonance (FMR) linewidth. Metallic thin-film ferromagnets are of int erest for use in these and related devices due to their high moments, integrability with CMOS processes, and potential for enhanced functionality from spin transport; low FMR linewidth has been demonstrated recently in metals through compensation by the spin Hall effect7. While some recent work has addressed nonlinear effects8–10and harmonic generation11–13in metallic ferromagnets and related devices14–16, these studies have generally used very high power or rf fields, and have not distinguished between effects above and b elow the Suhl instabil- ity threshold. In this manuscript, we demonstrate frequency dou bling below threshold in a metallic system (Ni 81Fe19) which is three orders of magnitude more efficient than that demonstrated previously in ferrite materials1. The results are in good quantitative agree- ment with an analytical expansion of the Landau-Lifshitz-Gilbert (L LG) equation. For all measurements shown, we used a metallic ferromagnetic thin fi lm structure, Ta(5 nm)/Cu(5 nm)/Ni 81Fe19(30 nm)/Cu(3 nm)/Al(3 nm). The film was deposited on an oxi- dized silicon substrate using magnetron sputtering at a base press ure of 2.0 ×10−7Torr. The bottom Ta(5 nm)/Cu(5 nm) layer is a seed layer to improve adhesion a nd homogeneity of the film and the top Cu(3 nm)/Al(3 nm) layer protects the Ni 81Fe19layer from oxidation. A diagram of the measurement configuration, adapted from a basic broadband FMR setup, is shown in Fig.1. The microwave signal is conveyed to and from the sam ple through a coplanar waveguide (CPW) with a 400 µm wide center conductor and 50 Ω characteristic impedance, which gives an estimated rf field of 2.25 Oe rms with the inpu t power of +30 dBm. We examined the second harmonic generation with fundamenta l frequencies at 6.1 GHz and 2.0 GHz. The cw signal from the rf source is first amplified by a solid state am- plifier, then the signal power is tuned to the desirable level by an adj ustable attenuator. Harmonics of the designated input frequency are attenuated by t he bandpass filter to less than the noise floor of the spectrum analyzer (SA). The isolator limit s back-reflection of the filtered signal from the sample into the rf source. From our ana lysis detailed in a later section of this manuscript, we found the second harmonic magnitud e to be proportional to 2FIG. 1. Experimental setup and the coordinate system, θ= 45◦; see text for details. EM: electromagnet; SA: spectrum analyzer. Arrows indicate the transmission of rf signal. the product of the longitudinal and transverse rf field strengths , and thus place the center conductor of CPW at 45◦from H Bto maximize the Hrf yHrf zproduct. The rf signal finally reaches the SA for measurements of the power of both the funda mental frequency and its second harmonic. Fig.2(a) demonstrates representative field-swept FMR absorptio n and the second har- monic emission spectra measured by the SA as 6.1 GHz and 12.2 GHz pea k intensities as a function of the bias field H B. We vary the input rf power over a moderate range of +4 - +18 dBm, and fit the peaks with a Lorentzian function to extract t he amplitude and the linewidth of the absorbed (∆ Pω) and generated (∆ P2ω) power. Noticeably, the second harmonic emission peaks have a much smaller linewidth, ∆ H1/2∼10 Oe over the whole power range, than those of the FMR peaks, with ∆ H1/2∼21 Oe. Plots of the absorption and emission peak amplitudes as a function of the input 6.1 GHz power, shown in Fig.2(b), clearly indicate a linear dependence of the FMR absorption and a quad ratic dependence of the second harmonic generation on the input rf power. Taking th e ratio of the radiated second harmonic power to the absorbed power, we have a convers ion rate of 3.7 ×10−5/W, as shown in Fig.2(c). Since the phenomenon summarized in Fig.2 is clearly not a threshold effe ct, we look into the second-harmonic analysis of the LLG equation with small rf fields , which is readily de- scribed in Gurevich and Melkov’s text for circular precession relevan t in the past for low-M s 3FIG. 2. Second harmonic generation with ω/2π= 6.1 GHz. a) left panel : 6.1 GHz input power +17.3 dBm; right panel : 6.1 GHz input power +8.35 dBm. b) amplitudes of the ω(FMR) and generated 2 ωpeaks as a function of input power Pω; right and top axes represent the data set in log-log plot (green), extracting the power index; c) rati o of the peak amplitudes of FMR and second harmonic generation as a function of the input 6.1 GHz power; green: log scale. 4ferrites18. For metallic thin films, we treat the elliptical case as follows. As illustra ted in Fig.1, the thin film is magnetized in the yzplane along /hatwidezby the bias field H B, with film- normal direction along /hatwidex. The CPW exerts both a longitudinal rf field hrf zand a transverse rf field hrf yof equal strength. First consider only the transverse field hrf y. In this well es- tablished case, the LLG equation ˙m=−γm×Heff+αm×˙mis linearized and takes the form ˙/tildewidermx ˙/tildewidermy = −α(ωH+ωM)−ωH ωH+ωM−αωH /tildewidermx /tildewidermy + γ/tildewiderhrf y 0 (1) , whereγis the gyromagnetic ratio, αis the Gilbert damping parameter, ωM≡γ4πMs, and ωH≡γHz. Introducing first order perturbation to mx,yunder additional longitudinal hrf z and neglecting the second order terms, we have ˙/tildewidermx+˙/tildewidest∆mx ˙/tildewidermy+˙/tildewidest∆my = −α(ωH+ωM)−ωH ωH+ωM−αωH /tildewidermx+/tildewidest∆mx /tildewidermy+/tildewidest∆my + γ/tildewiderhrf z/tildewidermy −γ/tildewiderhrf z/tildewidermx + γ/tildewiderhrf y 0 (2) Subtracting (1) from (2) and taking/tildewiderhrfy,z=Hrf y,ze−iωt,/tildewidemx,y= (Hrf y/Ms)e−iωt/tildewideχ⊥,/bardbl(ω), the equation for the perturbation terms is ˙/tildewidest∆mx ˙/tildewidest∆my = −α(ωH+ωM)−ωH ωH+ωM−αωH /tildewidest∆mx /tildewidest∆my +Hrf zHrf y Mse−i2ωt γ/tildewiderχ/bardbl(ω) −γ/tildewiderχ⊥(ω) (3) Sinceχ⊥is one order of magnitude smaller than χ/bardbl, we neglect the term −γ/tildewiderχ⊥(ω). In complete analogy to equation (1), the driving term could be viewed as an effective transverse field of Hrf z(Hrf y/Ms)/tildewiderχ/bardbl(ω)e−i2ωt, and the solutions to equation (3) would be /tildewidest∆mx= (Hrf zHrf y/M2 s)/tildewiderχ/bardbl(ω)/tildewiderχ⊥(2ω)e−i2ωt,/tildewidest∆my= (Hrf zHrf y/M2 s)/tildewiderχ/bardbl(ω)/tildewiderχ/bardbl(2ω)e−i2ωt. We can compare the power at frequency fand 2fnow that we have the expressions for both the fundamental and second harmonic components of the pr ecessing M. The time- averaged power per unit volume could be calculated as /angbracketleftP/angbracketright= [/integraltext2π ω 0P(t)dt]/(2π/ω), P(t) = −∂U/∂t= 2M∂H/∂twhere only the transverse components of MandHcontribute to P(t). Using the expression for /angbracketleftP/angbracketright,MandH, we have Pω=ωH2 y,rfχ(ω)′′ /bardbland P2ω= 2ωH2 z,rf(Hrf y/Ms)2|˜χ(ω)/bardbl|2χ(2ω)′′ /bardbl, from which we conclude that under H Bfor FMR at frequency f=ω/(2π), we should see a power ratio P2ω/Pω= 2(Hrf z/Ms)2χ(ω)′′ /bardblχ(2ω)′′ /bardbl (4) WithMs= 844 Oe, α= 0.007 as measured by FMR for our Ni 81Fe1930 nm sample and 2.25 Oe rf field amplitude at input power of 1 W for the CPW, we have a ca lculated 2 f/f 5power ratio of 1.72 ×10−5/W, which is in reasonable agreement with the experimental data 3.70×10−5/W as shown in Fig.2(c). To compare this result with the ferrite exper iment in ref.[1], we further add the factor representing the ratio of FMR ab sorption to the input rf power, which is 3 .9×10−2in our setup. This leads to an experimental upconversion ratio of 1.44×10−6/W in ref.[1]’s definition (∆ P2ω/Pω in2), compared with 7 .1×10−10/W observed in Mg 70Mn8Fe22O (Ferramic R-1 ferrite). Examining Eq.(4), we noticethatthereshouldbetwo peaksinthefield -swept 2femission spectrum: the first coincides with the FMR but with a narrower linewid th due to the term |˜χ(ω)/bardbl|2, and the second positioned at the H Bfor the FMR with a 2 finput signal due to the term χ(2ω)′′ /bardbl. The second peak should have a much smaller amplitude. Due to the fie ld limit of our electromagnet, we could not reach the bias field required f or FMR at 12.2 GHz under this particular configuration and continued to verify Eq.(4) a t a lower frequency of 2.0 GHz. We carried out an identical experiment and analysis and observ ed an upconversion efficiency of 0.39 ×10−3/W for the 4.0 GHz signal generation at 2.0 GHz input, again in reasonable agreement with the theoretical prediction 1.17 ×10−3/W. Fig.3 demonstrates the typical line shape of the4 GHzspectrum, in which the input 2 GHzpowe r being +18.9 dBm. A second peak at the H Bfor 4 GHz FMR is clearly visible with a much smaller amplitude and larger linewidth than the first peak, qualitatively consistent with Eq.(4). A theoretical line (dashed green) from equation (4) with fixed damping parameter α= 0.007 is drawn to compare with the experimental data. The observed second peak a t the 2fresonance H B shows a much lower amplitude than expected. We contribute this diffe rence to the possible 2fcomponent in the rf source which causes the 2 fFMR absorption. The blue line shows the adjusted theoretical line with consideration of this input signal impurity. Summary : We have demonstrated a highly efficient frequency doubling effect in thin- film Ni 81Fe19for input powers well below the Suhl instability threshold. An analysis of the intrinsically nonlinear LLG equation interprets the observed phe nomena quantitatively. The results explore new opportunities in the field of rf signal manipula tion with CMOS compatible thin film structures. We acknowledge Stephane Auffret for the Ni 81Fe19sample. We acknowledge support from the US Department of Energy grant DE-EE0002892 and Natio nal Science Foundation 6FIG. 3. 4 GHz generation with input signal at 2 GHz, +18.9 dBm. A second peak at the bias field for 4 GHz FMR is clearly present; red dots: experimental data ; dashed green: theoretical; blue: adjusted theoretical with input rf impurity. See text for de tails. ECCS-0925829. REFERENCES 1W. P. Ayres, P. H. Vartanian, and J. L. Melchor, J. Appl. Phys. 27, 188 (1956) 2N. Bloembergen and S. Wang, Phys. Rev. 93, 72 (1954) 3H. Suhl, J. Phys. Chem. Solids. 1, 209 (1957) 4J. D. Bierlein and P. M. Richards, Phys. Rev. B 1, 4342 (1970) 5G. P. Ridrigue, J. Appl. Phys. 40, 929 (1969) 6V. G. Harris, IEEE Trans. Magn. 48, 1075 (2012) 7V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. B aither, G. Schmitz and S. O. Demokritov, Nature Mat. 11, 1028 (2012) 8A. Berteaud and H. Pascard, J. Appl. Phys. 37, 2035 (1966) 9T. Gerrits, P. Krivosik, M. L. Schneider, C. E. Patton, and T. J. Silv a, Phys. Rev. Lett. 98, 207602 (2007) 10H. M. Olson, P. Krivosik, K. Srinivasan, and C. E. Patton, J. Appl. Ph ys.102, 023904 (2007) 711M. Bao, A. Khitun, Y. Wu, J. Lee, K. L. Wang, and A. P. Jacob, Appl. Phys. Lett. 93, 072509 (2008) 12Y. Khivintsev, J. Marsh, V. Zagorodnii, I. Harward, J. Lovejoy, P . Krivosik, R. E. Camley, and Z. Celinski, Appl. Phys. Lett. 98, 042505 (2011) 13J. Marsh, V. Zagorodnii, Z. Celinski, and R. E. Camley, Appl. Phys. Le tt.100, 102404 (2012) 14M. Yana, P. Vavassori, G. Leaf, F.Y. Fradin, and M. Grimsditch, J. M agn. Magn. Mater 320, 1909 (2008) 15V. E. Demidov, H. Ulrichs, S. Urazhdin, S. O. Demokritov, V. Besson ov, R. Gieniusz, and A. Maziewski, Appl. Phys. Lett. 99, 012505 (2011) 16C. Bi, X. Fan, L. Pan, X. Kou, J. Wu, Q. Yang, H. Zhang, and J. Q. Xia o, Appl. Phys. Lett.99, 232506 (2011) 17S. E. Bushnell, W. B. Nowak, S. A. Oliver, and C. Vittoria, Rev. Sci. In strum.63, 2021 (1992) 18A. G. Gurevich and G. A. Melkov, Magnetization Oscillation and Waves ( CRC, Boca Raton, 1996) 8

1910.11200v1.Spin_waves_in_ferromagnetic_thin_films.pdf

arXiv:1910.11200v1 [cond-mat.mes-hall] 24 Oct 2019Spin waves in ferromagnetic thin films Zhiwei Sun School of Mathematical Sciences, Soochow University, Suzh ou, China Jingrun Chen∗ School of Mathematical Sciences, Soochow University, Suzh ou, China and Mathematical Center for Interdisciplinary Research, Sooc how University, Suzhou, China (Dated: October 25, 2019) A spin wave is the disturbance of intrinsic spin order in magn etic materials. In this paper, a spin wave in the Landau-Lifshitz-Gilbert equation is obtained b ased on the assumption that the spin wave maintains its shape while it propagates at a constant ve locity. Our main findings include: (1) in the absence of Gilbert damping, the spin wave propagat es at a constant velocity with the increment proportional tothe strength of the magnetic field ; (2) in the absence of magnetic field, at a given time the spin wave converges exponentially fast to its initial profile as the damping parameter goes to zero and in the long time the relaxation dynamics of th e spin wave converges exponentially fast to the easy-axis direction with the exponent proportio nal to the damping parameter; (3) in the presence of both Gilbert damping and magnetic field, the s pin wave converges to the easy-axis direction exponentially fast at a small timescale while pro pagates at a constant velocity beyond that. These provides a comprehensive understanding of spin waves in ferromagnetic materials. PACS numbers: 05.45.Yv, 75.70.-i, 75.78.-n I. INTRODUCTION A spin wave is the disturbance of intrinsic spin order in magnetic mater ials. It is usually excited using magnetic fields and offers unique properties such as charge-less propagatio n and high group velocities, which are important for signal transformations and magnetic logic applications [1–6]. ThepropagationofspinwavesisdescribedbytheLandau-Lifshitz- Gilbert(LLG) equation[7,8]in thedimensionless form mt=−m×h−αm×(m×h), (1) where the magnetization m= (m1,m2,m3)Tis a three dimensional vector with unit length, αis the Gilbert damping parameter. The effective field hincludes the exchange term, the anisotropy term with easy axis alon g the x-axis and the anisotropy constant q, and the external field h= ∆m+qm1e1+hexte1. (2) hextis the strength of the external field applied along the x-axis with e1the unit vector. This model is often used to describe the magnetization dynamics in ferromagnetic thin films. From a theoretical perspective, a spin wave is known as a solitotary wave, which appears as the solution of a weakly nonlinear dispersive partial differential equation. In LLG equation ( 1)-(2), a soliton is caused by the cancellation of nonlinear and dispersive effects in the magnetic material. Solitons are of interests for quite a long time [9–13]. Most of works consider the one dimensional case and drop the damping te rm [9, 10, 13]. In [11], using the stereographic projection, the authors found that the presence of Gilbert damp ing was merely a rescaling of time by a complex constant. However, this was found to be valid only for a single spin in a constant magnetic field [12]. In this work, we give a comprehensive study of an explicit spin wave in t he LLG equation. Our starting point is that the spin wave maintains its shape while it propagates at a consta nt velocity and the derivation is based on the generalization of the method of characteristics. The main findings a re: (1) in the absence of Gilbert damping, the spin wave propagates at a constant velocity with the increment pro portional to the strength of the magnetic field; (2) in the absence of magnetic field, at a given time the spin wave converg es exponentially fast to its initial profile as the damping parameter goesto zeroand in the long time the relaxationdy namics of the spin waveconvergesexponentially ∗Electronic address: jingrunchen@suda.edu.cn2 fast to the easy-axis direction with the exponent proportional to the damping parameter; (3) in the presence of both Gilbert damping and magnetic field, the spin wave converges to the ea sy-axis direction exponentially fast at a small timescale while propagates at a constant velocity beyond that. II. DERIVATION AND RESULTS As mentioned above, we start with the assumption that a spin wave m aintains its shape while it propagates at a constant velocity. This can be seen from the method characterist ics in simple situations. In 1D when α=q=hext= 0, one can check that m(x,t) = cosθ0 sinθ0cos/parenleftig c cosθ0(x+ct)/parenrightig sinθ0sin/parenleftig c cosθ0(x+ct)/parenrightig (3) solvesmt=−m×mxx. Hereθ0is determined by the initial condition and u=x+ctis the characteristic line. (3) provides a solitary solution with the traveling speed c. A detailed derivation of (3) can be found in Chapter 2 of [13]. A generalization of the method of characteristics yields a spin wave t omt=−m×∆m m(x,t) = cosθ0 sinθ0cosv cosθ0 sinθ0sinv cosθ0 , (4) wherev=c1x+c2y+c3z+(c2 1+c2 2+c2 3)t=c·x+(c·c)twithc= (c1,c2,c3)T. The speed field is cwith magnitude |c|. Actually, both (3) and (4) can be rewritten as m(x,t) = cosθ0 sinθ0cos(w0·x+ϕ(t)) sinθ0sin(w0·x+ϕ(t)) , (5) wherew0=c/cosθ0andϕ(t) =/parenleftbig |c|2/cosθ0/parenrightbig t. (3)-(5) are obtained in the absence of Gilbert damping. In order to take the Gilbert damping and the other terms in (2) into account, we make an ansatz for the spin wave profile in the following form m(x,t) = cosθ(t) sinθ(t)cos(w0·x+ϕ(t)) sinθ(t)sin(w0·x+ϕ(t)) , (6) whereθandϕare independent of xand only depend on t. Substituting (6) into (2) and (1) and denoting w0·x+ϕ(t) byu(x,t), we have h= 0 −|w0|2sinθcosu(x,t) −|w0|2sinθsinu(x,t) +q cosθ 0 0 +hext 1 0 0 , m×h= 0 |w0|2sinθcosθsinu(x,t) −|w0|2sinθcosθcosu(x,t) +q 0 sinθcosθsinu(x,t) −sinθcosθcosu(x,t) +hext 0 sinθsinu(x,t) −sinθcosu(x,t) , m×(m×h) = −|w0|2sin2θcosθ |w0|2sinθcos2θcosu(x,t) |w0|2sinθcos2θsinu(x,t) +q −sin2θcosθ sinθcos2θcosu(x,t) sinθcos2θsinu(x,t) +hext −sin2θ sinθcosθcosu(x,t) sinθcosθsinu(x,t) , and mt= −θtsinθ θtcosθcosu(x,t)−ϕtsinθsinu(x,t) θtcosθsinu(x,t)+ϕtsinθcosu(x,t) . After algebraic simplifications, we arrive at /braceleftbigg θt=−α(|w0|2+q)sinθcosθ−αhextsinθ ϕt= (|w0|2+q)cosθ+hext. (7)3 A. The absence of Gilbert damping Whenα= 0, we have θ=θ0andϕ=/parenleftbig (|w0|2+q)cosθ0+hext/parenrightbig t. Therefore we have the solution m= cosθ0 sinθ0cos/parenleftbig w0·x+t/parenleftbig |w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig sinθ0sin/parenleftbig w0·x+t/parenleftbig |w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig . (8) Note that this recovers (5) when q= 0 and hext= 0. It is easy to see that the spin wave (8) propagates at a consta nt velocity. The increment of the velocity field is qcosθ0w0 |w0|2with magnitude|qcosθ0| |w0|, due to the magnetic anisotropy. The increment of the velocity field is hextw0 |w0|2with magnitude|hext| |w0|, due to the magnetic field. B. The absence of magnetic field Whenhext= 0, (7) reduces to /braceleftbigg θt=−α(|w0|2+q)sinθcosθ ϕt= (|w0|2+q)cosθ. (9) For the first equation in (9), assuming 0 ≤θ0< π/2, by separation of variables, we have α(|w0|2+q)t+C1= lncotθ, whereC1is a constant determined by the initial condition. Denote˜t=α(|w0|2+q)t+C1. It follows that tanθ=e−˜t, (10) from which one has cosθ=1/radicalbig 1+e−2˜t, (11) sinθ=1/radicalbig 1+e2˜t. (12) Whent= 0, (11) turns to cosθ0=1√ 1+e−2C1, (13) from which we can determine C1by the initial condition θ0. As forϕ, from the second equation in (9), one has that dϕ dθ=dϕ dt·dt dθ=−1 αsinθ. Therefore αϕ=−/integraldisplaydθ sinθ=1 2ln/parenleftbigg1+cosθ 1−cosθ/parenrightbigg +C2= lncot1 2θ+C2, (14) where C2=−1 2ln/parenleftbigg1+cosθ0 1−cosθ0/parenrightbigg . Substituting (11) and (13) into (14) yields ϕ=1 αln/parenleftigg e˜t+/radicalbig e2˜t+1 eC1+√ e2C1+1/parenrightigg =1 α/parenleftbigg lncot1 2θ−lncot1 2θ0/parenrightbigg . (15)4 In short summary, the spin wave when α/negationslash= 0 takes the form m=1/radicalbig 1+e2˜t e˜t cos(w0·x+ϕ) sin(w0·x+ϕ) . (16) The above derivation is valid when 0 ≤θ0< π/2. Ifπ/2< θ0≤π, we choose the other solution of (11) cosθ=−1/radicalbig 1+e−2˜t, (17) and ϕ=−1 αln/parenleftigg e˜t+/radicalbig e2˜t+1 eC1+√ e2C1+1/parenrightigg . (16) remains unchanged. Whenα→0, we have ˜t→C1and lim α→0m=1√ 1+e2C1 eC1 cos/parenleftig w0·x+ lim α→0ϕ/parenrightig sin/parenleftig w0·x+ lim α→0ϕ/parenrightig . By L’Hospital’s rule, one has that lim α→0ϕ= lim α→0d dα/parenleftigg ln/parenleftigg e˜t+/radicalbig e2˜t+1 eC1+√ e2C1+1/parenrightigg/parenrightigg = lim α→0e˜t /radicalbig e2˜t+1(|w0|2+q)t=eC1 √ eC1+1(|w0|2+q)t.(18) Therefore it follows that lim α→0m= cosθ0 sinθ0cos/parenleftbig w·x+cosθ0/parenleftbig |w0|2+q/parenrightbig t/parenrightbig sinθ0sin/parenleftbig w·x+cosθ0/parenleftbig |w0|2+q/parenrightbig t/parenrightbig . (19) This is exactly the solution (8) when hext= 0. In addition, when α→0, bothθandϕconverges exponentially fast to initial conditions; see equations (1 1), (12), (13), and (18). Therefore, at a giventime t, (19) convergesexponentiallyfast to the initial spin wave(8) when hext= 0 with the exponent proportionalto the damping parameter α. Moreover,in the long time, i.e., when t→+∞,˜t→+∞ andθ→0, (16) converges to (1 ,0,0)T(the easy-axis direction) exponentially fast with the rate proport ional to the damping parameter α. Whenπ/2< θ0≤π, from (17), we have that (16) converges to ( −1,0,0)T(again the easy-axis direction) exponentially fast with the rate proportional to the dam ping parameter α. It is easy to check that the right-hand side of (19) is the solution of (1) when hext= 0 with the initial condition θ0=π/2. Therefore, Gilbert damping does not have any influence on magne tization dynamics in this case. In [11], the authors used the stereographic projection and obser ved that the effect of Gilbert damping was only a rescaling of time by a complex constant. However, this was latter fo und to be valid only for a single spin in a constant magnetic field [12]. Our result provides an explicit characterization of magnetization dynamics in the presence of Gilbert damping. C. The presence of both Gilbert damping and magnetic field It is difficult to get the explicit solution of (7) in general. To understan d the magnetization dynamics, we use the method of asymptotic expansion. For small external magnetic field ,θandϕadmit the following expansions θ(t,hext) =θ0(t)+θ1(t)hext+θ2(t)h2 ext+···, ϕ(t,hext) =ϕ0(t)+ϕ1(t)hext+ϕ2(t)h2 ext+···.5 Therefore one has that θt(t,hext) =θ0 t(t)+θ1 t(t)hext+θ2 t(t)h2 ext+···, (20) ϕt(t,hext) =ϕ0 t(t)+ϕ1 t(t)hext+ϕ2 t(t)h2 ext+···. (21) On the other hand, from (7), it follows that θt=−α(|w0|2+q)sinθ0cosθ0−/parenleftbig α(|w0|2+q)θ1cos2θ0+αsinθ0/parenrightbig hext+···, (22) ϕt= (|w0|2+q)cosθ0+/parenleftbig −(|w0|2+q)θ1sinθ0+1/parenrightbig hext+···. (23) Combining (20) and (21) with (22) and (23), for the zero-order te rm, one has /braceleftbigg θ0 t=−α(|w0|2+q)sinθ0cosθ0 ϕ0 t= (|w0|2+q)cosθ0 , (24) which recovers (9) with solution (10) and (15). As for the first-order term, one has that /braceleftbigg θ1 t=−α(|w0|2+q)θ1cos2θ0−αsinθ0 ϕ1 t=−(|w0|2+q)θ1sinθ0+1. (25) Using variation of parameters, one can assume θ1=C(t) e˜t+e−˜tand it follows that C′(t) =−α(e˜t+e−˜t)sinθ0=−α(tanθ0+tan−1θ0)sinθ0. Since /integraldisplay −αtanθ0sinθ0dt=/integraldisplay −α(|w0|2+q)sinθ0cosθ0∗(|w0|2+q)−1sinθ0 cos2θ0dt =/integraldisplay (|w0|2+q)−1sinθ0 cos2θ0dθ0 =(|w0|2+q)−11 cosθ0, and /integraldisplay −αtan−1θ0sinθ0dt=/integraldisplay −αcosθ0dt =−α(|w0|2+q)−1ϕ0, one can get C(t) = (|w0|2+q)−1(1 cosθ0−αϕ0), and it follows that θ1= (|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0). (26) Substituting the first equation in (24) into the second equation in (2 5), one has −/integraldisplay (|w0|2+q)θ1sinθ0dt =1 α/integraldisplayθ1 cosθ0dθ0 =1 α(|w0|2+q)/integraldisplay tanθ0−sinθ0(lncot1 2θ0+C2)dθ0(using (26)) =−t+1 |w0|2+q(ϕ0cosθ0+α−1C1), and thus ϕ1=1 |w0|2+q(ϕ0cosθ0+α−1C1). (27)6 Therefore, when hextis small, it has the approximate solution /braceleftbigg θ∗=θ0+(|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0)hext ϕ∗=ϕ0+(|w0|2+q)−1(ϕ0cosθ0+α−1C1)hext(28) withθ0andϕ0satisfying (9). From (10) and (15), θ0converges exponentially fast to the easy-axis direction, while ϕ0grows linearly. Therefore, from (26), θ1converges exponentially fast to 0 as well with a larger exponent. Th is relaxation dynamics happens at a small timescale. Meanwhile, from (25), the difference between ϕ∗andϕ0satisfies ϕ∗ t−ϕ0 t=ϕ1 thext=−(|w0|2+q)θ1sinθ0hext+hext. (29) Sinceθ1sinθ0convergesto 0 at a small timescale, the dynamics of ϕ∗−ϕ0is determined by the external field at longer timescales. As a consequence, the increment of the velocity field is hextw0 |w0|2with magnitude|hext| |w0|. This validates the Walker’s ansatz [14] for a spin wave. Whenπ/2< θ0≤πand the magnetic field is applied along the negative x-axis, and if ( |w|2+q)|cosθ0| ≤hext, the result above will be correct. Note that θ0=π/2 does not fall into the above two cases since the magnetization dyn amics will change the spin wave profile. In fact, as t→+∞,θ→0 if the magnetic field is applied along the positive x-axis direction and θ→π if the magnetic field is applied along the negative x-axis direction. III. CONCLUDING REMARKS In this work, we study the magnetization dynamics in Landau-Lifshit z-Gilbert equation. By generalizing the method of characteristics, we are able to have an explicit characte rization of spin wave dynamics in the presence of both Gilbert damping and magnetic field. Gilbert damping drives the spin wave converge exponentially fast to the easy-axis direction with the exponent proportional to the damping parameter at a small timescale and the magnetic field drives the spin wave propagate at a constant velocity at longer timescales. It will be of interests whether the technique developed here applies to the antiferromagnetic case [15, 16] and how rigorous the results obtained here can be proved from a mathemat ical perspective. IV. ACKNOWLEDGEMENTS We thank Professor Yun Wang for helpful discussions. This work wa s partially supported by National Natural Science Foundation of China via grant 21602149 and 11971021. [1] M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Applied Physics Letters 87, 153501 (2005). [2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, et al., Nature 464, 262 (2010). [3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebra nds, Nature Physics 11, 453 (2015). [4] S. Woo, T. Delaney, and G. S. D. Beach, Nature Physics (201 7). [5] A. V. Chumak and H. Schultheiss, Journal of Physics D: App lied Physics 50, 300201 (2017). [6] M. Langer, R. A. Gallardo, T. Schneider, S. Stienen, A. Ro ld´ an-Molina, Y. Yuan, K. Lenz, J. Lindner, P. Landeros, and J. Fassbender, Physical Review B 99(2019). [7] L. Landau and E. Lifshitz, Physikalische Zeitschrift de r Sowjetunion 8, 153 (1935). [8] T. Gilbert, Physical Review 100, 1243 (1955). [9] K. Nakamura and T. Sasada, Physics Letters A 48, 321 (1974). [10] H. J. Mikeska, Journal of Physics C: Solid State Physics 11, L29 (1977). [11] M. Lakshmanan and K. Nakamura, Physical Review Letters 53, 2497 (1984). [12] E. Magyari, H. Thomas, and R. Weber, Physical Review Let ters56, 1756 (1986). [13] B. Guo and S. Ding, Landau-Lifshitz Equation (World Scientific, 2007). [14] N. L. Schryer and L. R. Walker, Journal of Applied Physic s45, 5406 (1974). [15] H. J. Mikeska, Journal of Physics C: Solid State Physics 13, 2913 (1980). [16] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Reviews of Modern Physics 90, 015005 (2018).

1501.05216v1.Lévy_walks_on_lattices_as_multi_state_processes.pdf

Lévy walks on lattices as multi-state processes Giampaolo Cristadoro y, Thomas Gilbert z, Marco Lenci y§ and David P. Sanders k yDipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy zCenter for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, C. P. 231, Campus Plaine, B-1050 Brussels, Belgium § Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy kDepartamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 México D.F., Mexico Abstract. Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolveaccordingtoasetofdelaydifferentialequations,amenabletoanalytictreatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion. Submitted to: J. Stat. Mech. Theor. Exp. PACS numbers: 05.40.Fb, 05.60.-k, 02.50.-r, 02.30.Ks, 02.70.-c E-mail: giampaolo.cristadoro@unibo.it, thomas.gilbert@ulb.ac.be, marco.lenci@unibo.it, dpsanders@ciencias.unam.mx 1. Introduction Stochastic processes in which independent particles scatter randomly at finite speed and may occasionally propagate over a long distance in single bouts are known as Lévy walks. Use of these models has become ubiquitous in the study of complex diffusive processes [1–5]. They are particularly relevant to situations such that the probability of a long jump decays slowly with its length [6]. The scale-free superdiffusive motion of Lévy walkers [7] has been identified as an efficient foraging strategy [8], spurring a large interest in these models, including with regards to human mobility patterns [9,10].arXiv:1501.05216v1 [cond-mat.stat-mech] 21 Jan 2015Lévy walks on lattices as multi-state processes 2 A central quantity in this formalism is the distribution of free path lengths, which gives the probability that a particle propagates over a distance xbetween two successive scattering events. Its asymptotic scaling determines how the moments of the displacement asymptotically scale with time. Assuming the probability of a free path of lengthxscales asymptotically as x1, one finds, in the so-called velocity picture of Lévy walks, the following scaling laws for the mean squared displacement after time t[11–15]: hr2it8 >>>>>>>< >>>>>>>:t2; 0<< 1; t2=logt;  = 1; t3; 1<< 2; tlogt;  = 2; t; > 2 ;(1) see below for a precise definition of this quantity. A scaling parameter value 0<2is suchthatthevarianceofthedistributionoffreepathsdiverges, inwhichcasetheprocess is often called scale-free [9]. Correspondingly, the asymptotic divergence of the mean squared displacement gives rise to anomalous transport in the form of superdiffusion. In a recent paper [16], we considered Lévy walks on lattices and generalized the standard description of the velocity picture of Lévy walks, according to which a new jump event takes place as soon as the previous one is completed, to include an exponentially-distributed waiting time separating successive jumps. As emphasised in reference [16], this additional phase induces a differentiation between the states of particles which are in the process of completing a jump and those that are waiting to start a new one. We call the former states propagating and the latter scattering. The distinction between these states leads to a new theoretical framework of Lévy walks in terms of multistate processes [17], whereby the generalized master equation approach to continuous time random walks [18,19] translates into a set of delay differential equations for the corresponding distributions. It is the purpose of this paper to show that a complete characterization of the solutions of such multistate processes can be obtained, which yields exact time- dependent analytic expressions of their mean squared displacement. These expressions can, on the one hand, be compared with the asymptotic solutions already reported in [16], and, on the other, also prove useful when the asymptotic regime is not reached, which is often the case with studies dealing with observational data. Such a situation arises when the probability of a transition from scattering to propagating states is small. A particle will then spend most of its time undergoing transitions among scattering states, performing a seemingly standard continuous-time random walk, only seldom undergoing a transition to a propagating state, during which it moves ballistically over a distance distributed according to the scaling parameter . Ifis small enough ( 2), these occurrences, although rare, have a dramatic effect on the asymptotic scaling properties of the mean squared displacement, such thatLévy walks on lattices as multi-state processes 3 a crossover from normal to anomalous diffusive transport is observed. This crossover time may in some situations, however, be much larger than the times accessible to measurements. Here we provide analytic results which give a precise characterisation of the emergence of an anomalous contribution out of a normally scaling one, showing how the asymptotic scaling (1) comes about. This is of particular relevance to the regime = 2, for which the logarithmic divergence in time of the mean squared displacement is indeed very slow. Our results are tested and confirmed by numerical simulations of the processes under consideration. The paper is organized as follows. In section 2, we introduce the multi-state description of Lévy walks with respect to scattering and propagating states, and define the transition probabilities between them in terms of two parameters, one relating to the probability of a transition from a scattering to a propagating state and the other characterizing the asymptotic scaling of the free path distribution. The fraction of particles in a scattering state evolves according to a delay differential equation which is derived and solved in section 3. These results are exploited in section 4 to obtain the mean squared displacement of the processes. A comparison with numerical simulations is provided in section 5. Conclusions are drawn in section 6. 2. Lévy walks as multi-state processes We consider a continuous-time random walk on a square lattice which generalizes the standard model of Montroll & Weiss [18] in that it assumes displacements to non- neighbouring sites occur in a time span given by the ratio of the distance traveled to the walker’s speed v, which itself remains fixed throughout. In this sense, the model is similar to the so-called velocity picture of Lévy walks [20], the difference being that successive propagations, irrespective of their lengths, are separated by random waiting times, which we assume have exponential distributions. The model itself is not new and, in some sense, is a simplification of other models of Lévy walks interrupted by rests; see reference [21]. As we show below, however, the combination of a discretized spatial structure and exponentially-distributed waiting times yields a novel description of the process in terms of delay differential equations amenable to analytic treatment. A natural distinction arises between the states of walkers which are moving across the lattice structure and those at rest. We call propagating the state of a particle which is undergoing a displacement phase and scattering the state of a particle at rest, waiting to start a new displacement. In the framework of intermittent random walks [22], the former state is usually referred to as ballistic and the latter as diffusive or reactive, depending on context. A scattering state may therefore be thought of as one associated with a local diffusive process bound to the scale `of the distance between neighbouring lattice sites. The interplay between the two states is as follows. A particle in a propagating state switches to a scattering state upon completing a displacement. A particle in a scattering state, on the other hand, can make transitions to both scattering andLévy walks on lattices as multi-state processes 4 propagating states; as soon as its randomly-distributed waiting time has elapsed, it moves on to a neighbouring site and, in doing so, may switch to a propagating state and continue its motion to the next site, or start anew in a scattering state. Consider a d-dimensional cubic lattice of individual cells n2(`Z)d. The state of a walker at position n(n1;:::;nd)`and timetcan take on a countable number of different values, (k;j), specified by an integer k2N, and direction j2f1;:::;zg, where z2ddenotes the coordination number of the lattice, that is the number of different lattice directions. States (0;j)are associated with a scattering state, irrespective of directionj, while states (k;j)withk1refer to propagating states, with kbeing the remaining number of lattice sites the walker will travel in direction jto complete its displacement. Time evolution proceeds in steps, characterized by a waiting-time density function and a transition probability. A particle at position nin a scattering state will wait for a random time t, exponentially distributed with mean zR, before updating its state to(k;j)with probability k=z, simultaneously changing its location to n+ej, where ejdenotes the lattice vector (of length `) associated with direction j. In contrast, a particle in a propagating state (k;j),k1, will change its state to (k1;j)after a timeB`=v, simultaneously moving from site nton+ej. The waiting times associated with scattering states are drawn from a standard Poisson process, while the renewal process generated by the combination of scattering and propagating states has arbitrary holding times, whose distribution is determined by the transition probabilities k. The waiting-time density of the process is thus the function k(t) =( 1 Ret=R; k = 0; D(tB); k6= 0;(2) whereD(:)denotes the Dirac delta function. When a step takes place, the transition probability to go from state (k;j)to state (k0;j0)is p(k;j);(k0;j0)=( z1k0; k = 0; k1;k0j;j0k6= 0;(3) where
;
is the Kronecker symbol. A particle which makes a transition from the scattering state to a propagating state (k;j)willthereforetraveladistance (k+1)`awayindirection j, untiliteventuallycomes back to the scattering state and may change directions at the next transition. We now introduce a characterization of the transition probabilities kin terms of two parameters. The first, which we refer to as the scattering parameter , is denoted by ,01, and gives the total probability of a transition from the scattering state, zThe subscript R in Rstands for residence as in “residence time.” In contrast, B in Bstands for ballistic as in “ballistic propagation time.”Lévy walks on lattices as multi-state processes 5 k= 0, into a propagating state, k1: 1X k=1k=; (4) The remaining transition probability, 01; (5) is that of a transition from a scattering state into another scattering state. The value = 0thus corresponds to the absence of propagating states: the process is then a simple continuous-time random walk with transitions to nearest neighbouring sites only and exhibits normal diffusion with coefficient `2=(zR). The opposite extreme, = 1, assigns zero probability to transitions from scattering to scattering states, which means that every transition involves a displacement over a distance of at least two sites. Scattering states remain populated, however, due to the decay of propagating states. Thesecondparameter, >0, isthe scaling parameter ofthetransitionprobabilities k, which controls their asymptotic behaviour, k/k1(k1); (6) and determines the scaling law of the mean squared displacement (1). The specific form ofkhas no effect on this scaling law, but is does affect the time-dependent properties of the mean squared displacement. To be specific, we consider in this paper the following double-telescopic form for the transition probabilities k: k=(121)1 k12(k+ 1)1+ (k+ 2)1 (k1); (7) its structure is particularly helpful for some of the computations presented below and is motivated by our study of anomalous transport in the infinite-horizon periodic Lorentz gas [23]. In this respect, the model is slightly different from that presented in reference [16], where a simple telescopic structure of the transition probabilities was used. For future reference, we define k=1X j=kj; (8) to be the probability of a transition to a state larger than or equal to k, such that, in particular, 0= 1and1=, and, for the choice of transition probabilities (7), k=(121)1 k1(k+ 1)1 : (9)Lévy walks on lattices as multi-state processes 6 We also note the following two identities: kX j=1j= 1(121)1[(k+ 1)1(k+ 2)1] ; kX j=1jj=(121)1[1(k+ 1)2+ (k+ 2)22(k+ 2)1]:(10) 3. Fraction of particles in the scattering state A quantity which plays a central role in the analysis of the process generated by the waiting-time density (2) and transition probabilities (3) is the average return time to the scattering state, =1X k=0k(R+kB); (11) The average return time is finite only when x > 1. The process is then said to be positive recurrent . In the remainder, we restrict our attention to this range of parameter values. The null-recurrent case, when 0<1, for which the average return time to the scattering state diverges, is considered in reference [16]. The occupation probability of particles at site nand timet,P(n;t), is a sum of the probabilities over the different states, Pk;j(n;t): P(n;t) =zX j=11X k=0Pk;j(n;t): (12) In reference [16], we obtained the following set of delay differential equations for the time-evolution of these occupation probabilities: @tP0;j(n;t) =1 zRzX j0=11X k=0kP0;j0(n(k+ 1)ej;tkB)1 RP0;j(n;t) +1X k=0k;j(nkej;tkB); (13) @tPk;j(n;t) =1 zRzX j0=11X k0=1k+k01h P0;j0(nk0ej;t(k01)B) P0;j0(nk0ej;tk0B)i +1X k0=0h k+k0;j(nk0ej;tk0B) xGenerally speaking, kmust decay asymptotically faster than k2for the average return time to be finite.Lévy walks on lattices as multi-state processes 7 k+k0;j(nk0ej;t(k0+ 1)B)i ; (14) where the inclusion of terms k;jaccounts for the possibility of external sources, such thatk;j(n;t)0is the rate of injection of particles at site nand timetin the state (k;j). To simplify matters, we assume that these source terms are independent of the lattice direction and write k;j(n;t)z1k(n;t). Furthermore, we will usually let k;j(n;t) = 0fork1, which amounts to assuming that particles are injected only in a scattering state. The injection of particles in a propagating state is easily treatable as well, and will be considered explicitly in order to initiate the process in a stationary state of equations (13)–(14). We will, however, limit such considerations to this specific choice and avoid the possibility that source terms interfere with the asymptotic scaling of the process generated by particles initially injected in a scattering state, as might arise from alternative choices of injection rates of propagating states. Of particular interest is the fraction of particles in the scattering state, S0(t)X n2ZdzX j=1P0;j(n;t); (15) whose time-evolution is described by the following delay differential equation: _S0(t) =1 R1X k=1kS0(tkB)1 RS0(t) +1X k=0k(tkB): (16) Here,k(tkB)denotes a source term for the rate of injection of particles in state k, irrespective of their positions, that is, k(tkB) =P nk(n;tkB). The first two terms on the right-hand side of this equation have the typical gain and loss structure of jump processes. Indeed, the second term corresponds to particles lost by scattering states due to transitions to propagating states, which occur at rate =R. Each such transition is gained back after a delay given by the length of the ballistic segment in a propagating state. The sum of those terms yields the first term on the right-hand side of equation (16). Before turning to the integration of this differential equation, we note that the asymptotic value of S0(t),t!1, has a simple expression. By ergodicity of the process, the equilibrium ratio of particles in the scattering state is given by the ratio between the average time spent in the scattering state, R, and the average return time to it, , equation (11), lim t!1S0(t) =R : (17) Assuming a positive recurrent process, such as when the transition probabilities kare specified by equation (7) with >1, the average return time =R+(121)1B is finite, so that the asymptotic fraction of particles in a scattering state is strictlyLévy walks on lattices as multi-state processes 8 positive. For the particular choice of the parameters k, equation (7), the return time is given by equation (11) and equation (17) becomes lim t!1S0(t) =R R+(121)1B: (18) Note, however, that the convergence to this asymptotic value follows a power law whose exponent goes to zero as !1; see figure 1a. Furthermore, S0(t)converges to 0when the scaling parameter falls into the null recurrent regime, 0<1. Correspondingly, the fraction of particles in the propagating states, Sk(t)X n2ZdzX j=1Pk;j(n;t); (19) tends to lim t!1Sk(t) =kB R+(121)1B; (20) which follows from the observation that a particle that makes a transition to state k spends an equal amount of time in all states jsuch that 1jk. 3.1. Time-dependent fraction of particles in the scattering state The integration of equation (16) relies on the specification of initial conditions. For instance, the choice k S0(t) =( 0; t< 0; 1; t= 0;(21) corresponds, when all particles are injected at the lattice origin, to the injection rate k(n;t) =D(t)k;0n;0; (22) and will be henceforth referred to as the all-scattering initial condition . The time-dependent fraction of particles in the scattering state, S0(t),t>0, may thenbeobtainedbythemethodofsteps[24],whichconsistsofintegratingthedifferential equation (16) successively over the intervals kBt(k+ 1)B,k2N, matching the solutions at the upper and lower endpoints of successive intervals. For the all-scattering initial condition (21), the fraction of particles in the scattering state is, for times t0, S0(t) =et=R+bt=BcX k=1e(tkB)=RkX n=1a(njk)n R(tkB)n; (23) wherebt=Bc(resp.dt=Be, used below) denotes the largest (resp. smallest) integer smaller (resp. larger) than or equal to t=B, and each coefficient a(njk)is the sum of all kThe discussion below can be easily generalized, by linear superposition, to processes where particles in a scattering state are continuously injected into the process.Lévy walks on lattices as multi-state processes 9 possible combinations of products i1:::in, where the sequences fijgn j=1are such that i1+


+in=k, a(njk)=1 n!X 1i1;:::;ink i1+


+in=knY j=1ij: (24) Thecontributionsto a(njk)consistofalldistinctwaysoftravellingadistance kinnsteps, divided by their number of permutations; the derivation of equation (23) is provided in Appendix A. A comparison between the time-dependent fraction of particles in the scattering state (23) and the asymptotic value (17) is illustrated in figure 1a, where the difference between the two expressions is plotted vs. time for different values of the scaling parameter and a specific choice of the scattering parameter , with the transition probabilities kspecified by equation (7). ��-������������-���-���-� ���(�)-τ�/[τ�+ϵ τ�/(�-��-α)] (a)(121)1= 1 ��-���������������������������������� ���(�) (b)(121)1= 5102 Figure 1: Convergence in time of the fraction of particles in a scattering state, S0(t), computed from equation (23), to the asymptotic value (18). The transition probabilities are taken according to equation (7), with B= 1andR= 1 +(121)1. The value of the scattering parameter is chosen such that =(121) = 1(left panel) and 5102 (rightpanel). Theleftpanelshowsthedifferencebetween S0(t)anditsasymptoticvalue. In both panels, the scaling parameter takes the values = 3=2(green curve), 2(cyan curve), and 5=2(red curve). The right panel corresponds to a perturbative regime, such thatS0(t)can be approximated by a first-order polynomial in (darker curves) whose first-order coefficient varies with time; see equation (29). The agreement between the exact and the approximate solutions improves as the value of decreases. 3.2. Small-parameter expansion Suppose that the scattering parameter is small, 1, so that the overwhelming majority of transitions occur between scattering states, and only rarely does a particle make an excursion into a propagating state, which may, however, last long, depending on the value of the scaling parameter >1.Lévy walks on lattices as multi-state processes 10 The first-order expansion of equation (23) yields the following approximation of S0: S0(t)'1t R+bt=BcX k=1ktkB R; =S0(bt=BcB)tbt=BcB R1X k=dt=Bek: (25) That is to say, for 1,S0(t)is a sequence of straight line segments joining the values the function takes at integer multiples of the propagation time B, S0(kB)'1B Rh kkX j=1j (kj)i : (26) In this regime, S0(t)converges asymptotically to the constant value (17), which, for the choice of parameters (7), is given by equation (18), i.e., with all other parameters being fixed, lim t!1S0(t)'1(121)1B R: (27) For these parameters, we make use of identities (10) to evaluate equation (26), S0(kB)'1(121)1[1(k+ 1)1]B R; (28) or, for general time values, S0(t)'1(121)1nt Rh bt=Bc+ 1
1 bt=Bc+ 2
1i +B Rh 1 bt=Bc+ 1
2+ bt=Bc+ 2
2 2 bt=Bc+ 2
1io : (29) A comparison between this approximate value and the exact one, equation (23), is displayed in figure 1b, where the value of was taken to be large enough that the curves remain distinguishable. 4. Mean squared displacement Assuming initial injection of particles at the origin, the mean squared displacement of particles as a function of time, given by the second moment of the displacement vector r`n,hr2it=`2hn2it=`2P n2Zdn2P(n;t), wherer2=r
randn2=n
n, evolves according to the differential equationLévy walks on lattices as multi-state processes 11 d dthn2it=1 R1X k=0(2k+ 1)kS0(tkB) +1X k=1(2k1)1X k0=0k+k0(0;tkB);(30) where, incontrasttotheexpressionderivedinreference[16], wehavehereaddedpossible contributions from source terms k;j(n;t),k1, which we assume to be concentrated at site n=0only. To conform with stationarity of the process, propagating states should be uniformly injected in the time interval B<t0so they decay uniformly in the time interval 0<tB. We thus let ( k1) k(n;t) =1 Bkn;0(t)(t+B); (31) wherefkgk1is a sequence of positive numbers such thatP kk= 1and(:)is the Heaviside step function, such that the product (t)(t+B)guarantees the uniform distribution of propagating source terms in the desired time interval. Equation (30) transforms to d dthn2it=1 R1X k=0(2k+ 1)kS0(tkB) +1 B(2bt=Bc+ 1)1X k=1bt=Bc+k:(32) Integrating over time, we obtain the mean squared displacement: hn2it=B Rbt=BcX k=0(2k+ 1)kZt=Bk 0dxS0(xB) +1X k=1Zt=B 0dx(2bxc+ 1)bxc+k:(33) There are two contributions to this expression; the first arises from the fraction of particles in a scattering state, and the second from particles initially distributed among propagating states. This term transforms to 1X k=1Zt=B 0dx(2bxc+ 1)bxc+k=1X k=0bt=BcX j=1(2j1)j+k + 2(t=Bbt=Bc)(bt=Bc+ 1=2)1X k=1bt=Bc+k:(34) 4.1. Stationary regimes The stationary initial condition , i.e. such that the initial fractions of particles in scattering and propagating states are stationary solutions of equations (13)-(14), are realized if, in equation (31), we identify kwith the stationary fraction of particles in statek, k=kB R+(121)1B; (35)Lévy walks on lattices as multi-state processes 12 cf. equation (20), and let 0(n;t) =0D(t)n;0;  0=R R+(121)1B: (36) With this choice, the mean squared displacement (33) yields the exact expression hn2it=t (121)R+B 121+ tbt=BcX k=1(tkB)(2k+ 1)[k1(k+ 1)1] +B (121)R+Bbt=BcX k=12k1 k1+ 2(t=Bbt=Bc)bt=Bc+ 1=2 (bt=Bc+ 1)1 : (37) Treatingnt=B1asanintegerandletting H() nPn k=1kdenotethegeneralized harmonic numbers, the two terms with sums over kin the above expression evaluate respectively to nX k=1n1(nk)(2k+ 1)[k1(k+ 1)1]'14n1H(2) n + 2H(1) n;(38) for the term arising from the fraction of particles in a scattering state, and to nX k=1(2k1)k1= 2H(2) nH(1) n; (39) for the term arising from particles initially distributed among the propagating states. Depending on the value of the scaling parameter , we have the following three asymptotic regimes. 4.1.1. Normal diffusion For parameter values  > 2, the asymptotic properties of the harmonic numbers, lim n!1n1H(2) n = 0; lim n!1H(1) n =(1);(40) are such that only the terms arising from the fraction of particles in a scattering asymptotically contribute to equation (37): lim t!11 thn2it=121+[1 + 2(1)] (121)R+B; (41) which, up to a factor z1, is the diffusion coefficient of the process.Lévy walks on lattices as multi-state processes 13 4.1.2. Weak superdiffusion For the marginal parameter value = 2, the harmonic numbers in equation (38) evaluate to H(0) n=n; H(1) n= logn+ + O(n1);(42) where '0:577216is Euler’s constant. We therefore have the asymptotic limit of equation (37), lim t!11 tlog(t=B)hn2it=4 R+ 2B; (43) which is due to the fraction of particles in a scattering state alone. This result is, however, of limited use because the asymptotic regime only emerges provided log(t=B)1(where 1corresponds to the order of the sub-leading term), which may not be attainable, especially when the scattering parameter is small. For this reason, it is preferable to retain also the next-order term in equation (37), 1 thn2it'1 R+ 2Bn 1 + 4 log(t=B) + 1=2 + O(t=B)1o ;(44) where a fraction 4=(R+ 2B)is contributed by particles initially distributed among propagating states. 4.1.3. Superdiffusion In the range of parameters 1<  < 2, we substitute the scaling properties of the harmonic numbers, lim n!1n3H(2) n = (3)1; lim n!1n2H(1) n = (2)1:(45) and obtain from equation (37) the limit lim t!11 (t=B)3hn2it=1 (2)(3)+1 32B (121)R+B; =1 (2)(3)2B (121)R+B: (46) As emphasized by the first line of this equation, the leading-order coefficient is the sum of two distinct non-trivial contributions from the two terms on the right-hand side of equation (37): one from the fraction of particles in a scattering state and the other from particles initially injected in propagating states. As a consequence, the transport coefficient corresponding to the all-scattering initial condition differs by a factor 1 from that above, corresponding to the stationary initial condition. This observation is consistent with the results of reference [25], where non-recurrent regimes of the scaling parameterwere also investigated.Lévy walks on lattices as multi-state processes 14 4.2. Exact time-dependent solution The following exact expression of the mean squared displacement (33) is obtained after substitution of the fraction of particles in a scattering state, equation (23), and applies to the all-scattering initial condition (22): hn2it=bt=BcX k=0(2k+ 1)1X l=k1l( 1e(tkB)=R +bt=Bck1X j=1jX m=1mX n=1na(njm)e(jm)B=RnX i=0n! i!iB Ri h (jm)i(jm+ 1)ieB=Ri +bt=BckX m=1mX n=1na(njm)e(bt=Bckm)B=RnX i=0n! i!iB Ri h (bt=Bckm)i(t=Bkm)ie(tbt=BcB)=Ri) ;(47) where, for consistency, one must interpret terms such as (jm)0as unity, even when j=m. ��-���������������������������� �〈��〉/� (a)(121)1= 1 ��-��������������������������������������� �〈��〉/� (b)(121)1= 5102 Figure 2: Growth in time of the rescaled mean squared displacement, hn2it=t, computed from equation (47). The transition probabilities were set according to equations (7), withB= 1andR= 1 +(121)1. The value of the scattering parameter was chosen such that =(121) = 1(left panel) and 5102(right panel). In both figures, the scaling parameter takes the values = 3=2(green curve), 2(cyan curve), and 5=2 (red curve). For comparison, in the left panel, the asymptotic scaling values predicted by equations (41), (43) and (46) are displayed in dashed lines with matching colors. The right panel shows a comparison between the exact solution and an approximated solution, exact to first order in (darker curves); see equation (53).Lévy walks on lattices as multi-state processes 15 Figure 2 displays the results of explicit evaluations of equation (47) for three different values of the scaling parameter,  > 1, with= 121. According to the asymptotic results (41), (43) and (46), the scaling parameter value = 2gives rise to a logarithmic divergence of hn2it=t, which separates the superdiffusive regime, for1<  < 2, from that of normal diffusion, for  > 2. A comparison between the exact solution and the asymptotic ones is displayed in figure 2a. On the time scale of the figure, convergence to the asymptotic regimes is convincingly observed only in the superdiffusive case. Note that while the computation of equation (47) up to t= 50B takes less than one half hour of CPU time on a reasonably fast computer, doubling the time range would increase the required CPU time to over ten days. Although equation (47) gives the exact time-dependence of the mean squared displacement of the process for the initial condition (21), it does not give much insight into its time development. In particular, how to extract the large-time behaviour of the mean squared displacement and connect this result to the asymptotic scalings (41), (43) and (46) is not transparent. As reflected by figure 2a, a numerical evaluation of equation (47) is necessarily limited to moderately large times, due to the number of terms involved. Aregimeofspecificinterestwhichallowsustoinfertheemergenceoftheasymptotic scalings (41), (43) and (46) from the solution (47) is, however, that of small values of the scattering parameter , i.e., such that transitions from scattering to propagating states occur only rarely. This is discussed below. We will otherwise have to resort to numerical simulations of the underlying stochastic processes to observe this convergence. Those results are presented in section 5. 4.3. Small-parameter expansion For small values of the scattering parameter , when transitions from scattering to propagating states are much rarer than transitions between scattering states, an expansion of equation (47) in powers of this parameter provides an approximate expression of the mean squared displacement from which the different asymptotic regimes discussed in section 4.1 can be inferred. In the anomalous regime of the scaling parameter, 2, terms constant in time, that carry a normal contribution to the mean squared displacement, may bring about contributions which, even for large times, may be much larger than that of terms diverging in time; an anomalous contribution to the mean squared displacement may then be masked by a normal one. To proceed, we substitute equation (28) into equation (33) and let tkB. Having dropped all terms of order 2and higher, we obtain hn2ikB=B RZk 0dxn 1(121)1B R[1(bxc+ 1)1] B R(xbxc)1X i=dxeio +B RkX j=1(2j+ 1)(kj)1X i=jiLévy walks on lattices as multi-state processes 16 =kB Rn 1B R(121)1 1k1(H(1) k1=2 + 1=2(k+ 1)1) +(121)1h 1(k+ 1)1+ (2k1)(H(1) k+1(k+ 1)2) 2k1(2H(2) k+1H(1) k+1(k+ 1)3)io ; 'kB Rn 1 +(121)1h 1 + 2H(1) k4k1H(2) kB Rio ; (48) where, in the last line, we have omitted terms that decay to zero as kgrows large. Plugging into this expression the asymptotic forms of the generalized harmonic numbers (40), (42) and (45), H(1) k2k1H(2) k'8 >>< >>:(1); > 2; logk+ 2;  = 2; 1 (2)(3)k2+(1);1<< 2;(49) we obtain approximations of the mean squared displacement (47), which can be compared with the asymptotic expressions (41), (43) and (46). In the regime of normal diffusion,  > 2, equation (48) reduces to the first order expansion in of equation (41). For 1< 2, the mean squared displacement (48) displays normal diffusion at short times and anomalous diffusion at large times. Thus in the weak superdiffusive regime, = 2, equation (48) reduces to 1 thn2it'1 Rn 1 + 2h 2 logt B+ 2 3B Rio ; (50) which differs from the stationary expression (44) by a factor 4=R, subleading with respect to the logarithmically diverging term, identical in both expression. This difference stems from the absence of particles populating propagating states in our choice of initial condition. In the superdiffusive regime, 1<  < 2, the divergence of the harmonic numbers, equation (49), dominates the large time behaviour, 1 thn2it'1 Rn 1 +(121)1h2(1) (2)(3)t B2 + 1 + 2(1)B Rio :(51) Here again we note a difference between this expression and equation (46), obtained in the stationary regime. At variance with the case = 2, however, the difference occurs in the coefficient of the leading term on the right-hand side of equation (51), whose numerator is 1instead of 1in the stationary regime. The transport coefficient on the right-hand side of equation (51) therefore changes, depending on the choice of initial conditions. At large times, after the mean squared displacement crosses over from normal diffusion to superdiffusion, the above expressions are equivalent to O()expansions of the asymptotic values (43) and (46). The value of the crossover time, tc, that separatesLévy walks on lattices as multi-state processes 17 the short time normal diffusion from the large time anomalous diffusion can be inferred from the above expressions: tcB8 < :h (2)(3) 2(1)121 i1 2;1<< 2; exp1 4
;  = 2;(52) which can be large, in particular when = 2. For short times, equation (48) can be improved by removing the assumption that t is an integer multiple of B. One then finds hn2it=t R+(121)1 t Rh 12(bt=Bc+ 1)2(bt=Bc+ 1)1+ 2H(1) bt=Bc+1i +B Rh 2(bt=Bc+ 1)3+ (bt=Bc+ 1)24H(2) bt=Bc+1+H(1) bt=Bc+1i B R2n bt=BcH(1) bt=Bc+ 1=21=2(bt=Bc+ 1)1 + 1=2 (t=B)2bt=Bc2 (bt=Bc+ 1)1(bt=Bc+ 2)2 + t=Bbt=Bc
 1(bt=Bc+ 1)2+ (bt=Bc+ 2)2 2(bt=Bc+ 2)1o : (53) A comparison between this approximate solution and the exact one, equation (47), is shown in figure 2b for different values of the scaling parameter, . 5. Numerical computations Numerical simulations of the process with rates (2) and probabilities (3) are based on a classic kinetic Monte Carlo algorithm [26], taking into account the possibility of ballistic motion of particles in a propagating phase. A collection of independent walkers are initialized at time t= 0at the origin of the two-dimensional square lattice Z2, either in the scattering state, k0= 0, for the all-scattering initial condition, or in state k00with relative weights k0specified by equations (35)–(36), for the stationary initial condition. For each walker, we generate a sequencef(kn;jn);tngn2Nof successive states (kn;jn),kn2N,jn= 1;:::;z4, and corresponding times tnas follows. Wheninastate kn1= 0, thenexttransitionisdeterminedbydrawingthefollowing three random numbers. This is referred to as a scattering step . (i) The first random number, 2[0;1], is drawn from a uniform distribution, and yields the state kn, such thatPkn1 a=0a<Pkn a=0a.Lévy walks on lattices as multi-state processes 18 (ii) The corresponding waiting time, i.e., the time tnthe particle waits in the state kn1= 0before the transition to state kntakes place, is obtained by drawing a second random number, whose distribution is exponential with mean R. (iii) Athirdrandominteger,withuniformdistributionamongthesetoflatticedirections f1;:::; 4g, identifies the direction jnof the corresponding displacement by one lattice unit. Depending on the new state kn, another scattering step is taken if kn= 0or, if kn1, a sequence of knpropagating steps f(km;jm);tmgn+kn m=n+1takes place, such that: (i) the state decreases by one unit, km=km11, (ii) the direction remains unchanged jm=jm1, and (iii) the time step takes value tm=B, corresponding to the propagation time over a single lattice cell. The above loop repeats itself until m=n+kn, that is until km= 0, at which point another scattering step is taken. Keeping track of the positions of all walkers as functions of time, which we measure at regular intervals on a logarithmic timescale, measurements of the mean squared displacementhn2itrescaled by 1=tare performed by taking averages of this quantity over the set of all walkers, which typically consists of 108trajectories. Such measurements are reported below for the different scaling regimes in the positive recurrent range of the scaling parameter values, >1. Throughout this section, we set the transition probabilities according to equation (7) and, for convenience, change the scattering parameter to 2 121+: (54) We further let B1; R2 2;(55) such that the asymptotic fraction of particles in the scattering state (18) becomes lim t!1S0(t) =2 2 +: (56)Lévy walks on lattices as multi-state processes 19 5.1. All-scattering vs. stationary initial conditions With the choice of parameters (54)-(55), The mean squared displacement, for tlarge and with the all-scattering initial condition, is expected to scale as hn2it t'2 2 +8 >>< >>:1 +(1); > 2; 1 +(logt+ 2);  = 2; 1 +h 1 (2)(3)t2+(1)i ;1<< 2;(all-scattering i. c.) (57) where, in the two anomalous cases, we kept terms constant in time to reflect the possibility that, when is small, the normal term may not be negligible with respect to the anomalous one over a large range of times. This is particularly relevant for the marginal case, = 2, where logtand 2remain of comparable sizes throughout the time range accessible to numerical computations. In comparison, for the stationary initial condition, the above expression remains unchanged in the normal diffusive case, but is modified in the two anomalous cases, hn2it t'2 2 +8 >>< >>:1 +(1); > 2; 1 +(logt+ 1);  = 2; 1 +h 1 (2)(3)t2+(1)i ;1<< 2:(stationary i. c.) (58) The first order expansion in of the two anomalous regimes in equation (57) (in all-scattering initial condition) yields results equivalent to equations (50) and (51) respectively. A computation of the time evolution of the mean squared displacement in regimes of normal ( >2) and anomalous ( 1<2) diffusion is shown in figure 3, providing a comparison between the two initial conditions analyzed in section 4, with the two corresponding sets of asymptotic regimes given by equations (57) and (58). The scaling parameter values are set to = 5=2(figure 3a), 2(figure 3b), and 3=2(figure 3c). The scattering parameter value is set to = 1throughout, such that transitions between scattering states are forbidden. As expected, the numerically computed mean squared displacement obtained from the stationary initial condition (magenta curves in figures 3a-3c) follow precisely the analytic result (37) in all three regimes. We further note that, in the regime of normal diffusion ( = 5=2), both data sets in figure 3a display consistent convergence to the same asymptotic regime, given by the first lines of equations (57) and (58), limt!1hn2it=t'2:597. A similar result is observed in figure 3b, where the convergence of the two data sets to the common leading value of equations (57) and (58), limt!1hn2it=tlogt= 4=5, is apparent. The effect of the differing subleading terms, the one in (57) negative, the other in (58) positive (the latter value is about one half the absolute value of the former for = 4=3), is, however,Lévy walks on lattices as multi-state processes 20 ��������������������� �〈�〉�/� �����������������������������(�) (a)= 5=2,'1:215(= 1) ������������������������ �〈�〉�/[����(�)] ��������������������������(�) (b)= 2,= 4=3(= 1) ������������������������������������ �〈�〉�/���� ��������������������������(�) (c)= 3=2,'1:547(= 1) Figure 3: Numerical measurement of the time-evolution of the normally or anomalously rescaled mean squared displacement in the three distinct regimes of the scaling parameter. Time evolutions obtained from the stationary initial condition (magenta curves) are compared with those generated by the all-scattering initial condition (cyan curves). The dashed black curves correspond to the exact result (37) and the dotted lines to the asymptotic regimes (57) and (58), identical for regimes of normal diffusion, but otherwise differing according to the type of initial conditions. Insets: time evolution of the fraction of particles in the scattering state compared with the stationary value (56) (black dotted line).Lévy walks on lattices as multi-state processes 21 manifest. Finally, in the superdiffusive regime, = 3=2, figure 3c exhibits the two asymptotic regimes of the anomalously rescaled mean squared displacement given by equations (57) and (58), limt!1hn2it=t3'0:582(all-scattering initial condition) or 1:163(stationary initial condition), whose values differ by a 1 : 2ratio for this value of the scaling parameter. 5.2. Perturbative regimes of the scattering parameter The time evolution of the mean squared displacement in the perturbative regime of the scattering parameter, 1, is analyzed in figure 4 for particles initially distributed in the scattering state at the origin, where the rescaled mean squared displacement is compared with the O()solution, equation (53). Excellent agreement between the analytic and numerical results is observed throughout the time range when the parameter is small enough ( = 102figures 4b, 4d, 4f). Numerical computations are also consistent with the asymptotic regime (57) for all cases. The scattering parameter value = 101, shown in figures 4a ( = 5=2), 4c (= 2), and 4e (= 3=2), is on the one hand small enough that, for short times, the computed mean squared displacement is barely distinguishable from the perturbative expansion (53). The effect of next order corrections is, on the other hand, apparent for larger times (t&102), beyond which a convergence to the asymptotic result (57) is observed. We note that the crossover time (52) is tc'1:3104for= 101(figure 4c) and much larger yet for = 102. For= 2, subleading terms in equation (57) therefore dominate the logarithmically divergent term throughout the time range of measurements. Similarly, in the superdiffusive case, = 3=2, the respective crossover times (52) are tc'2102(figure 4e) and tc'2:2104(figure 4f), such that subleading terms in equation (57) remain important throughout the time range of measurements in both cases. We conclude this section by pointing out that the time range of numerical measurements such as presented in figure 4 is limited by the finite precision of the random number generator used to draw the transition probabilities kand sets an effective maximal scale of free flights, kmax. This observation is analogous to the effect of machine-dependent limitations recently reported in reference [27] and is particularly relevanttotheregimeofweaksuperdiffusion. Theeffectivemaximalscale kmaxinducesa saturation of the logarithmic growth of the second moment for times larger than kmaxB. The process would thus become diffusive for times sufficiently large. Although this effect can be pushed upward to larger times by increasing the precision of the random number generator, it cannot be eliminated altogether. 6. Conclusions The inclusion of exponentially-distributed waiting times separating the successive jump events of Lévy walkers leads to a natural distinction between propagating and scatteringLévy walks on lattices as multi-state processes 22 ������������������������������������ �〈�〉�/� �����������������������������(�) (a)= 5=2,= 101('3:4102) ������������������������������������������ �〈�〉�/� �����������������������������(�) (b)= 5=2,= 102('3:25103) ������������������������������ �〈�〉�/� �����������������������������(�) (c)= 2,= 101('2:63102) ������������������������������������ �〈�〉�/� �����������������������������(�) (d)= 2,= 102('2:51103) ������������������������ �〈�〉�/� �����������������������������(�) (e)= 3=2,= 101('1:54102) �������������������������������� �〈�〉�/� �����������������������������(�) (f)= 3=2,= 102('1:47103) Figure 4: Numerical computations (cyan curves) of the normally rescaled mean squared displacement in the three different regimes of the scaling parameter for the all- scattering initial condition. The scattering parameter takes values in or near the perturbative regime: (a), (c), (e) = 101, and (b), (d), (f) = 102. The data are compared with the asymptotic solution (57) (black dotted curves) and the O() solution (53) (dashed curves). Insets: time evolution of the fraction of particles in the scattering state with the stationary value (18) (dotted lines) and the the O()solution (29) (dashed curves).Lévy walks on lattices as multi-state processes 23 states, whose respective concentrations evolve in time according to a set of generalized master equations. In particular, the fraction of walkers in a scattering state obeys a linear delay differential equation with a countable hierarchy of delays whose analytic solutions were obtained. AsopposedtoclassicmethodsbasedontheFourier–Laplacetransformofanintegral kernel and the use of Tauberian theorems [11,13], the approach presented in this paper is based solely on the solutions of such delay differential equations. Our method thus yields a simple expression of the mean squared displacement of Lévy walkers in terms of the distribution of free paths and the time integral of the fraction of particles in the scattering state. Bothexactandasymptoticexpressionsofthemeansquareddisplacementofwalkers were obtained in regimes ranging from normal to superdiffusive subballistic transport. In these regimes, the mechanism through which a walker can change directions between successive propagation phases plays an important role in determining the values of the transport coefficients, whether normal or anomalous. The transition through a scattering phase brings about a description in terms of two parameters, the first specifying the typical timescale of scattering events as opposed to the timescale of propagationacrossanelementarycell, thesecondweightingtheprobabilityoftransitions between scattering and propagating states. Relying on the stationary fractions of particles in scattering and propagating states, a detailed derivation of the transport coefficients, similar to that reported in [16], was given, exhibiting the precise effect of the two scattering parameters on these coefficients, as well as the influence of the initial distribution of walkers [25]. Our formalism also yieldstheexacttimeevolutionofthemeansquareddisplacement,whichwasinvestigated when particles are initially found in a scattering state. The comparison between the exact and asymptotic solutions is generally not immediate, but is fairly straightforward in perturbative regimes such that the likelihood of a transition from scattering to propagating states is small. A case in point – though by far not the only application of the present results – is given by the infinite-horizon Lorentz gas [28], for which the scaling parameter of the distribution of free paths takes onthemarginalvalue = 2. Asonemightexpect, measuringthelogarithmicdivergence in time of the rescaled mean squared displacement is typically hindered by dominant constant terms. This is particularly so in the regime of narrow corridors, which yields a stochastic description in terms of multistate Lévy walks such that the scattering parameter is small, 1[23]. The parameter induces a further separation between the time scales of the scattering and propagating phases, RB. To accurately model the scattering phase is therefore of paramount importance to understanding the dynamics of the infinite-horizon Lorentz gas in terms of a Lévy walk. Theresultsobtainedinthispaperprovidetheframeworktotransposingsuchresults to a range of applications exhibiting other regimes of transport, such as have been studied in the context of optimal intermittent search strategies [22]. Our theoretical framework brings about new perspectives to extend the study of intermittent walks toLévy walks on lattices as multi-state processes 24 power-law distributed ballistic phases, which may be relevant to the increasing body of literature on optimal search strategies [8,29,30]. Acknowledgments This work was partially supported by FIRB-Project No. RBFR08UH60 (MIUR, Italy), by SEP-CONACYT Grant No. CB-101246 and DGAPA-UNAM PAPIIT Grant No. IN117214 (Mexico), and by FRFC convention 2,4592.11 (Belgium). T.G. is financially supported by the (Belgian) FRS-FNRS. Appendix A. Time-dependent fraction of particles in the scattering state To derive equation (23), we note that a(1jk)=kand, for 2nk, a(njk)=1 nkn+1X j=1ja(n1jkj): (A.1) By recursive application of this formula, we obtain a(njk)=1 nkn+1X j1=1j1a(n1jkj1); =1 n(n1)kn+1X j1=1kj1n+2X j2=1j1j2a(n2jkj1j2); =1 n!kn+1X j1=1


k1j1


jn2X jn1=1ji:::jn1kj1


jn1; (A.2) which is equivalent to equation (24). Taking the derivative of equation (23), we verify equation (16): _S0(t) =1 RS0(t) +bt=BcX k=1e(tkB)=RkX n=1na(njk)n R(tkB)n1; =1 Rbt=BcX k=1kS0(tkB)1 RS0(t) +bt=BcX k=1e(tkB)=R"kX n=2na(njk)n R(tkB)n1 kbt=BckX j=1ejB=RjX n=1a(njj)n1 R(tkBjB)n# : (A.3)Lévy walks on lattices as multi-state processes 25 We have to show that the last term on the RHS vanishes, which amounts to proving the identity bt=BcX k=1ekB=R"kX n=2na(njk)n+1 R(tkB)n1 kbt=BckBX j=1ejB=RjX n=1a(njj)n R(tkBjB)n# = 0;(A.4) or, after rearrangement, bt=BcX k=2ekB=Rk1X n=1(n+ 1)a(n+1jk)n R(tkB)n =bt=BcX k=1kbt=BckX j=1e(k+j)B=RjX n=1a(njj)n R(tkBjB)n:(A.5) The second term in this expression, by successively changing the index jtojk, then swapping the sums over kandjand exchanging the indices jandk, transforms to bt=BcX k=1kbt=BckX j=1e(k+j)B=RjX n=1a(njj)n R(tkBjB)n =bt=BcX k=1kbt=BcX j=k+1ejB=RjkX n=1a(njjk)n R(tjB)n; =bt=BcX k=2ekB=Rk1X j=1jkjX n=1a(njkj)n R(tkB)n: (A.6) Now swapping the sums over jandnin the last line and plugging this expression back into equation (A.5), we obtain, after factorization of the common factors, the condition bt=BcX k=2ekB=Rk1X n=1" (n+ 1)a(n+1jk)knX j=1ja(njkj)# n R(tkB)n= 0;(A.7) which holds by identity (A.1), thus completing the proof that S0(t)as specified by equation (23) is the solution to equation (16) for the initial condition (21). References [1] Shlesinger M F, Zaslavsky G M and Frisch U (eds) 1995 Lévy Flights and Related Topics in Physics (Lecture Notes in Physics vol 450) (Berlin Heidelberg: Springer) URL http: //dx.doi.org/10.1007/3-540-59222-9Lévy walks on lattices as multi-state processes 26 [2] Klafter J, Shlesinger M F and Zumofen G 1996 Physics Today 4933–39 URL http://dx.doi. org/10.1063/1.881487 [3] Klages R, Radons G and Sokolov I M 2008 Anomalous transport: Foundations and applications (Weinheim: Wiley-VCH Verlag) URL http://dx.doi.org/10.1002/9783527622979 [4] Denisov S, Zaburdaev V Y and Hänggi P 2012 Physical Review E 8531148 URL http: //dx.doi.org/10.1103/PhysRevE.85.031148 [5] Zaburdaev V, Denisov S and Klafter J 2014 arXiv:1410.5100 URL http://arxiv.org/abs/1410. 5100 [6] Weiss G H and Rubin R J 1983 Advances in Chemical Physics 52363–505 URL http://dx.doi. org/10.1002/9780470142769.ch5 [7] Shlesinger M F, Zaslavsky G M and Klafter J 1993 Nature 36331–37 URL http://dx.doi.org/ 10.1038/363031a0 [8] Viswanathan G M, da Luz M G E, Raposo E P and Stanley H E 2011 The Physics of Foraging: an Introduction to Random Searches and Biological Encounters (Cambridge, UK: Cambridge University Press) URL http://www.cambridge.org/9781107006799 [9] Brockmann D, Hufnagel L and Geisel T 2006 Nature 439462–465 URL http://dx.doi.org/10. 1038/nature04292 [10] Raichlen D A, Wood B M, Gordon A D, Mabulla A Z, Marlowe F W and Pontzer H 2014 Proceedings of the National Academy of Sciences of the United States of America 111728– 733 URL http://dx.doi.org/10.1073/pnas.1318616111 [11] Geisel T, Nierwetberg J and Zacherl A 1985 Physical Review Letters 54(7) 616–619 URL http://dx.doi.org/10.1103/PhysRevLett.54.616 [12] Shlesinger M F and Klafter J 1985 Physical Review Letters 542551–2551 URL http://dx.doi. org/10.1103/PhysRevLett.54.2551 [13] Klafter J, Blumen A and Shlesinger M F 1987 Physical Review A 353081–3085 URL http: //dx.doi.org/10.1103/PhysRevA.35.3081 [14] Wang X J 1992 Physical Review A 45(12) 8407–8417 URL http://dx.doi.org/10.1103/ PhysRevA.45.8407 [15] Zumofen G and Klafter J 1993 Physical Review E 47(2) 851–863 URL http://dx.doi.org/10. 1103/PhysRevE.47.851 [16] Cristadoro G, Gilbert T, Lenci M and Sanders D P 2014 EPL (Europhysics Letters) 10850002 URL http://dx.doi.org/10.1209/0295-5075/108/50002 [17] Landman U, Montroll E W and Shlesinger M F 1977 Proceedings of the National Academy of Sciences of the United States of America 74430–433 URL http://www.pnas.org/content/ 74/2/430.abstract [18] Montroll E W and Weiss G H 1965 Journal of Mathematical Physics 6167 URL http://dx.doi. org/10.1063/1.1704269 [19] Kenkre V M, Montroll E W and Shlesinger M F 1973 Journal of Statistical Physics 945–50 URL http://dx.doi.org/10.1007/BF01016796 [20] Klafter J, Zumofen G and Shlesinger M F 1995 Lévy description of anomalous diffusion in dynamical systems Lévy Flights and Related Topics in Physics (Lecture Notes in Physics vol 450) ed Shlesinger M F, Zaslavsky G M and Frisch U (Berlin Heidelberg: Springer) pp 196–215 URL http://dx.doi.org/10.1007/3-540-59222-9_35 [21] Klafter J and Sokolov I M 2011 First steps in random walks: from tools to applications (Oxford, UK: Oxford University Press) ISBN 9780199234868 URL http://dx.doi.org/10. 1093/acprof:oso/9780199234868.001.0001 [22] Bénichou O, Loverdo C, Moreau M and Voituriez R 2011 Reviews of Modern Physics 8381 URL http://dx.doi.org/10.1103/RevModPhys.83.81 [23] Cristadoro G, Gilbert T, Lenci M and Sanders D P 2014 Physical Review E 90(5) 050102 URL http://dx.doi.org/10.1103/PhysRevE.90.050102 [24] Driver R D 1977 Ordinary and delay differential equations (New York, NY: Springer-Verlag) URLLévy walks on lattices as multi-state processes 27 http://dx.doi.org/10.1007/978-1-4684-9467-9 [25] Zumofen G and Klafter J 1993 Physica D 69436–446 URL http://dx.doi.org/10.1016/ 0167-2789(93)90105-A [26] Gillespie D T 1976 Journal of Computational Physics 22403–434 URL http://dx.doi.org/10. 1016/0021-9991(76)90041-3 [27] Radicchi F 2014 Physical Review E 90(5) 050801 URL http://dx.doi.org/10.1103/PhysRevE. 90.050801 [28] Cristadoro G, Gilbert T, Lenci M and Sanders D P 2014 Physical Review E 90(2) 022106 URL http://dx.doi.org/10.1103/PhysRevE.90.022106 [29] Lewis M A, Maini P K and Petrovskii S V (eds) 2013 Dispersal, individual movement and spatial ecology(Lecture Notes in Mathematics vol 2071) (Berlin: Springer) URL http://dx.doi.org/ 10.1007/978-3-642-35497-7 [30] MéndezV,CamposDandBartumeusF2013 Stochastic Foundations in Movement Ecology (Berlin, Heidelberg: Springer) URL http://dx.doi.org/10.1007/978-3-642-39010-4

2201.11498v3.Effect_of_vertex_corrections_on_the_enhancement_of_Gilbert_damping_in_spin_pumping_into_a_two_dimensional_electron_gas.pdf

Eect of vertex corrections on the enhancement of Gilbert damping in spin pumping into a two-dimensional electron gas M. Yama,1M. Matsuo,2;3;4;5T. Kato1, 1Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan 2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, China 3CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, China 4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan 5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, Japan (Dated: May 15, 2023) We theoretically consider the eect of vertex correction on spin pumping from a ferromagnetic insulator (FI) into a two-dimensional electron gas (2DEG) in which the Rashba and Dresselhaus spin-orbit interactions coexist. The Gilbert damping in the FI is enhanced by elastic spin- ipping or magnon absorption. We show that the Gilbert damping due to elastic spin- ipping is strongly enhanced by the vertex correction when the ratio of the two spin-orbit interactions is near a special value at which the spin relaxation time diverges while that due to magnon absorption shows only small modication. We also show that the shift in the resonant frequency due to elastic spin- ipping is strongly enhanced in a similar way as the Gilbert damping. I. INTRODUCTION In the eld of spintronics1,2, spin pumping has long been used as a method of injecting spins into various materials3{5. Spin pumping was rst employed to inject spins from a ferromagnetic metal into an adjacent normal metal (NM)6{9. Subsequently, it was used on ferromag- netic insulator (FI)/NM junctions10. Because spin injec- tion is generally related to the loss of the magnetization in ferromagnets, it aects the Gilbert damping measured in ferromagnetic resonance (FMR) experiments11. When we employ spin injection from the FI, the modulation of the Gilbert damping re ects the properties of the spin ex- citation in the adjacent materials, such as magnetic thin lms12, magnetic impurities on metal surfaces13, and su- perconductors14{17. This is in clear contrast with the Gilbert damping of a bulk FI, which re ects properties of electrons and phonons18{20. An attractive strategy is to combine spin pumping with spin-related transport phenomena in semiconduc- tor microstructures1,21. A two-dimensional electron gas (2DEG) in a semiconductor heterostructure is an easily controlled physical system that has been used in spin- tronics devices22{25. A 2DEG system has two types of spin-orbit interaction, i.e., Rashba26,27and Dresselhaus spin-orbit interactions28,29. In our previous work30, we theoretically studied spin pumping into a 2DEG in semiconductor heterostructures with both Rashba and Dresselhaus spin-orbit interac- tions, which can be regarded as a prototype for a 2DEG with a complex spin-texture near the Fermi surface [see Fig. 1 (a)]. In that study, we formulated the modu- lation of the Gilbert damping in the FI by using the 2DEG microwave FI (a) (b) totFIG. 1. (a) Schematic picture of junction composed of a fer- romagnetic insulator (FI) and a two-dimensional electron gas (2DEG) realized in a semiconductor heterostructure. Stotin- dicates the total spin of the FI. We consider a uniform spin precession of the FI induced by microwave irradiation. (b) Laboratory coordinates ( x;y;z ) and the magnetization-xed coordinates ( x0;y0;z0). The red arrow indicates the expec- tation value of the spontaneous spin polarization of the FI, hSi. second-order perturbation with respect to the interfa- cial coupling15,31{35and related it to the dynamic spin susceptibility of the 2DEG. We further calculated the spin susceptibility and obtained characteristic features of the Gilbert damping modulation. This modulation contains two contributions: elastic spin- ipping, which dominates at low resonant frequencies, and magnon ab- sorption, which dominates at high resonant frequencies. In addition, we claried that these contributions have dierent dependence on the in-plane azimuth angle of the ordered spin in the FI [see Fig. 1 (b)]. When the Rashba and Dresselhaus spin-orbit interac- tions have almost equal magnitudes, spin relaxation byarXiv:2201.11498v3 [cond-mat.mes-hall] 12 May 20232 nonmagnetic impurity scattering is strongly suppressed because the direction of the eective Zeeman eld gen- erated by the spin-orbit interactions is unchanged along the Fermi surface. Due to this substantial suppression of spin relaxation, there emerge characteristic physical phe- nomena such as the persistent spin helix state36{39. In general, the vertex corrections have to be taken into ac- count to treat various conservation laws, i.e., the charge, spin, momentum, and energy conservation laws in cal- culation of the response functions40{43. Therefore, for better description of realistic systems, we need to con- sider vertex correction, which captures eect of impurity more accurately by re ecting conservation laws. How- ever, the vertex corrections were neglected in our pre- vious work30. This means that our previous calculation should fail when the Rashba and Dresselhaus spin-orbit interactions compete. In this study, we consider the same setting, i.e., a junction composed of an FI and a 2DEG as shown in Fig. 1 (a), and examine eect of the spin conservation law by taking the vertex correction into account. We theo- retically calculate the modulation of the Gilbert damping and the shift in the FMR frequency by solving the Bethe- Salpeter equation within the ladder approximation. We show that the vertex correction substantially changes the results, in particular, when the strengths of the Rashba- and Dresselhaus-type spin-orbit interactions are chosen to be almost equal but slightly dierent; Specically, both the Gilbert damping and the FMR frequency shift are largely enhanced at low resonant frequencies re ect- ing strong suppression of spin relaxation. This remark- able feature should be able to be observed experimen- tally. In contrast, the vertex correction changes their magnitude only slightly at high resonant frequencies. Before describing our calculation, we brie y comment on study of the vertex corrections in a dierent context. In early studies of the spin Hall eect, there was a de- bate on the existence of intrinsic spin Hall eect44{46. By considering the vertex corrections, the spin Hall conduc- tivity, which is calculated from the correlation function between the current and spin current, vanishes in the presence of short-range disorder for simple models even if its strength is innitesimally small47{49. This seemingly contradictory result stimulated theoretical researches on realistic modied models50,51as well as denition of the spin current52{56. However, we stress that the vertex corrections for the dynamic spin susceptibility, which is calculated from the spin-spin correlation function, have no such subtle problem57because it does not include the spin current. The rest of this work is organized as follows. In Sec. II, we brie y summarize our model of the FI/2DEG junc- tion and describe a general formulation for the magnon self-energy following Ref. 30. In Sec. III, we formulate the vertex correction that corresponds to the self-energy in the Born approximation. We show the modulation of the Gilbert damping and the shift in the FMR frequency in Secs. IV and V, respectively, and discuss the eect ky kx (a) (b) FIG. 2. Schematic picture of the spin-splitting energy bands of 2DEG for (a) == 0 and (b) == 1. The red and blue arrows represent spin polarization of each band. In the case of (b), the spin component in the direction of the azimuth angle 3=4 is conserved. of the vertex correction in detail. Finally, we summa- rize our results in Sec. VI. The six Appendices detail the calculation in Sec. III. II. FORMULATION Here, we describe a model for the FI/2DEG junction shown in Fig. 1 (a) and formulate the spin relaxation rate in an FMR experiment. Because we have already given a detailed formulation on this model in our previous paper30, we will brie y summarize it here. A. Two-dimensional electron gas We consider a 2DEG whose Hamiltonian is given as HNM=Hkin+Himp, whereHkinandHimpdescribe the kinetic energy and the impurity, respectively. The kinetic energy is given as Hkin=X k(cy k"cy k#)^hkck" ck# ; (1) ^hk=k^Ihe(k)
; (2) whereckis the annihilation operator of conduction elec- trons with wave number k= (kx;ky) andzcomponent of the spin, (=";#),^Iis a 22 identity matrix, a (a=x;y;z ) are the Pauli matrices, k=~2k2=2m is the kinetic energy measured from the chemical poten- tial, andmis an eective mass. Hereafter, we assume that the Fermi energy is much larger than the other en- ergy scales such as the spin-orbit interactions, the tem- perature, and the ferromagnetic resonance energy. Then, the low-energy part of the spin susceptibility depends on the chemical potential and the eective mass monly through the density of states at the Fermi energy, D(F). The spin-orbit interaction is described by the eective3 Zeeman eld, he(k) =jkj(sin'cos';cos'+sin';0) 'kF(sin'cos';cos'+sin';0); (3) whereandrespectively denote the amplitudes of the Rashba- and Dresselhaus-type spin-orbit interactions and the electron wave number is expressed by polar coordi- nates as (kx;ky) = (jkjcos';jkjsin'). In the second equation of Eq. (3), we have approximated jkjwith the Fermi wave number kFassuming that the spin-orbit in- teraction energies, kFandkF, are much smaller than the Fermi energy58. When only the Rashba spin-orbit in- teraction exists ( = 0), the energy band is spin-splitted as shown in Fig. 2 (a). The spin polarization of each band depends on the azimuth angle 'because it is determined by the eective Zeeman eld hewhich is a function of 'as seen in Eq. (3). In the special case of == 1, the spin polarization always becomes parallel to the direction of the azimuth angle 3 =4 in thexyplane as shown in Fig. 2 (b). Then, the spin component in this direction is conserved. This observation indicates that eect of the spin conservation may become important when the two spin-orbit interactions compete ( '). The Hamiltonian of the impurity potential is given as Himp=uX i2impX  y (ri) (ri); (4) where (r) =A1=2P kckeik
r,Ais the area of the junction,uis the strength of the impurity potential, and riis the position of the impurity site. The nite-temperature Green's function for the con- duction electrons is dened by a 2 2 matrix ^g(k;i!m) whose elements are g0(k;i!m) =Z~ 0dei!mg0(k;); (5) g0(k;) =~1hck()cy k0i; (6) whereck() =eHNM=~ckeHNM=~,HNM=Hkin+ Himp,!m=(2m+ 1)=~is the fermionic Matsubara frequency, and is the inverse temperature. By em- ploying the Born approximation, the nite-temperature Green's function can be expressed as ^g(k;i!m) =(i~!mk+isgn(!m)=2)^Ihe
Q =(i~!mE k+isgn(!m)=2); (7) whereE k=kjhe(')jis the spin-dependent electron dispersion, = 2niu2D(F) (8) is level broadening, and niis the impurity concentration (see Appendix A and Ref. 30 for detailed derivation). As already mentioned, the case of = = 1 is special because the spin component parallel to the direction ofthe azimuth angle 3 =4 in thexyplane is conserved (see Fig. 2 (b)). By dening the spin component in this di- rection as s3=4 tot1 2X k(cy k+ck+cy kck); (9)  ck+ ck = 1=p 2ei3=4=p 2 ei3=4=p 2 1=p 2 ck" ck# ;(10) we can prove [ Hkin+Himp;s3=4 tot] = 0. When the value of = is slightly shifted from 1, the spin conservation law is broken slightly and this leads to a slow spin relaxation. As will be discussed in Secs. IV and V, this slow spin re- laxation, which is a remnant of the spin conservation at = = 1, strongly aects the spin injection from the FI into the 2DEG. To describe this feature, we need to con- sider the vertex correction to take the conservation law into account in our calculation as explained in Sec. III. B. Ferromagnetic insulator We consider the quantum Heisenberg model for the FI and employ the spin-wave approximation assuming that the temperature is much lower than the magnetic tran- sition temperature and the magnitude of the localized spins,S0, is suciently large. We write the expectation value of the localized spins in the FI as hSi, whose direc- tion is (cos ;sin;0) as shown in the Fig. 1 (b). Using the Holstein-Primakov transformation, the Hamiltonian in the spin-wave approximation is obtained as HFI=X k~!kby kbk; (11) wherebkis the magnon annihilation operator with wave numberk,~!k=Dk2+~ hdcis the energy dispersion of a magnon,Dis the spin stiness, is the gyromagnetic ratio, andhdcis the externally applied DC magnetic eld. We note that the external DC magnetic eld controls the direction of the ordered spins. We introduce new coor- dinates (x0;y0;z0) xed on the ordered spins by rotating the original coordinates ( x;y;z ) as shown in Fig. 1 (b). Then, the magnon annihilation operator is related to the spin ladder operator by the Holstein-Primakov transfor- mation asSx0+ kSy0 k+iSz0 k= (2S0)1=2bk. The spin correlation function is dened as G0(k;i!n) =Z~ 0dei!nG0(k;); (12) G0(k;) =1 ~hSx0+ k()Sx0 k(0)i; (13) where!n= 2n=~is the bosonic Matsubara fre- quency. The spin correlation function is calculated from the Hamiltonian (11), as G0(k;i!n) =2S0=~ i!n!kGj!nj; (14)4 whereG>0 is a phenomenological dimensionless pa- rameter that describes the strength of the Gilbert damp- ing in the bulk FI. C. Eect of the FI/2DEG interface The coupling between the FI and 2DEG can be ac- counted for by the Hamiltonian, Hint=X k(TkSx0+ ksx0 k+T ksx0+ kSx0 k); (15) whereTkis an exchange interaction at a clean interface, for which the momentum of spin excitation is conserved. The spin ladder operators for conduction electrons, sx0 k, are obtained using a coordinate rotation as30 sx0 k=1 2X ;0X k0cy k0(^x0)0ck0k0; (16) ^x0=sinx+ cosyiz; (17) where ^x0^y0i^z0and 0 @^x0 ^y0 ^z01 A=0 @cossin0 sincos0 0 0 11 A0 @x y z1 A: Assuming that the interfacial exchange interaction is much smaller than the spin-orbit interactions, kFand kF59,60, we perform a second-order perturbation theory with respect to the interfacial exchange interaction Hint. Accordingly, the spin correlation function of the FI is calculated as G(k;i!n) =1 (G0(k;i!n))1(k;i!n); (18) (k;i!n) =jTkj2A(k;i!n); (19) where (k;i!n) is the self-energy due to the interfacial exchange coupling and (k;i!n) is the spin susceptibility for conduction electrons per unit area, dened as (k;i!n) =Z~ 0dei!n(k;); (20) (k;) =1 ~Ahsx0+ k()sx0 k(0)i; (21) wheresx0 k() =eHNM=~sx0 keHNM=~. Within the second-order perturbation, we only need to calculate the spin susceptibility for pure 2DEG without considering the junction because the interfacial coupling is already taken into account in the prefactor of the self-energy in Eq. (19). The uniform component of the retarded spin correlation function is obtained by analytic continuation (a)(b) FIG. 3. Feynman diagrams of (a) the uniform spin susceptibil- ity and (b) the Bethe-Salpeter equation for the ladder-type vertex function derived from the Born approximation. The cross with two dashed lines indicates interaction between an electron and an impurity. i!n!!+i, as GR(0;!) =2S0=~ !(!0+!0) +i(G+G)!;(22) !0 !0'2S0jT0j2A ~!0ReR(0;!0); (23) G'2S0jT0j2A ~!0ImR(0;!0); (24) where the superscript Rindicates the retarded compo- nent,!0=!k=0(= hdc) is the FMR frequency, and !0andGare respectively the changes in the FMR frequency and Gilbert damping due to the FI/2DEG in- terface. We note that in contrast with the bulk Gilbert dampingG, the increase of the Gilbert damping, G, can be related directly to the spin susceptibility of 2DEG as shown by Eq. (24). In fact, measurement of G has been utilized as a qualitative indicator of spin cur- rent through a junction61,62. In Eqs. (23) and (24), we made an approximation by replacing !with the FMR frequency!0by assuming that the FMR peak is su- ciently sharp ( G+G1). Thus, both the FMR frequency shift and the modulation of the Gilbert damp- ing are determined by the uniform spin susceptibility of the conduction electrons, (0;!). In what follows, we include the vertex correction for calculation of (0;!), which was not taken into account in our previous work30. III. VERTEX CORRECTION We calculate the spin susceptibility in the ladder approximation42,43that obeys the Ward-Takahashi re- lation with the self-energy in the Born approximation57. The Feynman diagrams for the corresponding spin sus- ceptibility and the Bethe-Salpeter equation for the vertex function are shown in Figs. 3 (a) and 3 (b), respectively.5 The spin susceptibility of 2DEG is written as (0;i!n) =1 4AX k;i!mTrh ^g(k;i!m)^(k;i!m;i!n) ^g(k;i!m+i!n)^x0i ; (25) where the vertex function ^(k;i!m;i!n) is a 22 matrix whose components are determined by the Bethe-Salpeter equation [see Fig. 3 (b)], 0(k;i!m;i!n) = (^x0+)0+u2ni AX qX 12g02(q;i!m) 21(q;i!m;i!n)g1(q;i!m+i!n):(26) Since the right-hand side of this equation is indepen- dent ofk, the vertex function can simply be described as^(i!m;i!n). We express the vertex function with the Pauli matrices as ^(i!m;i!n)E^I+X^x0+Y^y0+Z^z0; (27) whereE,X,Y, andZwill be determined self- consistently later. The Green's function for the conduc- tion electrons can be rewritten as ^g(q;i!m) =A^I+B^x0+C^y0 D; (28) A(i!m) =i~!mq+i 2sgn(!m); (29) B=hecos(); (30) C=hesin(); (31) D(i!m) =Y =[i~!mE q+i 2sgn(!m)]; (32) whereis the azimuth angle by which the eective Zee- man eld is written as he= (hecos;hesin;0). Thisheis written as he'kFp 2+2+ 2sin 2' using the Fermi wave number kF. By substituting Eqs. (27) and (28) into the second term of Eq. (26) and by the algebra of Pauli matrices, we obtain u2ni AX q^g(q;i!m)^(q;i!m;i!n)^g(q;i!m+i!n) =E0^I+X0^x0+Y0^y0+Z0^z0; (33) where 0 B@E0 X0 Y0 Z01 CA=0 B@0+ 1 0 0 0 0  0+ 2 3 0 0  3 02 0 0 0 0  011 CA0 B@E X Y Z1 CA; (34)and j(i!m;i!n) (j= 0;1;2;3) are expressed as 0(i!m;i!n) =u2ni AX qAA0 DD0; (35) 1(i!m;i!n) =u2ni AX qh2 e DD0; (36) 2(i!m;i!n) =u2ni AX qh2 ecos 2() DD0; (37) 3(i!m;i!n) =u2ni AX qh2 esin 2() DD0; (38) using the abbreviated symbols, A=A(i!m),A0= A(i!m+i!n),D=D(i!m), andD0=D(i!m+i!n). Here, we have used the fact that the contributions of the rst-order terms of BandCbecome zero after replacing the sum with the integral with respect to qand perform- ing the azimuth integration. We can solve for E,X, Y, andZby combining Eq. (34) and the Bethe-Salpeter equation (26), which we rewrite as E^I+X^x0+Y^y0+Z^z0 = ^x0++E0^I+X0^x0+Y0^y0+Z0^z0; (39) with ^x0+= ^y0+i^z0. The solution is E= 0; (40) X=3 (10)22 22 3; (41) Y=102 (10)22 22 3; (42) Z=i 10+ 1: (43) By replacing the sum with an integral as q, 1 AX q(


)'D(F)Z1 1dZ2 0d' 2(


); (44) Eqs. (35)-(38) can be rewritten as j(i!m;i!n) =(!m)(!m+!n)~j(i!n); (45) ~j(i!n) =i 4Z2 0d' 2 X ;0=fj(;0;') i~!n+ (0)he(') +i; (46) where we have used Eq. (8), (x) is a step function, and f0(;0;') = 1; (47) f1(;0;') =0; (48) f2(;0;') =0cos 2((')); (49) f3(;0;') =0sin 2((')): (50)6 For detailed derivation, see Appendix B. Substituting the Green's function and the vertex function into Eq. (25), we obtain (0;i!n) =1 4AX k;i!m2 DD0h 2BCX + (AA0B2+C2)Yi(AA0B2C2)Zi :(51) By summing over kand!mand by analytical continu- ation,i!n!!+i, the retarded spin susceptibility is obtained as63 R(0;!) =D(F)~! 2i~R 0(1~R 0)~R 2(1~R 2) + ( ~R 3)2 (1~R 0)2(~R 2)2(~R 3)2 +~R 0~R 1 1~R 0+~R 1 D(F); (52) where ~R j=~R j(!) =~j(i!n!!+i) =i 4
0Z2 0d' 2 X 0fj(;0;') ~!=
0+ (0)he=
0+i=
0:(53) A detailed derivation is given in Appendix C. Here, we have introduced a unit of energy,
0=kF, for the con- venience of making the physical quantities dimensionless. Using Eqs. (23) and (24), we nally obtain the shift in the FMR frequency and the modulation of the Gilbert damping as !0 !0=G;0ReF(!0); (54) G=G;0ImF(!0); (55) F(!) =
0 2i~R 0(1~R 0)~R 2(1~R 2) + ( ~R 3)2 (1~R 0)2(~R 2)2(~R 3)2 +~R 0~R 1 1~R 0+~R 1
0 ~!; (56) whereG;0= 2S0jT0j2AD(F)=
0is a dimensionless parameter that describes the coupling strength at the interface. This is our main result. The spin susceptibility without the vertex correction can be obtained by taking the rst-order term with re-spect to ~R j: R(0;!)'~!D(F) 2i 2~R 0~R 1~R 2 D(F) =~!D(F)Zd' 2h1 ~!+i1cos2((')) 2 +1 ~!2he(') +i1 + cos2((')) 4 +1 ~!+ 2he(') +i1 + cos2((')) 4i D(F): (57) The imaginary part of R(0;!) reproduces the result of Ref. 30. Using this expression, the shift in the FMR frequency and the modulation of the Gilbert damping without the vertex correction are obtained as !nv 0 !0=G;0ReFnv(!0); (58) nv G=G;0ImFnv(!0); (59) Fnv(!) =
0 2i 2~R 0~R 1~R 2
0 ~!; (60) IV. MODULATION OF THE GILBERT DAMPING First, we show the result for the modulation of the Gilbert damping, G, for= = 0, 1, and 3 and dis- cuss the eect of the vertex correction by comparing it with the result without the vertex correction in Sec. IV A. Next, we discuss the strong enhancement of the Gilbert damping near == 1 in Sec. IV B. A. Eect of vertex corrections First, let us discuss the case of = = 0, i.e., the case when only the Rashba spin-orbit interaction exists64. Figure 4 (a) shows the eective Zeeman eld healong the Fermi surface. Figures 4 (b) and 4 (c) show the modulations of the Gilbert damping without and with the vertex correction. The horizontal axes of Figs. 4 (b) and 4 (c) denote the resonant frequency !0= hdcin the FMR experiment. Note that the modulation of the Gilbert damping, G, is independent of , i.e., the az- imuth angle ofhSi. The four curves in Figs. 4 (b) and 4 (c) correspond to =
0= 0:1, 0:2, 0:5, and 1:065. We nd that these two graphs have a common qualitative feature; the modulation of the Gilbert damping has two peaks at!0= 0 and!0= 2
0and their widths become larger as increases. The peak at !0= 0 corresponds to elastic spin- ipping of conduction electrons induced by the transverse magnetic eld via the exchange bias of the FI, while the peak at ~!0= 2
0is induced by spin excitation of conduction electrons due to magnon7 Without vertex corrections Without vertex corrections Without vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections
Without vertex corrections With vertex corrections With vertex corrections FIG. 4. (Left panels) Eective Zeeman eld heon the Fermi surface. (Middle panels) Modulation of the Gilbert damping, nv G, without vertex correction. (Right panels) Modulation of the Gilbert damping with vertex correction, G. In the middle and right panels, the modulation of the Gilbert damping is plotted as a function of the FMR frequency, !0= hdc. The spin-orbit interactions are as follows. (a), (b), (c): == 0. (d), (e), (f): == 1. (g), (h), (i): == 3. We note that (b), (e), (h) are essentially the same result as Ref. 30. absorption30. In the case of = = 0, the vertex cor- rection changes the modulation of the Gilbert damping moderately [compare Figs. 4 (c) with 4 (b)]. The widths of the two peaks at !0= 0 and!0= 2
0become nar- rower when the vertex correction is taken into account (see Appendix D for the analytic expressions). The case of = = 1 is special because the eective Zeeman eld healways points in the direction of ( 1;1) or (1;1), as shown in Fig. 4 (d). The amplitude of hedepends on the angle of the wave number of the conduction electrons, ', he(') = 2
0jsin('+=4)j; (61) and varies in the range of 0 2he4
0. Figures 4 (e) and 4 (f) show the modulation of the Gilbert dampingwithout and with the vertex correction for =
0= 0:5. The ve curves correspond to ve dierent angles of hSi, ==4;=8;0;=8, and=4. The most remarkable feature revealed by comparing Figs. 4 (f) with 4 (e) is that the peak at !0= 0 disappears if the vertex correction is taken into account (see Appendix E for the analytic expressions). In the subsequent section, we will show thatG(!0) has a-function-like singularity at !0= 0 for= = 1 due to the spin conservation law along the direction ofhe. In the case of = = 3, the direction of the eec- tive Zeeman eld hevaries along the Fermi surface [Fig. 4 (g)]. Figures 4 (h) and 4 (i) show the modula- tion of the Gilbert damping without and with the ver- tex correction for =
0= 0:5. For= = 3, a peak8 Without vertex corrections Without vertex corrections Without vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections
Without vertex corrections With vertex corrections With vertex corrections FIG. 5. Modulation of the Gilbert damping calculated for == 1:1 (a) without the vertex correction and (b) with the vertex correction. The horizontal axis is the FMR frequency !0and the ve curves correspond to ve dierent angles of hSi, i.e., ==4;=8;0;=8, and=4. (c) Enlarged plot of the modulations of the Gilbert damping as a function of the FMR frequency!0. The angle ofhSiis xed as==4 and the three curves correspond to = = 1:03, 1:05, and 1:1. In all the plots, we have chosen =
0= 0:5. at!0= 0 appears even when the vertex correction is taken into account. The broad structure in the range of 4
0~!08
0is caused by the magnon absorption process where its range re ects the distribution of the spin-splitting energy 2 healong the Fermi surface. By comparing Figs. 4 (h) and 4 (i), we nd that the vertex correction changes the result only moderately as in the case of= = 0; the peak structure at !0= 0 becomes sharper when the vertex correction is taken into account while the broad structure is slightly enhanced. B. Strong enhancement of the Gilbert damping Here, we examine the strong enhancement of the Gilbert damping for ='1. As explained in Sec. II A, the spin component in the direction of the azimuth angle 3=4 in thexyplane is exactly conserved at = = 1 [see also Fig. 4 (d)]. When the value of = is shifted slightly from 1, the spin conservation law is broken but the spin relaxation becomes remarkably slow. To see this eect, we show the modulation of the Gilbert damping without and with the vertex correction for = = 1:1 in Figs. 5 (a) and 5 (b), respectively. The ve curves correspond to ve dierent azimuth angles of hSi, and the energy broadening is set as =
0= 0:5. Figs. 5 (a) and 5 (b) indicate that the Gilbert damping is strongly enhanced at !0= 0 only when the vertex correction is taken into account. This is the main result of our work. Figure 5 (c) plots the modulation of the Gilbert damp- ing with the vertex correction for =
0= 0:5 and ==4, the latter of which corresponds to the case of the strongest enhancement at !0= 0. The three curves correspond to == 1:03, 1:05, and 1:1. As the ratio of =approaches 1, the peak height at !0= 0 gets larger. Without vertex corrections Without vertex corrections Without vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections Without vertex corrections With vertex corrections With vertex corrections





FIG. 6. Modulation of the Gilbert damping as a function of=. The ve curves correspond to ~!0=
0= 0;0:005, 0:01, 0:02, and 0:05. We have taken the vertex correction into account and have chosen =
0= 0:5. The inset illustrates maximum values of the modulation of the Gilbert damping, G;max, in varying = for a xed value of ~!0=
0. For='1,Gis calculated approximately as G G;0'
0 2s (~!0)2+ 2ssin2 + 4 ; (62) s2 Z2 0d' 2(hx+hy)2 1 + (2he=)2; (63) where sgives the peak width in Figs. 5 (b) and 5 (c) (see Appendix F for a detailed derivation). For == 1 + (1), sis proportional to 2and approaches zero in the limit of !0. This indicates that scorresponds to the spin relaxation rate due to a small breakdown of the spin conservation law away from the special point of = = 1. Note that the peak height of Gat!0= 0 diverges at = = 1. This indicates that for = = 1,G(!0) has a-function-like singularity at !0= 0, which is not drawn in Fig. 4 (f).9 FIG. 7. (Upper panels) Modulations of the Gilbert damping, G=G;0for (a)== 0, (b)== 1, and (c) == 3. (Lower panels) Shifts in the FMR frequency, !0=(G;0!0), for (d)= = 0, (e)= = 1, and (f) = = 3. The horizontal axes are the FMR frequency, !0= hdc, while the vertical axes show the azimuth angle of the spontaneous spin polarization, , in the FI. In all the plots, we have considered vertex corrections and have chosen =
0= 0:5. In (a), (c), and (e) there are regions in which the values exceed the upper limits of the color bar located in the right side of each plot; the maximum value is about 0:45 in (a), 0 :65 in (c), and about 10 in (e) (see also Fig. 8). In addition, (b) cannot express a -function-like singularity at !0= 0 (see the main text). Figure 6 plots the modulation of the Gilbert damping for =
0= 0:5 and==4 as a function of =. The ve curves correspond to ~!0=
0= 0;0:005;0:01;0:02, and 0:05, respectively. This gure indicates that when we x the resonant frequency !0and vary the ratio of =, the Gilbert damping is strongly enhanced when = is slightly smaller or larger than 1. We expect that this enhancement of the Gilbert damping is strong enough to be observed experimentally. We note that G=G;0ap- proaches 0:378 (0:318) for=!0 (=!1 ). The inset in Fig. 6 plots maximum values of G=G;0when =is varied for a xed value of ~!0=
0. In other words, the vertical axis of the inset corresponds to the peak height in the main panel for each value of ~!0=
0. We nd that the maximum value of G=G;0diverges as !0 approaches zero. V. SHIFT IN THE FMR FREQUENCY Next, we discuss the shift in the FMR frequency when the vertex correction is taken into account. The den- sity plots in Figs. 7 (a), 7 (b), and 7 (c) for == 0, 1, and 3 summarize the modulation of the Gilbert damping, G. These plots have the same features as in Figs. 4 (c), 4 (f), and 4 (i). Figures. 7 (d), 7 (e), and 7 (f) plot the shift in the FMR frequency !0=!0with density plots for= = 0, 1, and 3. By comparing Figs. 7 (a), 7 (b), Without vertex corrections Without vertex corrections Without vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections Without vertex corrections With vertex corrections With vertex corrections





FIG. 8. Shift in FMR frequency, !0=(G;0!0), as a func- tion of the resonance frequency !0for= = 1:1. The in- set shows the same quantities in the low-frequency range of 0~!0=
00:05 with a larger scale on the vertical axis. We have taken the vertex correction into account and have chosen =
0= 0:5. and 7 (c) with 7 (d), 7 (e), and 7 (f), we nd that some of the qualitative features of the FMR frequency shift are common to those of the modulation of the Gilbert damping,G; (i) they depend on for= > 0, while they do not depend on for= = 0, (ii) the structure10 at!0= 0 due to elastic spin- ipping appears, and (iii) the structure within a nite range of frequencies due to magnon absorption appears. We can also see a few dif- ferences between Gand!0=!0. For example, !0=!0 has a dip-and-peak structure at ~!0=
0= 2 whereG has only a peak. Related to this feature, !0=!0has a tail that decays more slowly than that for G. The most remarkable dierence is that !0=!0diverges at !0= 0 for= = 1 except for = 3=4;7=4, re ecting the - function-like singularity of Gat!0= 0. These features are reasonable because !0=!0andG, which are de- termined by the real and imaginary parts of the retarded spin susceptibility, are related to each other through the Kramers-Kronig conversion. The main panel of Fig. 8 shows the frequency shift !0=!0for= = 1:1 as a function of the resonant frequency!0. The ve curves correspond to = =4;=8;0;=8, and=4. Although the frequency shift appears to diverge in the limit of !0!0 in the scale of the main panel, it actually grows to a nite value and then goes to zero as !0approaches zero (see the inset of Fig. 8). For == 1 +(1), the frequency shift is calculated approximately as !0 G;0!0'
0 2~!0 (~!0)2+ 2ssin2 + 4 ; (64) where sis the spin relaxation rate dened in Eq. (63) (see Appendix F for the detailed derivation). We expect that this strong enhancement of the frequency shift near == 1 can be observed experimentally. VI. SUMMARY We theoretically investigated spin pumping into a two- dimensional electron gas (2DEG) with a textured eec- tive Zeeman eld caused by Rashba- and Dresselhaus- type spin-orbit interactions. We expressed the change in the peak position and the linewidth in a ferromag- netic resonance (FMR) experiment that is induced by the 2DEG within a second-order perturbation with re- spect to the interfacial exchange coupling by taking the vertex correction into account. The FMR frequency and linewidth are modulated by elastic spin- ipping or magnon absorption. We found that, for almost all of the parameters, the vertex correction modies the modula- tion of the Gilbert damping only moderately and does not change the qualitative features obtained in our previous paper30. However, we found that the Gilbert damping at low frequencies, which is caused by elastic spin- ipping, is strongly enhanced when the Rashba- and Dresselhaus- type spin-orbit interactions are chosen to be almost equal but slightly dierent. Even in this situation, the Gilbert damping at high frequencies, which is caused by magnon absorption, shows small modication. This strong en- hancement of the Gilbert damping at low frequencies ap- pears only when the vertex correction is taken into ac- count and is considered to originate from the slow spinrelaxation related to the spin conservation law that holds when the two spin-orbit interactions completely match. A similar enhancement was found for the frequency shift of the FMR due to elastic spin- ipping. We expect that this remarkable enhancement can be observed experi- mentally. Our work provides a theoretical foundation for spin pumping into two-dimensional electrons with a spin- textured Zeeman eld on the Fermi surface. Although we have treated a specic model for two-dimensional electron systems with both the Rashba and Dresselhaus spin-orbit interactions, our formulation and results will be helpful for describing spin pumping into general two- dimensional electron systems such as surface/interface states66{68and atomic layer compounds69,70. ACKNOWLEDGEMENTS The authors thank Y. Suzuki, Y. Kato, and A. Shi- tade for helpful discussion. T. K. acknowledges sup- port from the Japan Society for the Promotion of Sci- ence (JSPS KAKENHI Grant No. JP20K03831). M. M. is nancially supported by a Grant-in-Aid for Scientic Research B (Grants No. JP20H01863, No. JP21H04565, and No. JP21H01800) from MEXT, Japan. M. Y. is sup- ported by JST SPRING (Grant No. JPMJSP2108). Appendix A: Calculation of Green's function In our work, Green's function of conduction electrons is calculated by taking eect of impurity scattering into account. In general, the nite-temperature Green's func- tion ^g(k;i!m) after the impurity average is described by the Dyson equation with the impurity self-energy ^(k;!m) as ^g(k;i!m) =1 ^g0(k;i!m)1^(k;i!m); (A1) where ^g0(k;i!m)1is Green's function of electrons in the absence of impurities. In our work, we employ the Born approximation in which the self-energy is approximated by second-order perturbation with respect to an impurity potential. In the Born approximation, the self-energy is given as ^(k;i!m) =niu2Zd2k (2)2^g0(k;i!m); (A2) whereniis the impurity concentration. The correspond- ing Feynman diagram of the Dyson equation is shown in Fig. 9. By straightforward calculation, Eq. (7) can be derived. For a detailed derivation, see Ref. 30.11 FIG. 9. The Feynman diagram for Green's function within the Born approximation. Appendix B: Derivation of Equations. (45)-(50) Eqs. (35)-(38) can be rewritten with = 2 niu2D(F) as 0(i!m;i!n) =i 4Z2 0d' 2X ;0I0; (B1) 1(i!m;i!n) =i 4Z2 0d' 2X ;00I0; (B2) 2(i!m;i!n) =i 4Z2 0d' 2cos 2(')X ;00I0; (B3) 3(i!m;i!n) =i 4Z2 0d' 2sin 2(')X ;00I0; (B4) where I0=Z1 1d 2i1 i~!mhe+i(=2)sgn(!m) 1 i~(!m+!n)0he+i(=2)sgn(!m+!n): (B5) We note that one needs to calculate this integral only for !n>0 to obtain the retarded component by analytic continuation. Then, we can easily prove by the residue integral that I0= 0 for!m>0 and!m+!n>0 (!m<0 and!m+!n<0) because both of the two poles in the integrand are located only in the upper (lower) half of the complex plane of . For!m<0 and!m+!n>0, the integral is evaluated by the residue integral as I0=1 i~!n+ (0)he+i: (B6) By combining these results, Eqs. (45)-(50) can be de- rived. (a) (b) FIG. 10. Schematic picture of the change in the contour in- tegral. (a) The original contour. (b) The modied contour. Appendix C: Derivation of Eq. (52) In this Appendix, we give a detailed derivation of Eq. (52) from Eq. (51). First, we modify Eq. (51) as (0;i!n) =1 8AX kX ;0" 0sin 2()I0;1 +n 10cos 2()o I0;2 i(10)I0;3# ; (C1) where I0;j1 X i!mXj i~!mE k+i=2 sgn(!m) 1 i~!m+i~!nE0 k+i=2 sgn(!m+!n);(C2) and (X1;X2;X3) = (X;Y;Z ). A standard procedure based on the residue integral enables us to express the sumI0;jfor!n>0 as a complex integral on the con- tour C shown in Fig. 10 (a). This contour can be modied into a sum of the four contours, C l(l= 1;2;3;4), shown in Fig. 10 (b). Accordingly, I0;jis written as I0;j=4X l=1ICl 0;j; (C3) ICl 0;j=Z Cldz 2if(z)Xj(z;i!n) zE k+i=2 sgn(Imz) 1 z+i~!nE0 k+i=2 sgn(Imz+!n);(C4) wheref(z) = 1=(ez+ 1) is the Fermi distribution func- tion. The sum of the contributions from the two contours,12 C2and C 3, is calculated as IC2 0;j+IC3 0;j =~Xj(i!n)ZdE 2if(E) " 1 EE ki=21 E+i~!nE0 k+i=2 +1 Ei~!nE ki=21 EE0 k+i=2# :(C5) Here, we have used the fact that Xj(z;i!n) is indepen- dent ofzfor 0<Imz < !nfrom Eq. (45) and have dened its value as ~Xj(i!n) (j= 1;2;3). From Eqs. (41)- (43), ~Xj(i!n) are calculated as ~X1(i!n) =~3(i!n) (1~0(i!n))2~2(i!n)2~3(i!n)2;(C6) ~X2(i!n) =1~0(i!n)~2(i!n) (1~0(i!n))2~2(i!n)2~3(i!n)2;(C7) ~X3(i!n) =i 1~0(i!n) +~1(i!n): (C8) By changing the integral variable to E0=EE kin the rst term and to E0=(EE0 k) in the second term in Eq. (C5), we obtain IC2 0;j+IC3 0;j=~Xj(i!n)ZdE0 2i1 E0i=2 " f(E0+E0 k)f(E0+E k) E0+i~!n+E kE0 k+i=2# :(C9) Using formula (44), we replace the sum over kin Eq. (C1) by the integral with respect to and'. We can perform the-integral by using Z1 1d[f(E0+E k)f(E0+E0 k)] = 2E0+E kE0 k: (C10) Then, by performing the E0-integral, we obtain Z1 1d(IC2 0;j+IC3 0;j) =i~!n~Xj(i!n) E kE0 k+i~!n+i: (C11) Next, let us consider the contribution from C 1and C 4. On these two contours, Xj(z;i!n) is independent of z and its value is dened by ~X0 j(i!n) (j= 1;2;3). Because j(z;i!n) (j= 0;1;2;3) becomes zero for Im z < 0 or !n<Imzfrom Eq. (45), ~X0 j(i!n) are given as ~X0 1(i!n) = 0;~X0 2(i!n) = 1;~X0 3(i!n) =i:(C12)A similar calculation to that of C 2and C 3yields Z1 1d(IC1 0;j+IC4 0;j) =~X0 j(i!n): (C13) By substituting these results into Eq. (C1), we obtain (0;i!n) =D(F) 8X ;0Z2 0d' 2" 0sin 2() ~x1(i!n) + [10cos 2()][1 + ~x2(i!n)] i(10)[i+ ~x3(i!n)]# : (C14) where ~xj(i!n) =i~!n~Xj(i!n) E kE0 k+i~!n+i: (C15) Finally, Eq. (52) is derived by substituting the expres- sions for ~Xj(i!n) and by analytic continuation i!n! !+i. Appendix D: Analytic Expression for == 0 In this appendix, we derive analytic expressions of the modulation of the Gilbert damping when == 0, only the Rashba spin-orbit interaction exists, to see the quan- titative eect of taking the vertex correction into account. For== 0, the spin-splitting energy 2 he= 2
0is con- stant along the Fermi surface, and ~R j(!) (j= 0;1;2;3) is simplied as ~R 0(!) =i 4
0X 01 ~!=
0+ (0) +i=
0;(D1) ~R 1(!) =i 4
0X 00 ~!=
0+ (0) +i=
0;(D2) ~R 2(!) =~R 3(!) = 0: (D3) Then, we obtain the modulation of the Gilbert damping with the vertex corrections, G G;0'
0 2Re"~R 0(!0) 1~R 0(!0)+~R 0(!0)~R 1(!0) 1~R 0(!0) +~R 1(!0)# : (D4) The modulation of the Gilbert damping without the ver- tex correction is obtained by considering only the rst- order term with respect to ~R j(!0), nv G G;0'
0 2Re" 2~R 0(!0)~R 1(!0)# : (D5) When 
0, the contribution of =0is dominant for the peak at !0= 0 and the modulation of the Gilbert13 damping can be analytically calculated as G G;0'
0 4
=2 (~!0)2+ (=2)2; (D6) nv G G;0'
0 4
(~!0)2+ 2: (D7) This indicates that the peak width is halved by taking the vertex correction into account, which is consistent with the results shown in Figs. 4 (b) and 4 (c). In a similar way, we can evaluate the modulation of the Gilbert damping near the peak at !0= 2
0=~as G G;0'
0 4
1 23=4 (~!02
0)2+ (3=4)2 +=2 (~!02
0)2+ (=2)2 ; (D8) nv G G;0'
0 4
3=2 (~!02
0)2+ 2: (D9) As well, for the peak at !0= 2
0=~, the peak width becomes smaller when the vertex correction is taken into account. This observation is consistent with the results shown in Figs. 4 (b) and 4 (c). For a nite value of , a sum of Eqs. (D6) and (D8) [Eqs. (D7) and (D9)] gives a better analytic form which ts the numerical result with (without) the vertex correction. Note that Gandnv G depend on the impurity potential strength, u, and im- purity concentration, ni, through = 2 niu2D(F) [see Eq. (8)]. As shown in Eqs. (D6)-(D9), the peak widths of the Lorentzian functions in Gandnv Gare deter- mined by [see Figs. 4 (b) and 4 (c)]. It is remarkable that the peak width in Gis reduced from to =2 by taking the vertex correction into account. To summarize the eect of the vertex correction, we show Gnv G and!0!nv 0in Figs. 11 (a) and 11 (d), respectively. We nd that the vertex correction modies mainly the peak width around ~!0=
0= 0 and 2, in consistent with the above analytic expressions. Finally, we note that the same analytical expressions forGandnv Gcan be obtained for the case of == 0, i.e., when only the Dresselhaus spin-orbit interaction exists. We also note that for general values of =,G andnv Gdepend on in a more complicated way. Appendix E: Analytic Expression for == 1 In this Appendix, we derive analytic expressions of the modulation of the Gilbert damping when = = 1. In this case, the eective Zeeman eld is parallel to the (1;1;0) direction and its amplitude is given as he(') = 2
0jsin('+=4)j: (E1)Then, ~R j(!) (j= 0;1;2;3) becomes ~R 0(!) =i 4
0X 0J0 (E2) ~R 1(!) =i 4
0X 00J0; (E3) ~R 2(!) =sin 2~R 1(!); (E4) ~R 3(!) =cos 2~R 1(!) (E5) where J0(!)Z2 0d' 2
0 ~!+ (0)he(') +i:(E6) In the case of ==4, the modulation of the Gilbert damping with the vertex correction is expressed as G G;0=
0 2Re" 2 +1 1~R 0(!0) +~R 1(!0) +1 1~R 0(!0)~R 1(!0)# : (E7) The third term of the above equation is calculated as 1 1~R 0(!0)~R 1(!0)=1 1i ~!0+i=~!0+i ~!0:(E8) This indicates that the expansion with respect to ~R jcan not be allowed for !0. This is why the modula- tion without the vertex correction, which is obtained by taking from the rst-order term of ~R jin Eq. (E7) as nv G G;0=
0 2Re 2~R 0(!0) ; (E9) gives a dierent result near !0'0. Actually, for ==4, Gandnv Gare calculated as G G;0=
0 2Re" i 2
0(J++J+) 1i 2
0(J++J+)# ; (E10) nv G G;0=1 4Re" i J++J++J+++J# :(E11) Note that Eq.(E10) is not valid for !0= 0. As indicated from the absence of J++andJ, the graph of G(!0) has no peak at zero frequency even though nv G(!0) has a peak there. This observation is consistent with Figs. 4 (e) and 4 (f). In the case of ==4, the modulations of the Gilbert damping with and without the vertex correction are G G;0=
0 Re" i 2
0(J++J+) 1i 2
0(J++J+)# ; (E12) nv G G;0=1 2Re" i(J++J+)# : (E13)14 FIG. 11. (Upper panels) Change of the Gilbert damping due to the vertex correction, Gnv G, for (a)== 0, (b)== 1, and (c)== 3. (Lower panels) Change of the FMR frequency due to the vertex correction, !0!nv 0, for (d)== 0, (e) == 1, and (f) == 3. The horizontal axes are the FMR frequency, !0= hdc, whereas the vertical axes are the azimuth angle of the spontaneous spin polarization, , in the FI. In all the plots, we have set =
0= 0:5. Note thatnv Gis obtained by taking the rst-order term in Eq. (E12). As indicated by the absence of the terms, J++andJ, neitherGnornv Ghas any structure around!0= 0. It can be checked that these two expres- sions give almost the same result when .
0, which is consistent with Figs. 4 (e) and 4 (f). Note as well that Gis just doubled compared with the result for ==4 in Eq. (E10). To summarize the eect of the vertex correction, we showGnv Gand!0!nv 0in Figs. 11 (b) and 11 (e), respectively. We nd that the vertex correction modies mainly the peak width around ~!0=
0= 0. In addition, the broad peak in the range of 0 <~!0< 2
0is enhanced or suppressed depending on the azimuth angle of the ordered spin. These features are consistent with the above analytic expressions. We note that similar features are observed for == 3 as seen in Figs. 11 (c) and 11 (f). Appendix F: Approximate Expressions near == 1 In this Appendix, we derive the approximate expres- sions Eqs. (62) and (64) for = = 1 +(1) and !'0. For== 1 +(1), we can use the approx- imation, cos 2()'sin 2 1 +(hx+hy)2 h2 e ; (F1) sin 2()'cos 2 1 +(hx+hy)2 h2 e : (F2)Then, we obtain ~R 2'Xsin 2; (F3) ~R 3'Xcos 2; (F4) Xi 4Z2 0d' 2X 00 1 +(hx+hy)2 h2 eff ~!+ (0)he+i(F5) in the low-frequency region. Here, the contribution of the second term of the bracket in Eq. (56) does not have a singularity at !0= 0 because ~R 0and ~R 1do not de- pend on the eective Zeeman eld he. Therefore, the singularity comes from the rst term of the bracket in Eq. (56) and we can approximate F(!) as F(!)'
0 2i~R 0(1~R 0)~R 2(1~R 2) + ( ~R 3)2 (1~R 0)2(~R 2)2(~R 3)2 =
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1408.0349v1.Machta_Zwanzig_regime_of_anomalous_diffusion_in_infinite_horizon_billiards.pdf

Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards Giampaolo Cristadoro,1Thomas Gilbert,2Marco Lenci,1, 3and David P. Sanders4 1Dipartimento di Matematica, Universit `a di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy 2Center for Nonlinear Phenomena and Complex Systems, Universit ´e Libre de Bruxelles, C. P . 231, Campus Plaine, B-1050 Brussels, Belgium 3Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy 4Departamento de F ´ısica, Facultad de Ciencias, Universidad Nacional Aut´onoma de M ´exico, Ciudad Universitaria, 04510 M ´exico D.F ., Mexico We study diffusion on a periodic billiard table with infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a L ´evy walk com- bining exponentially-distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [Phys. Rev. Lett. 50, 1959 (1983)]. PACS numbers: 05.60.-k ,05.40.Fb,05.45.-a,02.50.-r The study of stochastic processes exhibiting anomalous transport has attracted considerable attention in the recent past, in particular in the context of continuous-time random walks [1–6]. It has since evolved into a highly interdisci- plinary field, and includes a large array of applications [7]. From a fundamental perspective, a problem of specific in- terest concerns the emergence of such stochastic processes from deterministic dynamics [8, 9], and the derivation of their limiting distributions [10], which typically differ from Gaus- sian, as, for instance, L ´evy stable distributions [11]. Infinite-horizon periodic billiard tables, which are the fo- cus of this Letter, provide an example of mechanical systems whose transport properties exhibit a weak form of anomalous behavior, in the sense that there is a logarithmic correction to the linear growth in time of the mean squared displacement [12–14]; this is in fact a marginal case, separating diffusive and superdiffusive regimes [15]. It is known, in particular, that the distribution of the anomalously rescaled displacement vector, [r(t)r(0)]=ptlogt, converges weakly to a normal distribution, whose covariance matrix is given in terms of a scalar expression of the model’s parameter, the so-called Ble- her variance [16–18]. As opposed to finite-horizon billiards, which are such that the displacement between two successive collisions of a point particle with obstacles remains bounded, infinite-horizon bil- liards admit, among their solutions, collisionless trajectories; point particles can therefore move arbitrarily far through in- finite corridors devoid of obstacles. When the widths of these corridors is taken to be vanishingly small, a separa- tion between two timescales occurs, the first associated with the propagation phases of a trajectory across the billiard table along the corridors and the second with an effective trapping mechanism which consists of a scattering phase that takes place within a single billiard cell, separating successive prop- agation phases. While the former timescale is independent of the widths of the corridors, the latter grows unbounded in the narrow-corridor regime. The purpose of this Letter is to show that, in the limit ofnarrow corridors, a novel regime emerges, such that a normal contribution to the finite-time diffusion coefficient may be- come arbitrarily larger than the anomalous component, even though the latter diverges logarithmically as time increases while the former remains asymptotically constant. Due to its connection with the Machta-Zwanzig approximation of the diffusion coefficient of finite-horizon billiard tables [19], which is obtained in a limit similar to that of narrow corridors, we refer to this limit as a Machta-Zwanzig regime of anoma- lous diffusion in infinite-horizon billiards. To analyze this regime, we model the process on the infinite-horizon billiard table by a L ´evy walk [20, 21]. Our approach is based on the framework of continuous-time ran- dom walks and relies on the distinction between the states of particles in propagating and scattering phases and the deriva- tion of a multistate generalized master equation [22, 23]. The process whereby a walker on a lattice remains in a scatter- ing phase during an exponentially-distributed waiting time and then propagates in a random direction over a free path of random length is such a multistate process and it can be solved analytically [24]. When applied to the infinite-horizon billiard, the probability distribution of free paths decays al- gebraically with the third power of their lengths [13, 16]. The mean squared displacement then has two relevant compo- nents: The first associated with the anomalous rescaling of the displacement vector [16–18], and the second with a normal diffusion component, similar to the Machta-Zwanzig approx- imation obtained in the finite-horizon regime [19]. Whereas the former contribution is second order in the widths of the corridors, the latter is first order and, for finite time, becomes much larger than the former. We report numerical measurements of the mean square dis- placement of infinite-horizon periodic billiard tables which exhibit those two contributions, in agreement with the results predicted by the L ´evy walk model. Model. We consider the simplest kind of billiard in the plane, which is defined by a periodic array of square cells of sides `with identical circular scatterers of radii r, 0<arXiv:1408.0349v1 [cond-mat.stat-mech] 2 Aug 20142 Hm-3,nLHm-2,nLHm-1,nLHm,nLHm+1,nLHm+2,nL {d FIG. 1. (Color online) Infinite-horizon periodic billiard table of square cells of sides `. The discs centered at the cells’ corners have radii r=0:45`. Infinite corridors of widths d=`2rspan along horizontal and vertical axes, through the centers of each cell. The broken red line shows a trajectory. r< `= 2, whose centers are placed at the corners of the cells. A point-particle moves freely on the exterior of these obsta- cles, performing specular collisions upon their boundaries; see Fig. 1. This is an infinite-horizon configuration, which refers to the existence of collisionless trajectories along the vertical and horizontal corridors that separate the obstacles. In contrast, afinite-horizon configuration would occur if, for instance, an additional circular obstacle of radius r, with`=2r<r< `=p 2r, were placed at the center of each cell [25], such that a particle would have to perform at least one collision in each visited cell. The widths of the corridors is d=`2r, which we take to be a control parameter. Transport of particles on such billiard tables can be studied in terms of the time-evolution of the coarsegrained distribu- tion of particles in cell n2Z2, whose changes are determined by the transitions particles make as they go from one cell to another. Generally speaking, this is a complicated process which is affected by correlations between successive collision events. The situation however simplifies when these correla- tions become negligible, which occurs when d`. Infinite vs finite horizon. In finite-horizon tables, where diffusion is always normal [26], this regime yields an approx- imation of the process by a continuous-time random walk for displacements on the lattice structure [27]. The associated timescale is given in terms of the residence time, tR=pA 4p0d; (1) where Adenotes the area of the elementary cell, viz.A= `2p(r2+r2 ), and p0is the speed of point particles [28]. When tRis large with respect to the intercollisional time, a particle typically spends a long time rattling about the same cell, making many collisions with its obstacles, before reach- ing one of the four slits of widths d. Under these conditions, the velocity vectors of a particle entering and subsequently exiting a cell will effectively be uncorrelated. The motion of a particle on the billiard table can thus be approximated by a succession of scattering phases with random waiting times, exponentially-distributed with scale tR, punctuated by instan- taneous random hops from one cell to a neighboring one.In this regime, the mean squared displacement grows lin- early in time, with coefficient approximated by 4 DMZ, where DMZ=`2 4tR(2) is a dimensional expression, known as the Machta-Zwanzig approximation to the diffusion coefficient [19]. To leading order, the actual diffusion coefficient differs from Eq. (2) by a correction which is expected to be linear in d=`[29]. The validity of the Machta-Zwanzig approximation thus relies on the separation d`. In the case of infinite horizon, the presence of corridors ren- ders the Machta-Zwanzig regime more complex. In addition to the residence time tR, Eq. (1), one has to take into consid- eration the timescale of propagation across a cell, tF, which, in the narrow-corridor regime is small with respect to tR[30], tF` p0tR: (3) When the phase-space coordinates of a particle are such that it crosses the boundary between two cells with velocity almost perpendicular to it, the next scattering phase may take place at a remote distance, after a time approximately equal to tF multiplied by the number of cells traveled free of collisions. The assumption that correlations between successive scat- tering phases become negligible amounts to approximating the transport process on the infinite-horizon billiard table by a L´evy walk, whose distribution of free paths, i.e. the dis- tance a particle propagates free of collisions, matches that of the infinite-horizon billiard, taking into account the timede- lay induced by the propagation along free paths. The general framework for analyzing L ´evy walks where a distinction oc- curs between the states of particles in scattering and propa- gating phases was described in Ref. [24]. Whereas particles in a scattering state make transitions at random times, expo- nentially distributed with scale tR, and move in a random di- rection to a neighboring cell, the transitions of particles in a propagating state are deterministic: They take place after ex- actly time tFand are accompanied by displacements along the same direction as the previous transition. Parameter values. We label the internal state of a walker byk2Z, with k=0 denoting the scattering state and k1 a propagating state, where kcorresponds to the total number of steps of duration tFuntil the propagating state returns to a scattering one. The direction of propagation is fixed in the propagating state. When the particle is in the scattering state, we let mkdenote the probability of a transition to either a scat- tering state, k=0, or a propagating state, k1, for any of the four lattice directions the walker can move in. In the narrow-corridor regime, the overwhelming major- ity of transitions are from scattering to scattering states; only rarely do particles perform transitions from scattering to prop- agating states. When they do, however, as mentioned ear- lier, the probability of a long excursion to a propagating state, k1, decays with the third power of their lengths, mkk3.3 GjH0LGjH0L GjH1L GjH1L GjH2LGjH2L -d {-d 2{0d 2{d {-d 2-d 40d 4d 2 p°p0s FIG. 2. (Color online) Graph showing two-dimensional phase-space sets according to the distance separating phase points at the boundary between two cells from their next collision on a disc. The darkest color encodes sets of points whose next collision takes place in the cell they move into; the brighter the colors the more distant the next collision. The two axes correspond, respectively, to pk=p0, which denotes the fraction of the velocity component directed along the boundary,1<pk=p0<1 (here restricted to d=` < pk=p0<d=`) and to the position salong the boundary, d=2<s<d=2, where d=`2r(r=0:45`), is the width of the corridor. With these coordinates, the area corresponds to the natural invariant measure, which is normalized if divided by 2 d. In fact, to first order in d=`, we can write mk=( 1d 4`; k=0; d k(k+1)(k+2)`;k1:(4) To obtain this result, consider the phase-space sets that lie at the boundary between two neighboring cells nandn+ej, where ejis a unit vector in one of the four lattice directions, j2f1;:::; 4g. A particle in cell nwhich crosses over to cell n+ejis mapped at this boundary to a phase point with coor- dinatejsj<d=2, along the direction of the separation between the two cells, with velocity psuch that p
ejp?>0. LetGj(k)denote the set of such phase points which are mapped by the flow to the next collision event on an obstacle in cell n+kej. Since the points of these sets can be mapped back to their preceding collision on an obstacle in cell nlej, for some l0, they are associated with point particles in a propagation phase of length at least k. We must measure these points by means of the natural invariant measure, the one in- duced by the Liouville measure on the cross-sections defined by collisions or cell crossings, which is known to be the area in phase space as parameterized in Fig. 2. Therefore, up to a normalization factor, the area of Gj(k)is given by å¥ i=kmi. By geometric arguments, and as can also be seen from Fig. 2, the measure of[¥ i=kGj(i)is, to leading order in d=`, proportionalto the area of a right triangle of base d=(k`). We therefore have, for k1, ¥ å i=k¥ å j=imj=d 2k`; (5) which implies, and thus justifies, Eq. (4). Anomalous diffusion. We let r=n`denote the displace- ment of L ´evy walkers on the two-dimensional square lattice, measured from the origin where they are initially located, and obtain an expression of their mean squared displacement as a function of time, hr2it, in terms of the time-integral of the overall fraction of particles in a scattering state, s(t), hr2it `2=1 tRZt 0dss(s)+d 2`tRbt=tFc å k=12k+1 k(k+1)ZtktF 0dss(s); (6) see Ref. [24] for further details. In the stationary state, the fraction of particles in a scattering state is simply given by the ratio of the average time spent in the scattering state, tR, to the average return time to this state, tR+å¥ k=1kmktF. Substituting the transition probabilities, Eq. (4), yields lim t!¥s(t) =tR tR+d 2`tF'1d 2`tF tR: (7) Plugging this expression into Eq. (6), we obtain an expres- sion of the mean square displacement of walkers in terms of harmonic numbers, which, in the long-time limit, reduces to hr2it 4t=`2 4tR+d` 4tR[logt+O(1)]: (8) This is our main result. It generalizes to infinite-horizon billiard tables in the narrow-corridor regime the Machta- Zwanzig approximation of the diffusion coefficient for finite- horizon tables, Eq. (2). For short times, the right-hand side of Eq. (8) is dominated by the constant term, which, to leading order in d=`, corre- sponds to a regime of normal diffusion, with the coefficient (2), consistent with the Machta-Zwanzig approximation of the (normal) diffusion coefficient. The term carrying the logarithmic correction has coefficient d`=(4tR), which is identical to the Bleher limiting variance of the anomalously rescaled limiting distribution [16–18], i.e. such that the displacement vector rescaled by the square root of tlogtconverges in distribution to a centered normal distribution whose covariance matrix reduces to a scalar given by this coefficient [31]. Whereas the coefficient of the constant term on the right- hand side of Eq. (8) is first order in the small parameter d=`, the coefficient of the logarithmic term is second order. Such a contribution thus becomes significant only for times such that log t`=d. In the narrow-corridor regime, however, the constraints on the integration times are such that log tremains small with respect `=d.4 0.000.020.040.060.080.100.121.01.11.21.31.41.51.6 d4tRa,4tRbdr=0.485 1011021035253545556 tXrHtL2\H4xflow tL FIG. 3. (Color online) Computed slopes, b(blue, lower curve), and intercepts, a(red, upper curve), of the mean squared displacement of point particles on the infinite-horizon Lorentz gas, fitted according to Eq. (9) for different values of the parameter d. The values found are here normalized by those predicted by Eqs. (10) and (11). To illustrate the fitting procedure, the inset shows the mean squared dis- placement (blue curve, including error bars) measured, as a function of time, for d=0:03. The red dashed line is the result of a linear fit performed in the interval marked by the two vertical lines; see Ref. [32] for further details on this procedure. The units are chosen such that `p01. Numerical results. Following the results presented in Ref. [32], we perform numerical measurements of the mean squared displacement of infinite-horizon billiard tables such as shown in Fig. 1 and determine a range of time values such that the distribution of free paths is accurately sampled, which, according to Eq. (4), scales like the square root of the to- tal number of initial conditions taken (typically 109). In that range, we seek an asymptotically affine fitting function of log t for the normally rescaled mean squared displacement, hr2it 4ta+blogt; (9) where aandbare implicit functions of time, and are expected to converge to the values predicted by Eq. (8) as t!¥, i.e. lim d!0lim t!¥4tRa `2=1; (10) lim t!¥4tRb d`=1; (11) where, in the first line, the narrow-corridor limit takes care of d-dependent corrections to aour theory does not account for. Values found for these fitting parameters are reported in Fig. 3 for different values of d. For the parameter b, on the one hand, one expects Eq. (11) to hold for all values of d in the range shown and, in view of the difficulties presented by such measurements [32], the agreement is indeed rather good, especially given the prevalence of finite-time effects when d!0. There is, on the other hand, no analytic pre- diction for the value of the parameter a, other than that given in the narrow-corridor limit, Eq. (10). Nevertheless, our data provides clear evidence in support of this result.We should note that, in contrast to the approximating L ´evy walk, for which corrections to the zeroth order result (8) are found to be negative, the corrections to the zeroth order result for measurements performed for billiards appear to be positive at first order in d=`. This is to be expected, since memory ef- fects should indeed bring about corrections of the same order, as is the case with finite-horizon billiards [29]; such correc- tions may well predominate. Conclusions Infinite-horizon billiard tables with narrow corridors display anomalous transport properties such that the logarithmic divergence in time of the mean squared displace- ment must effectively be treated as a subleading contribution with respect to a normally diffusive one. Our stochastic analysis of the process in terms of a L ´evy walk with both scattering phases, characterized by random waiting times with exponential distributions, and propagat- ing phases along the table’s corridors provides two quanti- tative predictions which match the Machta-Zwanzig dimen- sional prediction of the (normal) diffusion coefficient, on the one hand, and the Bleher variance of the anomalously rescaled process, on the other hand. Their physical interpretations is, moreover, transparent: (i) the overwhelming majority of tran- sitions taking place on the billiard table are similar to those observed in finite-horizon billiard tables, giving rise to the pre- dominant normal contribution to the mean squared displace- ment, and (ii) rare scattering events allow propagation along the billiard’s corridors over long distrances whose lengths fol- low a precise distribution, at the origin of the anomalous con- tribution to the mean squared displacement. As our numerical results make clear, ignoring the first of these two contributions would obstruct the accurate measurement of the second. We conclude by observing that the scaling properties of the transition probabilities, Eq. (4), can be generalized to other values, extending the relevance of our approach well beyond the regime studied in this Letter. As discussed in Ref. [24], tuning the parameter values allows the description of both normal and anomalous transport regimes, including ballistic transport. Further applications will be reported elsewhere. This work was partially supported by FIRB-Project No. RBFR08UH60 (MIUR, Italy), by SEP-CONACYT Grant No. CB-101246 and DGAPA-UNAM PAPIIT Grant No. IN117214 (Mexico), and by FRFC convention 2,4592.11 (Belgium). T.G. is financially supported by the (Belgian) FRS-FNRS. [1] J. W. Haus and K. W. Kehr, Physics Reports 150, 263 (1987). [2] J.-P. Bouchaud and A. Georges, Physics Reports 195, 127 (1990). [3] G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994). [4] M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, eds., L´evy Flights and Related Topics in Physics , Lecture Notes in Physics, V ol. 450 (Springer, Berlin, Heidelberg, 1995). [5] S. Benkadda and G. M. Zaslavsky, eds., Chaos, Kinetics and5 Nonlinear Dynamics in Fluids and Plasmas , Lecture Notes in Physics, V ol. 511 (Springer, Berlin Heidelberg, 1998). [6] G. M. Zaslavsky, Physics Reports 371, 461 (2002). [7] R. Klages, G. Radons, and I. M. Sokolov, Anomalous trans- port: Foundations and applications (Wiley-VCH Verlag, Wein- heim, 2008). [8] M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature 363, 31 (1993). [9] G. M. Zaslavsky, Physics Today 52, 39 (1999). [10] J. Klafter, M. F. Shlesinger, and G. Zumofen, Physics Today 49, 33 (1996). [11] B. V . Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables , revised ed. (Addison- Wesley, New York, 1968). [12] B. Friedman and R. F. Martin, Jr., Physics Letters A 105, 23 (1984). [13] J.-P. Bouchaud and P. Le Doussal, Journal of Statistical Physics 41, 225 (1985). [14] A. Zacherl, T. Geisel, J. Nierwetberg, and G. Radons, Physics Letters A 114, 317 (1986). [15] T. Geisel, J. Nierwetberg, and A. Zacherl, Physical Review Letters 54, 616 (1985). [16] P. M. Bleher, Journal of Statistical Physics 66, 315 (1992). [17] D. Sz ´asz and T. Varj ´u, Journal of Statistical Physics 129, 59 (2007). [18] D. I. Dolgopyat and N. I. Chernov, Russian Mathematical Sur- veys 64, 651 (2009). [19] J. Machta and R. Zwanzig, Physical Review Letters 50, 1959 (1983). [20] J. Klafter, A. Blumen, and M. F. Shlesinger, Physical ReviewA35, 3081 (1987). [21] G. Zumofen and J. Klafter, Physical Review E 47, 851 (1993). [22] V . M. Kenkre, E. W. Montroll, and M. F. Shlesinger, Journal of Statistical Physics 9, 45 (1973). [23] U. Landman, E. W. Montroll, and M. F. Shlesinger, Proceed- ings of the National Academy of Sciences of the United States of America 74, 430 (1977). [24] G. Cristadoro, T. Gilbert, M. Lenci, and D. P. Sanders, arXiv:1407.0227 (2014). [25] Without loss of generality, we assume rr. For definiteness of a unique relevant timescale, Eq. (1), the separation between the central disc and the outer ones should also be substantially larger than the separation between two outer discs. [26] L. A. Bunimovich, Y . G. Sinai, and N. Chernov, Russian Math- ematical Surveys 46, 47 (1991). [27] P. Gaspard and T. Gilbert, Chaos 22, 026117 (2012). [28] N. Chernov, Journal of Statistical Physics 88, 1 (1997). [29] T. Gilbert and D. P. Sanders, Physical Review E 80, 041121 (2009). [30] Strictly speaking, the residence time, Eq. (1), accounts for the possibility of collisionless motion inside a cell. However in the narrow-corridor regime, the difference between tRand the ac- tual residence time conditioned on particles performing colli- sions inside the cell is next order in the small parameter d=` and will thus be neglected. [31] The second moment actually scales with coefficient given by twice the Gaussian value. See discussion in Ref. [32]. [32] G. Cristadoro, T. Gilbert, M. Lenci, and D. P. Sanders, Physical Review E, in press; arXiv:1405.0975 (2014).

2106.14858v3.Stability_of_a_Magnetically_Levitated_Nanomagnet_in_Vacuum__Effects_of_Gas_and_Magnetization_Damping.pdf

Stability of a Magnetically Levitated Nanomagnet in Vacuum: Eects of Gas and Magnetization Damping Katja Kustura,1, 2Vanessa Wachter,3, 4Adri an E. Rubio L opez,1, 2and Cosimo C. Rusconi5, 6 1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria. 2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria. 3Max Planck Institute for the Science of Light, Staudtstrae 2, 91058 Erlangen, Germany 4Department of Physics, University of Erlangen-N urnberg, Staudtstrae 7, 91058 Erlangen, Germany 5Max-Planck-Institut f ur Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany. 6Munich Center for Quantum Science and Technology, Schellingstrasse 4, D-80799 M unchen, Germany. (Dated: June 1, 2022) In the absence of dissipation a non-rotating magnetic nanoparticle can be stably levitated in a static magnetic eld as a consequence of the spin origin of its magnetization. Here we study the eects of dissipation on the stability of the system, considering the interaction with the background gas and the intrinsic Gilbert damping of magnetization dynamics. At large applied magnetic elds we identify magnetization switching induced by Gilbert damping as the key limiting factor for stable levitation. At low applied magnetic elds and for small particle dimensions magnetization switching is prevented due to the strong coupling of rotation and magnetization dynamics, and the stability is mainly limited by the gas-induced dissipation. In the latter case, high vacuum should be sucient to extend stable levitation over experimentally relevant timescales. Our results demonstrate the possibility to experimentally observe the phenomenon of quantum spin stabilized magnetic levitation. I. INTRODUCTION The Einstein{de Haas [1, 2] and Barnett eects [3] are macroscopic manifestations of the internal angular mo- mentum origin of magnetization: a change in the mag- netization causes a change in the mechanical rotation and conversely. Because of the reduced moment of in- ertia of levitated nano- to microscale particles, these ef- fects play a dominant role in the dynamics of such sys- tems [4{10]. This oers the possibility to harness these eects for a variety of applications such as precise magne- tometry [11{16], inertial sensing [17, 18], coherent spin- mechanical control [19, 20], and spin-mechanical cool- ing [21, 22] among others. Notable in this context is the possibility to stably levitate a ferromagnetic parti- cle in a static magnetic eld in vacuum [23, 24]. Stable levitation is enabled by the internal angular momentum origin of the magnetization which, even in the absence of mechanical rotation, provides the required angular mo- mentum to gyroscopically stabilize the system. Such a phenomenon, which we refer to as quantum spin stabi- lized levitation to distinguish it from the rotational stabi- lization of magnetic tops [25{27], relies on the conserva- tive interchange between internal and mechanical angular momentum. Omnipresent dissipation, however, exerts additional non-conservative torques on the system which might alter the delicate gyroscopic stability [26, 28]. It thus remains to be determined if stable levitation can be observed under realistic conditions, where dissipative eects cannot be neglected. In this article, we address this question. Speci- cally, we consider the dynamics of a levitated magnetic nanoparticle (nanomagnet hereafter) in a static magnetic eld in the presence of dissipation originating both fromthe collisions with the background gas and from the intrinsic damping of magnetization dynamics (Gilbert damping) [29, 30], which are generally considered to be the dominant sources of dissipation for levitated nano- magnets [8, 13, 31{33]. Conned dynamics can be ob- served only when the time over which the nanomagnet is levitated is longer than the period of center-of-mass os- cillations in the magnetic trap. When this is the case, we dene the system to be metastable . We demonstrate that the system can be metastable in experimentally feasible conditions, with the levitation time and the mechanism behind the instability depending on the parameter regime of the system. In particular, we show that at weak ap- plied magnetic elds and for small particle dimensions (to be precisely dened below) levitation time can be signicantly extended in high vacuum (i.e. pressures be- low 103mbar). Our results evidence the potential of unambiguous experimental observation of quantum spin stabilized magnetic levitation. We emphasize that our analysis is particularly timely. Presently there is a growing interest in levitating and con- trolling magnetic systems in vacuum [9, 34, 35]. Current experimental eorts focus on levitation of charged para- magnetic ensembles in a Paul trap [19, 36, 37], diamag- netic particles in magneto-gravitational traps [38{40], or ferromagnets above a superconductor [14, 20, 41]. Lev- itating ferromagnetic particles in a static magnetic trap oers a viable alternative, with the possibility of reaching larger mechanical trapping frequencies. The article is organized as follows. In Sec. II we in- troduce the model of the nanomagnet, and we dene two relevant regimes for metastability, namely the atom phase and the Einstein{de Haas phase. In Sec. III and IV we analyze the dynamics in the atom phase and thearXiv:2106.14858v3 [cond-mat.mes-hall] 31 May 20222 Figure 1. (a) Illustration of a spheroidal nanomagnet levi- tated in an external eld B(r) and surrounded by a gas at the temperature Tand the pressure P. (b) Linear stability dia- gram of a non-rotating nanomagnet in the absence of dissipa- tion, assuming a= 2b. Blue and red regions denote the stable atom and Einstein{de Haas phase, respectively; hatched area is the unstable region. Dashed lines show the critical values of the bias eld which dene the two phases. In particular, BEdH,15=[4 2 0(a2+b2)M],BEdH,23 [B02=(4 0M)]1=3, andBatom = 2kaV=. Numerical values of physical parame- ters used to generate panel (b) are given in Table I. Einstein{de Haas phase, respectively. We discuss our re- sults in Sec. V. Conclusions and outlook are provided in Sec. VI. Our work is complemented by three appendices where we dene the transformation between the body- xed and laboratory reference frames (App. A), analyze the eect of thermal uctuations (App. B), and provide additional gures (App. C). II. DESCRIPTION OF THE SYSTEM We consider a single domain nanomagnet levitated in a static1magnetic eld B(r) as shown schematically in Fig. 1(a). We model the nanomagnet as a spheroidal rigid body of mass density Mand semi-axes lengths a;b (a > b ), having uniaxial magnetocrystalline anisotropy, with the anisotropy axis assumed to be along the major semi-axisa[42]. Additionally, we assume that the mag- netic response of the nanomagnet is approximated by a point dipole with magnetic moment of constant mag- nitudejj, as it is often justied for single domain particles [42, 43]. Let us remark that such a simplied model has been considered before to study the classical dynamics of nanomagnets in a viscous medium [31, 44{ 49], as well as to study the quantum dynamics of mag- netic nanoparticles in vacuum [5, 13, 50, 51]. Since the model has been successful in describing the dynamics of single-domain nanomagnets, we adopt it here to inves- tigate the stability in a magnetic trap. In particular, our study has three main dierences as compared with 1We denote a eld static if it does not have explicit time depen- dence, namely if @B(r)=@t= 0.Table I. Physical parameters of the model and the values used throughout the article. We calculate the magnitude of the magnetic moment as =V, where=MB=(50amu), withBthe Bohr magneton and amu the atomic mass unit. Parameter Description Value [units] M mass density 104[kg m3] a;b semi-axes see main text [m]  magnetization 2 :2106[J T1m3] ka anisotropy constant 105[J m3] 0 gyromagnetic ratio 1 :761011[rad s1T1] B0 eld bias see main text [T] B0eld gradient 104[T m1] B00eld curvature 106[T m2]  Gilbert damping 102[n. u.] T temperature 101[K] P pressure 102[mbar] M molar mass 29 [g mol1] c re ection coecient 1 [n. u.] previous work. (i) We consider a particle levitated in high vacuum, where the mean free path of the gas parti- cles is larger than the nanomagnet dimensions (Knudsen regime [52]). This leads to gas damping which is gen- erally dierent from the case of dense viscous medium mostly considered in the literature. (ii) We consider center-of-mass motion and its coupling to the rotational and magnetic degrees of freedom, while previous work mostly focuses on coupling between rotation and mag- netization only (with the notable exception of [48]). (iii) We are primarily interested in the center-of-mass conne- ment of the particle, and not in its magnetic response. Within this model the relevant degrees of freedom of the system are the center-of-mass position r, the linear momentum p, the mechanical angular momentum L, the orientation of the nanomagnet in space , and the mag- netic moment . The orientation of the nanomagnet is specied by the body-xed reference frame Oe1e2e3, which is obtained from the laboratory frame Oexeyez according to ( e1;e2;e3)T=R( )(ex;ey;ez)T, where = (;; )Tare the Euler angles and R( ) is the rotational matrix. We provide the expression for R( ) in App. A. The body-xed reference frame is chosen such thate3coincides with the anisotropy axis. The magnetic momentis related to the internal angular momentum F according to the gyromagnetic relation = 0F, where 0is the gyromagnetic ratio of the material2. 2The total internal angular momentum Fis a sum of the individ- ual atomic angular momenta (spin and orbital), which contribute to the atomic magnetic moment. For a single domain magnetic particle, it is customary to assume that Fcan be described as a vector of constant magnitude, jFj== 0(macrospin approxi- mation) [43].3 A. Equations of Motion We describe the dynamics of the nanomagnet in the magnetic trap with a set of stochastic dierential equa- tions which model both the deterministic dissipative evo- lution of the system and the random uctuations due to the environment. In the following it is convenient to de- ne dimensionless variables: the center-of-mass variables ~rr=a,~p 0ap=, the mechanical angular momen- tum` 0L=, the magnetic moment m=, and the magnetic eld b(~r)B(a~r)=B0, whereB0denotes the minimum of the eld intensity in a magnetic trap, which we hereafter refer to as the bias eld. Note that we choose to normalize the position r, the magnetic mo- mentand the magnetic eld B(r) with respect to the particle size a, the magnetic moment magnitude , and the bias eld B0, respectively. The scaling factor for an- gular momentum, = 0, and linear momentum, =(a 0), follow as a consequence of the gyromagnetic relation. The dynamics of the nanomagnet in the laboratory frame are given by the equations of motion _~r=!I~p; (1) _e3=!e3; (2) _~p=!Lr~r[m
b(~r)]cm~p+p(t); (3) _`=!Lmb(~r)_mrot`+l(t); (4) _m=m 1 +2[!em(!+!e+!m) +b(t)]: (5) Here we dene the relevant system frequencies: !I =( 0Ma2) is the Einstein{de Haas frequency, with M the mass of the nanomagnet, !L 0B0is the Larmor frequency,!AkaV 0=is the anisotropy frequency, withVthe volume of the nanomagnet and kathe ma- terial dependent anisotropy constant [43], !I1L is the angular velocity, with Ithe tensor of inertia, and!e2!A(m
e3)e3+!Lb(~r). Dissipation is parametrized by the dimensionless Gilbert damping pa- rameter[29, 53], and the center-of-mass and rotational friction tensors cmand rot, respectively [32]. The eect of stochastic thermal uctuations is represented by the random variables p(t) andl(t) which describe, respec- tively, the uctuating force and torque exerted by the surrounding gas, and by b(t) which describes the ran- dom magnetic eld accounting for thermal uctuations in magnetization dynamics [54]. We assume Gaussian white noise, namely, for X(t)(p(t);l(t);b(t))Twe havehXi(t)i= 0 andhXi(t)Xj(t0)iij(tt0). Equations (1-4) describe the center-of-mass and rota- tional dynamics of a rigid body in the presence of dis- sipation and noise induced by the background gas [32]. The expressions for cmand rotdepend on the parti- cle shape { here we take the expressions derived in [32]for a cylindrical particle3{, and on the ratio of the sur- face and the bulk temperature of the particle, which we assume to be equal to the gas temperature T. Fur- thermore, they account for two dierent scattering pro- cesses, namely the specular and the diusive re ection of the gas from the particle, which is described by a phenomenological interpolation coecient c. The or- der of magnitude of the dierent components of cmand rotis generally well approximated by the dissipation rate (2Pab=M )[2M=(NAkBT)]1=2, wherePand Mare, respectively, the gas pressure and molar mass, kBis the Boltzmann constant and NAis the Avogadro number. The magnetization dynamics given by Eq. (5) is the Landau-Lifshitz-Gilbert equation in the laboratory frame [8, 57], with the eective magnetic eld !e= 0. We remark that Eqs. (1-5) describe the classical dynam- ics of a levitated nanomagnet where the eect of the quantum spin origin of magnetization, namely the gy- romagnetic relation, is taken into account phenomeno- logically by Eq. (5). This is equivalent to the equations of motion obtained from a quantum Hamiltonian in the mean-eld approximation [24]. Let us discuss the eect of thermal uctuations on the dynamics of the nanomagnet at subkelvin temper- atures and in high vacuum. These conditions are com- mon in recent experiments with levitated particles [58{ 60]. The thermal uctuations of magnetization dy- namics, captured by the last term in Eq. (5), lead to thermally activated transition of the magnetic mo- ment between the two stable orientations along the anisotropy axis [54, 61]. Such process can be quan- tied by the N eel relaxation time, which is given by N(=! A)p kBT=(kaV)ekaV=(kBT). Thermal acti- vation can be neglected when Nis larger than other timescales of magnetization dynamics, namely the pre- cession timescale given by L1=j!ej, and the Gilbert damping timescale given by G1=(j!ej). Con- sidering for simplicity j!ej2!A, for a particle size a= 2b= 1 nm and temperature T= 1 K, and the values of the remaining parameters as in Table I, the ra- tio of the timescales is of the order N=L103, and it is signicantly increased for larger particle sizes and at smaller temperatures. We remark that, for the val- ues considered in this article, Nis much larger than the longest dynamical timescale in Eqs. (1-5) which is associ- ated with the motion along ex. Thermal activation of the magnetic moment can therefore be safely neglected. The stochastic eects ascribed to the background gas, cap- tured by the last terms in Eqs. (3-4), are expected to be important at high temperatures (namely, a regime where MkBT 2 0a2=2&1 [32]). At subkelvin temperatures and in high vacuum these uctuations are weak and, con- sequently, they do not destroy the deterministic eects 3The expressions for cmand rotfor a cylindrical particle capture the order of magnitude of the dissipation rates for a spheroidal particle [55, 56].4 captured by the remaining terms in Eqs. (1-5) [33]. In- deed, for the values of parameters given in Table I and fora= 2b,MkBT 2 0a2=20:8T=(a[nm]). For sub- kelvin temperatures and particle sizes a>1 nm, thermal uctuations due to the background gas can therefore be safely neglected. In the following we thus neglect stochastic eects by settingp=l=b= 0, and we consider only the de- terministic part of Eqs. (1-5) as an appropriate model for the dynamics [8, 33, 54]. In App. B we carry out the analysis of the dynamics including the eects of gas uctuations in equations (1-5), and we show that the results presented in the main text remain qualitatively valid even in the presence of thermal noise. For the mag- netic eld B(r) we hereafter consider a Ioe-Pritchard magnetic trap, given by B(r) =ex B0+B00 2 x2y2+z2 2 ey B0y+B00 2xy +ez B0zB00 2xz ;(6) whereB0;B0andB00are, respectively, the eld bias, gra- dient and curvature [62]. We remark that this is not a fundamental choice, and dierent magnetic traps, pro- vided they have a non-zero bias eld, should result in similar qualitative behavior. B. Initial conditions The initial conditions for the dynamics in Eqs. (1-5), namely at time t= 0, depend on the initial state of the system, which is determined by the preparation of the nanomagnet in the magnetic trap. In our analysis, we consider the nanomagnet to be prepared in the thermal state of an auxiliary loading potential at the temperature T. Subsequently, we assume to switch o the loading potential at t= 0, while at the same time switching on the Ioe-Pritchard magnetic trap. The choice of the auxiliary potential is determined by two features: (i) it allows us to simply parametrize the initial conditions by a single parameter, namely the temperature T, and (ii) it is an adequate approximation of general trapping schemes used to trap magnetic particles. Regarding point (i), we assume that the particle is lev- itated in a harmonic trap, in the presence of an external magnetic eld applied along ex. This loading scheme provides, on the one hand, trapping of the center-of-mass degrees of freedom, with trapping frequencies denoted by !i(i=x;y;z ). On the other hand, the magnetic moment in this case is polarized along ex. The Hamiltonian of the system in such a conguration reads Haux=p2=(2M) +P i=x;y;zM!2 ir2 i=2+LI1L=2kaVe2 3;xxBaux, where Bauxdenotes the magnitude of the external magnetic eld, which we for simplicity set to Baux=B0in all our simulations. At t= 0 the particle is released in the mag- netic trap given by Eq. (6). For the degrees of freedomx(~r;~p;`;mx)T, we take as the initial displacement from the equilibrium the corresponding standard devia- tion in a thermal state of Haux. More precisely, xi(0) = xi;e+ (hx2 iihxii2)1=2, wherexi;edenotes the equilib- rium value, andhxk iiZ1R dxxk iexp[Haux=(kBT)], withk= 1;2 and the partition function Z. For the Eu- ler angles we use 1(0)cos1[p hcos2 1i] and i(0)cos1[p hcos2 ii] (i= 2;3). The initial condi- tions for e3follow from using the transformation given in App. A. Regarding point (ii), the initial conditions obtained in this way describe a trapped particle prepared in a ther- mal equilibrium in the presence of an external loading potential where the center of mass is decoupled from the magnetization and the rotational dynamics. It is outside the scope of this article to study in detail a particular loading scheme. However, we point out that an auxil- iary potential given by Hauxcan be obtained, for exam- ple, by trapping the nanomagnet using a Paul trap as demonstrated in recent experiments [19, 21, 37, 63{70]. In particular, trapping of a ferromagnetic particle has been demonstrated in a Paul trap at P= 102mbar, with center-of-mass trapping frequency of up to 1 MHz, and alignment of the particle along the direction of an applied eld [19]. We note that particles are shown to remain trapped even when the magnetic eld is varied over many orders of magnitudes or switched o. We re- mark further that alignment of elongated particles can be achieved using a quadrupole Paul trap even in the absence of magnetic eld [55, 71]. C. Linear stability In the absence of thermal uctuations, an equilibrium solution of Eqs. (1-5) is given by ~re=~pe=`e= 0 and e3;e=me=ex. This corresponds to the conguration in which the nanomagnet is xed at the trap center, with the magnetic moment along the anisotropy axis and anti- aligned to the bias eld B0. Linear stability analysis of Eqs. (1-5) shows that the system is unstable, as expected for a gyroscopic system in the presence of dissipation [28]. However, when the nanomagnet is metastable, it is still possible for it to levitate for an extended time before being eventually lost from the trap, as in the case of a classical magnetic top [25{27]. As we show in the fol- lowing sections, the dynamics of the system, and thus its metastability, strongly depend on the applied bias eld B0. We identify two relevant regimes: (i) strong-eld regime, dened by bias eld values B0> B atom, and (ii) weak-eld regime, dened by B0< B atom, where Batom2kaV=. This dierence is reminiscent of the two dierent stable regions which arise as a function of B0in the linear stability diagram in the absence of dis- sipation [see Fig. 1(b)] [23, 24]. In Sec. III and Sec. IV we investigate the possibility of metastable levitation by solving numerically Eqs. (1-5) in the strong-eld and weak-eld regime, respectively.5 III. DYNAMICS IN THE STRONG-FIELD REGIME: ATOM PHASE The strong-eld regime, according to the denition given in Sec. II C, corresponds to the blue region in the linear stability diagram in the absence of dissipation, shown in Fig. 1(b). This region is named atom phase in [23, 24], and we hereafter refer to the strong-eld regime as the atom phase. This parameter regime corre- sponds to the condition !L!A;!I. In this regime, the coupling of the magnetic moment and the anisotropy axise3is negligible, and, to rst approximation, the nanomagnet undergoes a free Larmor precession about the local magnetic eld. In the absence of dissipation, this stabilizes the system in full analogy to magnetic trap- ping of neutral atoms [72, 73]. In Fig. 2(a-c) we show the numerical solution of Eqs. (1-5) for nanomagnet dimensions a= 2b= 20 nm and the bias eld B0= 200 mT. As evidenced by Fig. 2(a), the magnetization mxof the particle changes direction. During this change, the mechanical angu- lar momentum lxchanges accordingly in the manifesta- tion of the Einstein{de Haas eect, such that the to- tal angular momentum m+`is conserved4. The dy- namics observed in Fig. 2(a) is indicative of Gilbert- damping-induced magnetization switching, a well-known phenomenon in which the projection of the magnetic mo- ment along the eective magnetic eld !e= 0changes sign [30]. This is expected to happen when the applied bias eldB0is larger than the eective magnetic eld associated with the anisotropy, given by !A= 0. Mag- netization switching displaces the system from its equi- librium position on a timescale which is much shorter than the period of center-of-mass oscillations, estimated from [24] to be cm1s. The nanomagnet thus shows no signature of connement [see Fig. 2(b)]. The timescale of levitation in the atom phase is given by the timescale of magnetization switching, which we estimate as follows. As evidenced by Fig. 2(a-b), the dynamics of the center of mass and the anisotropy axis are approximately constant during switching, such that !e!e(t= 0). Under this approximation and as- suming1, the magnetic moment projection mk !e
m=j!ejevolves as _mk[!L+ 2!Amk](1m2 k): (7) According to Eq. (7) the component mkexhibits switch- ing ifmk(t= 0)&1 and!L=2!A>1 [30], both of which are fullled in the atom phase. Integrating Eq. (7) we obtain the switching time [dened as mk()0], which can be well approximated by ln 1 +jmk(t= 0)j
2(!L+ 2!A)ln 1jmk(t= 0)j
2(!L2!A):(8) 4We always nd the transfer of angular momentum to the center of mass angular momentum rpto be negligible. Figure 2. Dynamics in the atom phase. (a) Dynamics of the magnetic moment component mx, the mechanical angular momentum component lx, and the anisotropy axis component e3;xfor nanomagnet dimensions a= 2b= 20 nm and the bias eldB0= 200 mT. For the initial conditions we consider trapping frequencies !x= 22 kHz and!y=!z= 250 kHz. Unless otherwise stated, for the remaining parameters the numerical values are given in Table I. (b) Center-of-mass dynamics for the same case considered in (a). (c) Dynamics of the magnetic moment component mk. Line denoted by circle corresponds to the case considered in (a). Each remaining line diers by a single parameter, as denoted by the legend. Dotted vertical lines show Eq. (8). (d) Switching time given by Eq. (8) as a function of the bias eld B0and the major semi-axisa. In the region left of the thick dashed line the deviation from the exact value is more than 5%. Hatched area is the unstable region in the linear stability diagram in Fig. 1.(b). The estimation Eq. (8) is in excellent agreement with the numerical results for dierent parameter values [see Fig. 2(c)]. Magnetization switching characterizes the dynamics of the system in the entire atom phase. In particular, in Fig. 2(d) we analyze the validity of Eq. (8) for dierent values of the bias eld B0and the major semi-axis a, as- sumingb=a=2. The thick dashed line shows the region where Eq. (8) diers from the exact switching time, as estimated from the full dynamics of the system, by 5%; left of this line the deviation becomes increasingly more signicant, with Eq. (8) predicting up to 20% larger val- ues close to the stability border (namely, for bias eld close toBatom = 90 mT). We believe that the signi- cant deviation close to the border of the atom phase is due to the non-negligible coupling to the anisotropy axis,6 Figure 3. Dynamics in the Einstein{de Haas phase. (a) Motion of the system in the ey-ezplane until time t= 5s for nanomagnet dimensions a= 2b= 2 nm and the bias eld B0= 0:5 mT. For the initial conditions we consider trapping frequencies !x= 22 kHz and !y=!z= 21 MHz. For the remaining parameters the numerical values are given in Table I. (b) Dynamics of the projection mkand (c) dynamics of the anisotropy axis component e3;x, for the same case considered in (a). (d) Dynamics of the center-of-mass component ryand (e) dynamics of the magnetic moment component mxon a longer timescale, for the same values of parameters as in (a). (f) Escape time t?as a function of gas pressure P, for dierent congurations in the Einstein{de Haas phase. Circles correspond to the case considered in (a). Each remaining case diers by parameters indicated by the legend. (g) Escape time t?as a function of the major semi-axis a, with the values of the remaining parameters as in (a). Dashed vertical line denotes the upper limit of the Einstein{de Haas phase, given by the critical eld BEdH,1 [see Fig. 1(b)]. which results in additional mechanisms not captured by the simple model Eq. (7). In fact, it is known that cou- pling between magnetization and mechanical degrees of freedom might have an impact on the switching dynam- ics [74]. As demonstrated by Fig. 2(d), the switching time is always shorter than the center-of-mass oscillation periodcm, and thus no metastability can be observed in the atom phase. Let us note that the conclusions we draw in Fig. 2 remain valid if one varies the anisotropy constant ka, Gilbert damping parameter , and the temperature T, as we show in App. C. Finally, we note that the dis- sipation due to the background gas has negligible ef- fects. In particular, for the values assumed in Fig. 2(a-b) the timescale of the gas-induced dissipation is given by 1= = 440s. IV. DYNAMICS IN THE WEAK-FIELD REGIME: EINSTEIN{DE HAAS PHASE We now focus on the regime of weak bias eld, cor- responding to the condition !L!A. In this regime magnetization switching does not occur, and the dynam- ics critically depend on the particle size. In the follow- ing we focus on the regime of small particle dimensions,i.e.!L!I, which, as we will show, is benecial for metastability. In the absence of dissipation, this regime corresponds to the Einstein{de Haas phase [red region in Fig. 1(b)] [23, 24]. The hierarchy of energy scales in the Einstein{de Haas phase (namely, !L!A;!I) man- ifests in two ways: (i) the anisotropy is strong enough to eectively \lock" the direction of the magnetic moment  along the anisotropy axis e3(!A!L), and (ii) accord- ing to the Einstein{de Haas eect, the frequency at which the nanomagnet would rotate if switched direction is signicantly increased at small dimensions ( !I!L), such that switching can be prevented due to energy con- servation [4]. In the absence of dissipation, the combina- tion of these two eects stabilizes the system. In Fig. 3(a-c) we show the numerical solution of Eqs. (1-5) for nanomagnet dimensions a= 2b= 2 nm and the bias eld B0= 0:5 mT. The nanomagnet is metastable, as evidenced by the conned center-of-mass motion shown in Fig. 3(a). In Fig. 3(b-c) we show the dynamics of the magnetic moment component mkand the anisotropy axis component e3;x, respectively, which indicates that no magnetization switching occurs in this regime. We remark that the absence of switching can- not be simply explained on the basis of Eqs. (7-8). In fact, the simple model of magnetization switching, given7 by Eq. (7), assumes that the dynamics of the rotation and the center-of-mass motion happen on a much longer timescale than the timescale of magnetization dynam- ics. However, in this case rotation and magnetization dynamics occur on a comparable timescale, as evidenced by Fig. 3(b-c). The weak-eld condition alone ( !L!A) is thus not sucient to correctly explain the absence of switching, and the role of particle size ( !L!I) needs to be considered. Let us analyze the role of Gilbert damping in this case. Since in the Einstein{de Haas phase mk1, we dene me3+m, wheremrepresents the deviation of m from the anisotropy axis e3, and we assumejmjje3j [see Fig. 3(b)]. This allows us to simplify Eq. (5) as _m!em[2!A+!3e3
(m+`)]m;(9) where!3=( 0I3), withI3the principal moment of inertia along e3. As evidenced by Eq. (9), the only eect of Gilbert damping is to align mande3on a timescale given by01=[(2!A+!3)], irrespective of the dy- namics of e3. For the values of parameters considered in Fig. 3(a-c), 0= 5 ns, and it is much shorter than the timescale of center-of-mass dynamics, given by cm1 s. For all practical purposes, the magnetization in the Einstein{de Haas phase can be considered frozen along the anisotropy axis. The nanomagnet in the presence of Gilbert damping is therefore equivalent to a hard magnet (i. e.ka!1 ) [24]. The main mechanism behind the instability in the Einstein{de Haas phase is thus gas-induced dissipation. In Fig. 3(d-e) we plot the dynamics of the center-of- mass component ryand the magnetic moment compo- nentmxon a longer timescale, for two dierent values of the pressure P. The eect of gas-induced dissipation is to dampen the center-of-mass motion to the equilibrium position, while the magnetic moment moves away from the equilibrium. Both processes happen on a timescale given by the dissipation rate . When ex=mx0, the system becomes unstable and ultimately leaves the trap [see arrow in Fig. 3(d)]. We dene the escape time t?as the time at which the particle position is y(t?)5y(0), and we show it in Fig. 3(f) as a function of pressure Pfor dierent congurations in the Einstein{de Haas phase, and forb=a=2. Fig. 3(f) conrms that the dissipation aects the system on a timescale which scales as 1=P. The metastability of the nanomagnet in the Einstein{de Haas phase is therefore limited solely by the gas-induced dissipation, which can be signicantly reduced in high vacuum. Finally, in Fig. 3(g) we analyze the eect of particle size on metastability. Specically, we show the escape time t?as a function of the major semi-axis aat the bias eld B0= 0:5 mT, forb=a=2. The escape time is signicantly reduced at increased particle sizes. This conrms the advantage of the Einstein{de Haas phase to observe metastability, even in the presence of dissipation.V. DISCUSSION In deriving the results discussed in the preceding sec- tions, we assumed (i) a single-magnetic-domain nanopar- ticle with uniaxial anisotropy and constant magnetiza- tion, with the values of the physical parameters summa- rized in Table I, (ii) deterministic dynamics, i. e. the absence of thermal uctuations, (iii) that gravity can be neglected, and (iv) a non-rotating nanomagnet. Let us justify the validity of these assumptions. We rst discuss the values of the parameters given in Table I, which are used in our analysis. The material pa- rameters, such as M,,kaand, are consistent with, for example, cobalt [75{78]. We remark that the uniax- ial anisotropy considered in our model represents a good description even for materials which do not have an in- trinsic magnetocrystalline uniaxial anisotropy, provided that they have a dominant contribution from the uniaxial shape anisotropy. This is the case, for example, for fer- romagnetic particles with a prolate shape [75]. We point out that the values used here do not correspond to a spe- cic material, but instead they describe a general order of magnitude corresponding to common magnetic materi- als. Indeed, our results are general and can be particular- ized to specic materials by replacing the above generic values with exact numbers. As we show in App. C, the re- sults and conclusions presented here remain unchanged even when dierent values of the parameters are con- sidered. The values used for the eld gradient B0and the curvature B00have been obtained in magnetic mi- crotraps [62, 79{82]. The values of the gas pressure P and the temperature Tare experimentally feasible, with numerous recent experiments reaching pressure values as low asP= 106mbar [58, 68, 70, 83{85]. All the values assumed in our analysis are therefore consistent with cur- rently available technologies in levitated optomechanics. Thermal uctuations can be neglected at cryogenic conditions (as we argue in Sec. II A), as their eect is weak enough not to destroy the deterministic eects cap- tured by Eqs. (1-5). In particular, thermal activation of the magnetization, as quantied by the N eel relaxation time, can be safely neglected due to the large value of the uniaxial anisotropy even for the smallest particles considered. As for the mechanical thermal uctuations, we conrm that they do not modify the deterministic dynamics in App. B, where we simulate the associated stochastic dynamics. Gravity, assumed to be along ex, can be safely ne- glected, since the gravity-induced displacement of the trap center from the origin is much smaller than the length scale over which the Ioe-Pritchard eld signi- cantly changes [24]. Specically, the gravitational poten- tialMgx shifts the trap center from the origin r= 0 along exby an amount rgMg= (B00), wheregis the gravitational acceleration. On the other hand, the characteristic length scales of the Ioe-Pritchard eld are given by
r0p B0=B00for the variation along ex, and
r0B0=B00for the variation o-axis. When-8 everrg
r0;
r0, gravity has a negligible role in the metastable dynamics of the system. In the parameter regime considered in this article, this is always the case. We note that the condition to neglect gravity is the same as for a magnetically trapped atom, since both Mand scale with the volume. Finally, we remark that the analysis presented here is carried out for the case of a non-rotating nanomag- net5. The same qualitative behavior is obtained even in the presence of mechanical rotation (namely, considering a more general equilibrium conguration with `e6= 0). The analysis of dynamics in the presence of rotation is provided in App. C. In particular, the dynamics in the Einstein{de Haas phase remains largely unaected, pro- vided that the total angular momentum of the system is not zero. In the atom phase, mechanical rotation leads to dierences in the switching time , as generally expected in the presence of magneto-mechanical coupling [74, 88]. VI. CONCLUSION In conclusion, we analyzed how the stability of a nano- magnet levitated in a static magnetic eld is aected by the most relevant sources of dissipation. We nd that in the strong-eld regime (atom phase) the system is un- stable due to the Gilbert-damping-induced magnetiza- tion switching, which occurs on a much faster timescale than the center-of-mass oscillations, thereby preventing the observation of levitation. On the other hand, the sys- tem is metastable in the weak-eld regime and for small particle dimensions (Einstein{de Haas phase). In this regime, the connement of the nanomagnet in a mag- netic trap is limited only by the gas-induced dissipation. Our results suggest that the timescale of stable levitation can reach and even exceed several hundreds of periods of center-of-mass oscillations in high vacuum. These nd- ings indicate the possibility of observing the phenomenon of quantum spin stabilized magnetic levitation, which we hope will encourage further experimental research. The analysis presented in this article is relevant for the community of levitated magnetic systems. Speci- cally, we give precise conditions for the observation of the phenomenon of quantum spin stabilized levitation under experimentally feasible conditions. Levitating a magnet in a time-independent gradient trap represents a new direction in the currently growing eld of magnetic levitation of micro- and nanoparticles, which is interest- ing for two reasons. First, the experimental observation of stable magnetic levitation of a non-rotating nanomag- net would represent a direct observation of the quantum nature of magnetization. Second, the observation of such 5Rotational cooling might be needed to unambiguously identify the internal spin as the source of stabilization. Subkelvin cooling of a nanorotor has been recently achieved [86, 87], and cooling toK temperatures should be possible [56].phenomenon would be a step towards controlling and us- ing the rich physics of magnetically levitated nanomag- nets, with applications in magnetometry and in tests of fundamental forces [9, 11, 34, 35]. ACKNOWLEDGMENTS We thank G. E. W. Bauer, J. J. Garc a-Ripoll, O. Romero-Isart, and B. A. Stickler for helpful discussions. We are grateful to O. Romero-Isart, B. A. Stickler and S. Viola Kusminskiy for comments on an early ver- sion of the manuscript. C.C.R. acknowledges funding from ERC Advanced Grant QENOCOBA under the EU Horizon 2020 program (Grant Agreement No. 742102). V.W. acknowledges funding from the Max Planck So- ciety and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project- ID 429529648-TRR 306 QuCoLiMa ("Quantum Cooper- ativity of Light and Matter"). A.E.R.L. thanks the AMS for the nancial support. Appendix A: Rotation to the body frame In this appendix we dene the transformation ma- trix between the body-xed and the laboratory reference frames according to the ZYZ Euler angle convention, with the Euler angles denoted as = (;; )T. We dene the transformation between the laboratory frame Oexeyezand the body frame Oe1e2e3as follows, 0 @e1 e2 e31 A=R( )0 @ex ey ez1 A; (A1) where R( )Rz()Ry()Rz( ) =0 @cos sin 0 sin cos 0 0 0 11 A 0 @cos0sin 0 1 0 sin0 cos1 A0 @cossin0 sincos0 0 0 11 A:(A2) Accordingly, the components vj(j= 1;2;3) of a vector vin the body frame Oe1e2e3and the components v (=x;y;z ) of the same vector in the laboratory frame Oexeyezare related as 0 @v1 v2 v31 A=RT( )0 @vx vy vz1 A: (A3) The angular velocity of a rotating particle !can be writ- ten in terms of the Euler angles as != _ez+_e0 y+ _ e3, where ( e0 x;e0 y;e0 z)T=Rz()(ex;ey;ez)Tdenotes the frameOe0 xe0 ye0 zobtained after the rst rotation of the9 laboratory frame Oexeyezin the ZYZ convention. By using (A1) and (A2), we can rewrite angular velocity in terms of the body frame coordinates, != _2 4R( )10 @e1 e2 e31 A3 5 3+_2 4R( )10 @e1 e2 e31 A3 5 2+ _ e3; (A4) which is compactly written as ( !1;!2;!3)T=A( )_ , with A( ) =0 @cos sinsin 0 sinsin cos 0 cos 0 11 A: (A5) Appendix B: Dynamics in the presence of thermal uctuations In this appendix we consider the dynamics of a lev- itated nanomagnet in the presence of stochastic forces and torques induced by the surrounding gas. The dy- namics of the system are described by the following set of stochastic dierential equations (SDE), d~r=!I~pdt; (B1) de3=!e3dt; (B2) d~p= [!Lr~r[m
b(~r)]cm~p] dt+p DcmdWp;(B3) d`= [!Lmb(~r)_mrot`] dt+p DrotdWl; (B4) dm=m 1 +2[!em(!+!e+!m)]dt; (B5) where we model the thermal uctuations as uncorrelated Gaussian noise represented by a six-dimensional vector of independent Wiener increments (d Wp;dWl)T. The corresponding diusion rate is described by the tensors DcmandDrotwhich, in agreement with the uctuation- dissipation theorem, are related to the corresponding dis- sipation tensors cmand rotasDcm2cm;D rot 2rot, whereMkBT 2 0a2=2. In the following we numerically integrate Eqs. (B1-B5) using the stochastic Euler method implemented in the stochastic dierential equations package in MATLAB. As the eect of thermal noise is more prominent for small particles at weak elds, we focus on the Einstein-de Haas regime considered in Sec. IV. We show that even in this case the eect of thermal uctuations leads to dynamics which are qualitatively very close to the results obtained in Sec. IV. In Fig. 4 we present the results of the stochas- tic integrator by averaging the solution of 100 dierent trajectories calculated using the same parameters consid- ered in Fig. 3(a-c). The resulting average dynamics agree qualitatively with the results obtained by integrating the corresponding set of deterministic equations Eqs. (1-5) Figure 4. Stochastic dynamics of a nanomagnet for the same parameter regime as considered in Fig. 3. (a) Average motion of the system in the y-zplane until time t= 5s. (b) Dy- namics of center of mass along the ey(top) and ez(bottom) directions. (c) Dynamics of the anisotropy axis component e3;x. (d) Numerical error as function of time. The simulations show the results of the average of 100 dierent realizations of the system dynamics. In panels (b-d) the solid dark lines are the average trajectories, while the shaded area represents the standard deviation. [cfr. Fig. 3(a-c)]. The main eect of thermal excitations is to shift the center of oscillations of the particle's de- grees of freedom around the value given by the thermal uctuations. This is more evident for the dynamics of e3[cfr. Fig. 4(c) and Fig. 3(c)]. We thus conclude that the deterministic equations Eqs. (1-5) considered in the main text correctly capture the metastable behavior of the system. We emphasize that the results presented in this section include only the noise due to the surround- ing gas. Should one be interested in simulating the ef- fect of the uctuations of the magnetic moment, the Eu- ler method used here is not appropriate, and the Heun method should be used instead [89]. Let us conclude with a technical note on the numerical simulations. In the presence of dissipation and thermal uctuations the only conserved quantity of the system is the magnitude of the magnetic moment ( jmj= 1). We thus use the deviation 1 jmj2as a measure of the numer- ical error in both the stochastic and deterministic sim- ulations presented in this article. For the deterministic simulations the error stays much smaller than any other physical degree of freedom of the system during the whole simulation time. The simulation of the stochastic dynam- ics shows a larger numerical error [see Fig. 4(d)], which can be partially reduced by taking a smaller time-step size. We note that, for the value of magnetic anisotropy given in Table I, the system of SDE is sti. This, together with the requirement imposed on the time-step size by10 the numerical error, ultimately limits the maximum time we can simulate to a few microseconds. However, this is sucient to validate the agreement between the SDE and the deterministic simulations presented in the article. Appendix C: Additional gures In this appendix we provide additional gures. 1. Dynamics in the atom phase In Fig. 5 we analyze magnetization dynamics in the atom phase as a function of dierent system parame- ters. In Fig. 5(a) we show how magnetization switching changes as the anisotropy constant kais varied. We con- sider the bias eld B0= 1100 mT, which is larger than the value considered in the main text. This is done to en- sure thatB0>B atom for all anisotropy values. Fig. 5(a) demonstrates that the switching time , given by Eq. (8), is an excellent approximation for the dynamics across a wide range of values for the anisotropy constant ka. The larger discrepancy between Eq. (8) and the line showing the case with ka= 106J/m3is explained by the prox- imity of this point to the unstable region (in this case given by the critical eld Batom = 900 mT), and better agreement is recovered at larger bias eld values. In Fig. 5(b) we analyze the validity of Eq. (8) for dif- ferent values of the Gilbert damping parameter and the temperature T. The thick dashed line shows the region where Eq. (8) diers from the exact switching time by 5%; below this line the deviation becomes increasingly more signicant. As evidenced by Fig. 5(b), shows lit- tle dependence on T; its order of magnitude remains con- stant over a wide range of cryogenic temperatures. On the other hand, the dependence on is more pronounced. In fact, reducing the Gilbert parameter signicantly de- lays the switching time, leading to levitation times as long as1s. Additionally, we point out that depends on the eldgradientB0and curvature B00through the initial con- ditionmk(t= 0). In particular, magnetization switch- ing can be delayed by decreasing B0, as this reduces the initial misalignment of the magnetization and the anisotropy axis (i. e. jmk(t= 0)j!1). 2. Dynamics in the presence of rotation In Fig. 6 we consider a more general equilibrium con- guration, namely a nanomagnet initially rotating such that in the equilibrium point Le=I3!Sex, with!S>0 denoting the rotation in the clockwise direction. This equilibrium point is linearly stable in the absence of dis- sipation [23, 24], with additional stability of the system provided by the mechanical rotation, analogously to the classical magnetic top [25{27]. In Fig. 6(a) we analyze how magnetization switching in the atom phase changes in the presence of rotation for dierent values of parameters. The rotation has a slight eect on the switching time , shifting it forwards (back- wards) in case of a clockwise (counterclockwise) rotation. This is generally expected in the presence of magneto- mechanical coupling [74, 88]. In Fig. 6(b) we show the motion in the y-zplane in the Einstein{de Haas phase for both directions of rota- tion. This can be compared with Fig. 3(a). The rotation does not qualitatively aect the dynamics of the system. The dierence in the two trajectories can be explained by a dierent total angular momentum in the two cases, as in the case of a clockwise (counterclockwise) rotation the mechanical and the internal angular momentum are parallel (anti-parallel), such that the total angular mo- mentum is increased (decreased) compared to the non- rotating case. This asymmetry arises from the linear stability of a rotating nanomagnet, and it is not a conse- quence of dissipation. In fact, we conrm by numerical simulations that the escape time t?as a function of the pressurePshows no dependence on the mechanical ro- tation!S. Namely, even in the presence of mechanical rotation one recovers the same plot as shown in Fig. 3(f). [1] A. Einstein and W. J. de Haas, Experimental proof of the existence of Amp ere's molecular currents, Proc. K. Ned. Akad. Wet. 18, 696 (1915). [2] O. W. 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Dynamics of a nanomagnet initially rotat- ing around the axis exwith frequencyj!Sj=(2) = 100 MHz. (a) Magnetization switching in the atom phase. Line denoted by circle corresponds to the same set of pa- rameters as in Fig. 2(a). Each remaining line diers by a single parameter, as denoted by the legend. Dotted verti- cal lines show Eq. (8). (b) Motion in the y-zplane in the Einstein{de Haas phase, using the same numerical values of the parameters as in Fig. 3. Left panel: Clockwise rotation. Right panel: counterclockwise rotation. [11] D. F. Jackson Kimball, A. O. Sushkov, and D. Budker, Precessing ferromagnetic needle magnetometer, Phys. Rev. Lett. 116, 190801 (2016). [12] P. Kumar and M. Bhattacharya, Magnetometry via spin- mechanical coupling in levitated optomechanics, Opt. Express 25, 19568 (2017). [13] Y. B. Band, Y. Avishai, and A. Shnirman, Dynamics of a magnetic needle magnetometer: Sensitivity to Landau- Lifshitz-Gilbert damping, Phys. Rev. Lett. 121, 160801 (2018). [14] T. 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2008.08043v2.A_class_of_Finite_difference_Methods_for_solving_inhomogeneous_damped_wave_equations.pdf

A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO Abstract. In this paper, a class of nite dierence numerical techniques is presented to solve the second-order linear inhomogeneous damped wave equation. The consistency, stability, and convergences of these numerical schemes are discussed. The results obtained are compared to the exact solution, ordinary explicit, implicit nite dierence methods, and the fourth-order compact method (FOCM). The general idea of these methods is developed by using C0-semigroups operator theory. We also showed that the stability region for the explicit nite dierence scheme depends on the damping coecient. 1.introduction The damped wave equation is an important evolution model. Physicists and engineers widely use it in describing the propagation of water waves, sound waves, electromagnetic waves, etc. For instance, a model that describes the transverse vibrations of a string of a nite length in the presence of an external force proportional to the velocity satises the following partial dierential equation (PDE) utt=
u (x)ut+g(x;t);foraxb; t2R; (1) with initial conditions u(x;0) =(x); ut(0;x) = (x);foraxb; and boundary conditions u(a;t) =ua(t)u(b;t) =ub(t); t2R; where 0 is the damping force, u(x;t) is the position of a point xin the string, at instantt. The functions (x); (x) and their derivatives are continuous functions of xand the forcing function g(x;t)2L1 x(R). The study of the numerical solution of this model will be our main focus in this article. In general, the damping reduces the amplitude of vibration, and therefore, it is desirable to have some amount of damping to achieve stability in the system. One can nd a detailed study in [4, 2, 14] of the eect of damping in the long-time stability of the equation (1). Also, for practical purposes, it is important to know how much damping is needed in the system to ensure the fastest decay rate in the amplitude of the wave as time evolves. For example, in the case of the 3D tsunami wave, we would like to know the size and structure of the damping force to bring the amplitude of the tsunami to a safe level before it hits the Date : December 23, 2021. 2000 Mathematics Subject Classication. 65M06, 37N30, 65N22 . Key words and phrases. damped wave equation, numerical method, Pad e approximation, compact nite dierence scheme, unconditionally stable, convergence. 1arXiv:2008.08043v2 [math.NA] 22 Dec 20212 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO shore (see [17] and references within). In the case of the damping terms as a function of time and space, obtaining an analytic solution is a challenging problem. There comes the numerical study to nd the approximate solution to such problems. In recent years, much attention has been given to studying the behaviours of the numerical solution of (1); see for example [18, 3, 13, 5, 8]. In this manuscript, we develop a class of methods based on the properties of C0- semigroups of the evolution equations, as well as the nite dierence method (FD). Gen- erally speaking, the FD methods are easy to apply to partial dierential equitations, but they may not lead to optimal results depending on the type of equation. The techniques used in this article take advantage of the C0-semigroup property and Pad e approximation, which lead to a better performance of new numerical schemes presented in this article. At the time of writing this paper, we became aware of [12] that have a similar approach in which the authors drive a fourth-order implicit nite dierence scheme to solve a second- order telegraph equation with constant coecients. However, the author of [12] did not consider the explicit nite dierence schemes and used the higher-order approximation terms of the space derivative and time integration to attain higher-order accuracy of the numerical solution. In this manuscript, in addition to driving a class of explicit and implicit methods, we discussed the issue of the instability of the explicit nite dierence methods. Moreover, this paper explains the importance of the non-zero damping term in the existence of the stability region as well. We have also shown that the explicit nite dierence method produces better results and costs a lot fewer calculations in its stability region. An outline of the contents of this paper is as follows. In section 2, we set our numerical schemes and derive our method. Section 3 is devoted to the analytical properties of the method, i.e., consistency, stability, and convergence. Finally, in section 4, the numerical results of our method are compared with some of the existing methods. 2.The semigroup approach To present a more convenient form of (1), we dene a new vector function U(x;t) = (u;ut)T; U 0= ((x); (x))T: (2) With these changes, the equation (1) turns into an evolution equation of rst-order in time Ut=AU+G; (3) where A=0 @0I
(x)1 A; G (x;t) =0 @0 g(x;t)1 A; with initial condition U(x;0) = (u(x;0);ut(x;0))T: The system above is dened on a Hilbert space H=H1[a;b]L2(R). The domain of A isD(A) =H2[a;b]H1(R). SinceAis a dissipative and invertible operator on a Hilbert space, it generates a C0-semigroup of contractions for t0 by the Lumber-Phillips theoremA CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 3 [10]. Also, note that the inclusion D(A),! H is compact by the Rellich-Kondrachiv theorem. Thus, the spectrum of Acontains only eigenvalues of nite multiplicity. 2.1.Discretization. We use the central discretization for the Laplacian operator
as
u(x;t) =u(xh;t)2u(x;t) +u(x+h;t) h2: We set the mesh points xi=a+ih; i = 0;1;2:::N; whereh=ba N of the interval [ a;b]. Then the continuous operator Acan be approximated by the matrix operator M(2N2)=2 40I 1 h2A3 5; whereIis the identity matrix of order N1, and A=2 66666666642 1 0


0 12 1


0 0 12...0 ............... 0


0 123 7777777775 (N1)(N1); =2 66666666664 (x1) 0


0 0 (x2)::: 0 0 0


0 ............ 0


0 (xN1)3 77777777775 (N1)(N1):(4) The discrete operator M(2N2)is dened on the nite-dimensional Banach space X(2N2)= C(2N2)[a;b]. LetV2N2(t) =u(x1;t);u(x2;t):::u(xN1;t);ut(x1;t);


ut(xN1)Tbe a vector which discretizes the function U(x;t) = (u(x;t);@tu(x;t)) over the interval [ x1;xN1], then (3) leads us to the following dynamical system dV2N2(t) dt=2 40I 1 h2A3 5V2N2(t) +2 40 G(t)3 5+2 40 1 h2B(t)3 5; (5) whereG(t) = [g(x1;t);g(x2;t);:::;g (xN1;t)]T,B(t) =ua(t);0;0;:::; 0;0;ub(t) and the initial condition V2N2(0) =(x1);(x2):::(xN1); (x1);


(xN1)T: We will now drop the subscript 2 N2 and write V2N2(x;t) byV(t), andM2N2byM in the rest of our presentation. SinceMis a bounded linear operator on a nite-dimensional space X(2N2)H1 0(R), it generates a C0-semigroup for each N. Then, by using the C0-semigroup theory of inhomo- geneous evolution equations, we can construct the sequences of approximating solutions to (5) as V(t) =eMtV(0) +Zt 0eM(ts)F(s)ds;4 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO where F(t) =2 40 G(t)3 5+2 40 1 h2B(t)3 5: We replace tbyt+kin the above equation and use the C0-semigroup property, eM(t+k)= eMteMk, we get V(t+k) =eM(t+k)V(0) +Zt+k 0eM(t+ks)F(s)ds =eMkeMtV(0) +eMkZt 0eM(ts)F(s)ds+eMkZt+k teM(ts)F(s)ds =eMk V(t)Zt 0eM(ts)F(s)ds +eMkZt 0eM(ts)F(s)ds +eMkZt+k teM(ts)F(s)ds: Thus, V(t+k) =eM(k)V(t) +eMkZt+k teM(ts)F(s)ds: (6) To approximate the term eMk, we make use of the rational approximation of exponential functions, i.e., the Pad e approximation. 2.2.Pad e Approximant. The Pad e approximation is a rational approximation of a function of a given order [1]. The technique was developed around 1890 by Henri Pad e, but it goes back to F. G. Frobenius who introduced the idea and investigated the features of rational approximations of power series. The Pad e approximation is usually superior when functions contain poles because the use of rational function allows them to be well represented. The Pad e approximation often gives a better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. Pad e approximation gives the exponential functions eas e=1 +a1+a22+


+aTT 1 +b1+b22+


+TS+cS+T+1S+T+1+O(S+T+2); whereCS+T+1,ai's andb's are constants. The rational function RS;T() :=1 +a1+a22+


+aTT 1 +b1+b22+


+TS=PT() QS()(7) is the so-called Pad e approximation of order ( S;T) toewith the leading error cS+T+1S+T+1. The table below gives some Pad e approximations of the exponential function[18].A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 5 (S,T) RS;T() Leading error (0,1) 1 +1 22 (0,2) 1 ++1 22 1 63 (1,0) 1 11 22 (1,1) 1 +1 2 11 21 123 Now combining (6) and (7), we get QS(Mk)V(t+k) =PT(Mk)V(t) (8) +PT(Mk)Zt+k tPT(M(ts))(QS(M(ts)))1F(s)ds: For the integration term on the right-hand side, one can use the numerical integration formula. Here, we will use the Trapezoidal approximation of integration to get the following numerical scheme QS(Mk)V(t+k) =PT(Mk)V(t) +k 2PT(Mk)F(t) +k 2QS(Mk)F(t+k): (9) This is the general form of our scheme, and each choice of QSandPTproduces explicit and implicit nite dierence methods to the solution of the damped wave equation (1). Next, we present two schemes by taking ( S;T) = (0;1) and (S;T) = (1;1). Similarly, we can develop more schemes of dierent order by taking dierent values of SandTmentioned in the table above. Explicit Method ( FD(0;1)): If we set ( S;T) = (0;1) i.e.Q0() = 1 andP1() = 1+ in (9), we will obtain the FD-(0,1) as ( V(t+k) = (1 +Mk)V(t) +k 2(I+Mk)F(t) +k 2F(t+k); V0= [u1(0);


;uN1(0);@tu1(0);


;@tuN1(0)]:(10) Implicit Method ( FD(1;1)): By a choice of P1() = 1 +1 2andQ1() = 11 2in (9), we will obtain the FD-(1,1) as 8 >< >: 11 2Mk
V(t+k) = 1 +1 2Mk
V(t) +k 2 I+1 2Mk
F(t) +k 2 I1 2Mk
F(t+k); V0= [u1(0);


;uN1(0);@tu1(0);


;@tuN1(0)]:(11)6 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO In the case of the implicit method, we need to solve a more extensive system of equations in each time step due to the implicit nature of the system. However, the analysis and numerical results suggest that the implicit scheme gives us an accurate approximation and, more importantly, an unconditionally stable scheme. 3.Consistency, Stability and Convergence In this section, we will investigate the analytical properties of our numerical schemes (10) and (11). We will prove that the numerical methods (10) and (11) are consistent, stable, and hence convergent. We will use the direct analysis to prove the consistency, the matrix method to prove the stability, and the Lax-equivalence theorem to prove the convergence of our numerical schemes. 3.1.Consistency. Given a partial dierential equation Lu=fand a nite dierence scheme,Fh;kv=f, we say that the nite dierence scheme is consistent with the partial dierential equation if for any smooth function (x;t), LFh;k!0 ash;k!0; or in other words, the local truncation goes to zero as the mesh size handktends to zero. The partial dierential equation Ut0 @0I
(x)1 AU0 @0 g(x;t)1 A= 0; is approximated at the point ( xi;t) by thenthrow of the following dierence equations 1 k(QS(Mk)V(t+k)PT(Mk)V(t))1 2PT(Mk)F(t)1 2QS(Mk)F(t+k) = 0; forn= 1;2;


;(2N2): Then the local truncation error Ti;t(U) is dened as the nthrow of 1 k(QS(Mk)U(t+k)PT(Mk)U(t))1 2PT(Mk)F(t)1 2QS(Mk)F(t+k); forn= 1;


;(2N2). The truncated error depends on the choice of QSandPT. Therefore, we should consider them case by case. Here we consider FD(0;1) andFD(1;1). The remaining cases follow the same path. 3.1.1.FD(0;1).The local truncation error T0;1 i;t(U) of the explicit FD(0;1) is dened as thenthrow of 1 k(U(t+k)(I+Mk)U(t))1 2(I+Mk)F(t)1 2F(t+k) forn= 1;2


;(2N2). Thus fori= 1;2;


N1, we get the following system of (2 N2) equations T0;1 i;t(U) =1 k(u(xi;t+k)u(xi;t))ut(xi;t)k 2g(xi;t);A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 7 and T0;1 i+N1;t(U) =1 kut(xi;t+k)1 h2(u(xih;t)2u(xi;t) +u(xi+h;t)) 1 k(1k (xi))ut(xi;t)(1 (xi)k) 2g(xi;t)1 2g(xi;t+k): By Taylor series expansion, we get T0;1 i;t(U) =k 2!utt(xi;t) +k2 3!uttt(xi;t) +


k 2g(xi;t) and T0;1 i+N1;t(U) = (utt(xi;t)uxx(xi;t) + (xi)ut(xi;t)g(xi;t)) +k 2!uttt(xi;t) +O(k2)2h2 4!uxxxx(xi;t) +O(h4) + (xi)k 2g(xi;t) k 2gt(xi;t) +O(k2): fori= 1;2;


;(N1). By (1), the last ( N1), equations can be written as T0;1 i;t(U) =k 2!uttt(xi;t) +O(k2)2h2 4!uxxxx(xi;t) +O(h4) + (xi)k 2g(xi;t) k 2gt(xi;t) +O(k2): We observe as handkgo to zero, the truncation error Ti;t(U)!0. Hence, the numerical scheme is consistent. . 3.1.2.FD(1;1).The local truncation error T1;1 i;t(U) of the explicit FD(1;1) is dened as thenthrow of 1 k I1 2Mk U(t+k) I+1 2Mk U(t) 1 2 I+1 2Mk F(t)1 2 I1 2Mk F(t+k) forn= 1;2;


;(2N2). Thus fori= 1;2


;N1, we get the following system of (2 N2) equations T1;1 i;t(U) =1 k(u(xi;t+k)u(xi;t))ut(xi;t)k 4(g(xi;t+k)g(xi;t)); and T1;1 i+N1;t(U) = 1 2h2(u(xih;t+k)2u(xi;t+k) +u(xi+h;t+k)) +1 k 1 + k 2 ut(xi;t+k) 1 2h2(u(xih;t)2u(xi;t) +u(xi+h;t)) +1 k 1 k 2 ut(xi;t) 1 2 (1 + k 2)g(xi;t) + (1 k 2)g(xi;t+k) :8 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO By Taylor series expansion, we get T1;1 i;t(U) =k 2utt(xi;t)k2 4gt(xi;t) +O(k3); and T1;1 i+N1;t(U) = (utt(xi;t)uxx(xi;t) + (xi)ut(xi;t)g(xi;t)) +k 2uttt(xi;t) +O(k2)k 2uxxt(xi;t) +O(k2)h2 2uxxxx(xi;t) +O(h4) kh2 6uxxxxt +h2O(k2)k 2gt(xi;t) +O(k2) +k3 (xi) 4gt(xi;t) +O(k3); fori= 1;2;


;(N1). By (1), the last ( N1), equations can be written as T1;1 i;t(U) =k 2uttt(xi;t) +O(k2)k 2uxxt(xi;t) +O(k2)h2 2uxxxx(xi;t) +O(h4) kh2 6uxxxxt +h2O(k2)k 2gt(xi;t) +O(k2) +k3 (xi) 4gt(xi;t) +O(k3): Ashandkgo to zero, the truncation error Ti;t(U)!0. Hence, the numerical scheme is consistent. 3.2.Stability. To prove the stability of our numerical schemes, we show that there exists a region  so that for every h;k2, all the eigenvalues of the amplication matrix related to the numerical schemes lie in or on the unit circle. Proposition 1. The explicit FD-(0,1) approximation dened in (9)is stable for k <2  andp k h<p  2, where = maxx2[a;b] (x). The following lemma will be used to prove Proposition 1. Lemma 1. Letp(x) =ax2+bx+cbe a polynomial function with a >0, then necessary and sucient conditions for the polynomial p(x)to have the modulus of its roots less or equal to 1 are (i)jcj<a (ii)p(1)>0andp(1)>0. One can nd the proof of the above lemma in [7, 16]. Proof of proposition 1. The eigenvalues of the amplication matrix I+kMare the roots of the following quadratics equation 2+ (2 + (xn)k)+ 1k (xn) + 4r2sin2n 2N = 0; n= 1;


;(N1); wherer=k=h. Note for each n, there are two roots of the above polynomial, and hence we have 2 N2 eigenvalues for the matrix I+kM. Next, in order to satisfy the conditions ( i) and (ii) of lemma (1), we impose restrictions on andr. Indeed, the assumption ( i) is satised if 1<1k (xn) + 4r2sin2n 2N <1; n= 1;2;


;N1:A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 9 The right-hand inequality gives us 4r2sin2n 2N <k (xn)k ; r2<k  4 sin2n 2N
: Thus,p k h<p  2. Now, the rst part of the assumption (ii) is satised if p(1) = 4r2sin2n 2N >0; which is true as long as r>0. Now, the second part of assumption (ii) is satised if p(1) = 42k (xn) + 4r2sin2n 2N >0; which is true if k <2: Hence the second part of the assumption (ii) of lemma (1) is satised for k<2 . Proposition (1) tells us that the damping term plays an important role in the stability of the explicit method (9). The nite dierence scheme (9) will be unstable for any values ofhandkif the damping term (x) is identically zero or handkare out of the required bounds of the proposition (1). Proposition 2. The implicit FD-(1,1) approximation dened by (11) is unconditionally stable. Proof. The eigenvalues of the matrix Mare given by  n= (xn) 21 2r (xn)216 h2sin2(n 2N); n= 1;


;N1: Then, by using functional calculus, the eigenvalues  nof the matrix ( I1 2kM)1((I+ 1 2kM)) are given by  n=1 +k 2 n 1k 2 n; n= 1;


;N1: Also, we have Re(n)0 because 0. Thus, for any values of n;h;k , and (xn), we getj nj1. Hence, the implicit method (11) is unconditionally stable.  A direct application of the Lax Equivalence Theorem [9, 15] leads to the convergence of our models. Corollary 1. The nite dierence explicit FD(0;1)of(9)and implicit FD(1;1)of (11) are convergent.10 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO 4.Performance of Numerical schemes In this section, we will see the performance of each nite dierence scheme on a sample problem. Sample Problem: We consider the following damped wave equation utt=uxx2ut; over the region = [0 x](t>0) with initial conditions u(x;0) = sin(x); u t(x;0) =sin(x); and boundary conditions u(0;t) = 0 =u(;t);fort>0: The exact solution of the above problem is u(x;t) =etsin(x). Figure 1. The approximate solution given by explicit FD-(0,1) of (10) at t= 1 withk= 0:05,h= 0:13464. Figure 2. The approximate solution given by implicit FD-(1,1) of (11) at t= 1 withk= 0:05,h= 0:13464. FIGURE (1) and (2) show the numerical solutions using nite dierence methods (9) and (11) at t= 1. From the obtained numerical results, we can conclude that the numerical solutions are in good agreement with the exact solution.A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 11 4.1.Comparison with other methods. In this section, we compare our result with the ordinary explicit and implicit nite dierence methods mentioned below. We also compare our result with the FOCM method of [6]. We take the same test example mentioned above for this comparison. Ordinary Explicit Finite Dierence Scheme (OEFD): The ordinary explicit nite dierence scheme in the matrix form is (1 + k 2)u(t+k) = (2Ir2A)u(t) + k 21 un1+r2B(t); (12) wherer=k=h,B(t) =ua(t);0;0;:::; 0;0;ub(t) , and the matrix Ais dened in equation (4). Ordinary Implicit Finite Dierence Scheme(OIFD): The ordinary implicit nite dierence scheme in the matrix form is  1 + (xn)k 2r2 2A u(t+k) = 2 +r2 2A u(t) + (xn)k 21 u(tk) (13) +r2 2(B(t+k) +B(t)); wherer=k=h,B(t) =ua(t);0;0;:::; 0;0;ub(t) , and the matrix Ais dened in equation (4). The derivation of these schemes can be found in [11]. FIGURE (3) and (4) show the performances of our methods (10) and (11) in comparison with nite dierence schemes (12) and (13) using k= 0:01 andh= 0:063. The implicit FD-(1,1) produces a much better result even for a large value of r. When the values of h andkfail to satisfy the stability conditions of the explicit FD-(0,1), it can be seen that the numerical solution became unstable after some time iterations. However, it is interesting to see that even in this case the global numerical solution fails to exist, the local numerical solution does exist for a small time and it was very close to the exact solution. It is apparent that the explicit nite dierence scheme (12) and (9) are not stable for large values ofr. The implicit FD-(1,1) is very stable and produces a much better result when compared to the ordinary implicit nite dierence scheme (13). Figure 3. The absolute er- ror of the method (10) and (11) forr= 1:5915. Figure 4. The absolute er- ror of the method (12) and (13) forr= 1:5915.12 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO Figure 5. The absolute error of the method (10), (11), (12) and (13). In the FIGURE (5), we plotted the absolute error at the four dierent values of r=:016, r=:159,r=:995, andr= 1:45. One can see for a small values of r= 0:016, all the four schemes produce fairly stable results. This shows that when our explicit nite dierence FD-(0,1) satises the assumptions of proposition (1), it is stable and produces better results than the other three. However, the performances of the explicit nite dierence method (9) and implicit nite dierence FD-(1,1) (11) are very similar for small values of r. TABLE 1 shows the comparison between the errors generated by FOCM, OEFD, OIFD, EX(0;1) andIM(1;1) att= 0:3 withh= 10andk=1 10. TABLE 2 shows the magnitude of the maximum error at time t= 6 between the exact solution and the numerical solution obtained by using FOCM, OEFD, OIFD, FD(0;1), andFD(1;1) discussed above with dierent values of handk. 5.Conclusion In this paper, a class of nite dierence methods using the C0-semigroup operator theory for solving the inhomogeneous damped wave equation is presented. The stability and consistency of the implicit and explicit methods are proved. Test examples are presented, and the results obtained are compared with the exact solutions. The comparison certiesA CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 13 x FOCM OEFD OIFD EX-(0,1) IM-(1,1) 0 0 0 0 0 0 0.314159265 0.00012256 6.29067E-05 0.000135485 0.001494844 1.23932E-05 0.628318531 0.00022777 0.000119656 0.000257708 0.002843363 2.35734E-05 0.942477796 0.00031458 0.000164692 0.000354705 0.003913553 3.24459E-05 1.256637061 0.00036955 0.000193607 0.000416981 0.004600658 3.81425E-05 1.570796327 0.00038865 0.00020357 0.000438439 0.004837418 4.01054E-05 1.884955592 0.00036955 0.000193607 0.000416981 0.004600658 3.81425E-05 2.199114858 0.00031458 0.000164692 0.000354705 0.003913553 3.24459E-05 2.513274123 0.00022777 0.000119656 0.000257708 0.002843363 2.35734E-05 2.827433388 0.00012256 6.29067E-05 0.000135485 0.001494844 1.23932E-05 3.141592654 0 0 0 0 0 Table 1. Absolute Error r EFD IFD EX-(0,1) IM-(1,1) 1.59 1.00967E+34 0.002547509 9.08234E+13 2.231E-06 0.53 3.05424E-05 0.00079153 2.18322E+11 1.36036E-05 0.32 2.04246E-05 0.000473008 3410.243641 1.43835E-05 0.23 1.76452E-05 0.000339697 0.011310925 1.45754E-05 0.18 1.64986E-05 0.000266457 7.84447E-05 1.46457E-05 Table 2. Maximum Error at t= 6 that implicit FD-(1,1) gives good results. Summarizing these results, we can say the general form of the new nite dierence methods has a reasonable amount of calculations and the form is easy to use. All results are obtained by using MATLAB version 9.7.14 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO References [1] GA Baker and PR Graves-Morris. Pad e approximants, part i, encycl. math., vol. 13. Reading, MA: Addison-Wesley , 7:233{236, 1981. [2] Nicolas Burq and Romain Joly. Exponential decay for the damped wave equation in unbounded domains. Communications in Contemporary Mathematics , 18(06):1650012, 2016. [3] Ian Christie, David F Griths, Andrew R Mitchell, and Olgierd C Zienkiewicz. Finite element meth- ods for second order dierential equations with signicant rst derivatives. International Journal for Numerical Methods in Engineering , 10(6):1389{1396, 1976. [4] Lawrence C. Evans. Partial dierential equations . American Mathematical Society, Providence, R.I., 2010. [5] Feng Gao and Chunmei Chi. Unconditionally stable dierence schemes for a one-space-dimensional linear hyperbolic equation. Applied Mathematics and Computation , 187(2):1272{1276, 2007. [6] M.T. Hussain, A. Pervaiz, Zainulabadin Zafar, and M.O. Ahmad. Fourth order compact method for one dimensional homogeneous damped wave equation. Pakistan Journal of Science , 64(2):122, 2012. [7] Eliahu Jury. On the roots of a real polynomial inside the unit circle and a stability criterion for linear discrete systems. IFAC Proceedings Volumes , 1(2):142{153, 1963. [8] Stig Larsson, Vidar Thom ee, and Lars B Wahlbin. Finite-element methods for a strongly damped wave equation. IMA journal of numerical analysis , 11(1):115{142, 1991. [9] Peter D Lax and Robert D Richtmyer. Survey of the stability of linear nite dierence equations. Communications on pure and applied mathematics , 9(2):267{293, 1956. [10] Gunter Lumer and Ralph S Phillips. Dissipative operators in a banach space. Pacic Journal of Mathematics , 11(2):679{698, 1961. [11] Andrew Ronald Mitchell and David Francis Griths. The nite dierence method in partial dieren- tial equations. Wiley. New York , 1980. [12] Akbar Mohebbi. A fourth-order nite dierence scheme for the numerical solution of 1d linear hyper- bolic equation. Commun. Numer. Anal , 2013. [13] Ahmet Ozkan Ozer and E _Inan. One-dimensional wave propagation problem in a nonlocal nite medium with nite dierence method. In Vibration Problems ICOVP 2005 , 383{388. Springer, 2006. [14] Jerey Rauch, Michael Taylor, and Ralph Phillips. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana university Mathematics journal , 24(1):79{86, 1974. [15] Robert D Richtmyer and Keith W Morton. Dierence methods for initial-value problems. dmiv , 1994. [16] Paul A Samuelson. Conditions that the roots of a polynomial be less than unity in absolute value. The Annals of Mathematical Statistics , 12(3):360{364, 1941. [17] Harvey Segur. Waves in shallow water, with emphasis on the tsunami of 2004. In Tsunami and nonlinear waves , 3{29. Springer, 2007. [18] Gordon Smith. Numerical solution of partial dierential equations: nite dierence methods . Oxford university press, 1985. Fazel Hadadifard, Department of Mathematics, Drexel University Email address :fh352@drexel.edu Satbir Malhi, Department of Mathematics, Saint Mary's College of California Email address :smalhi@stmarys-ca.edu Zhengyi Xiao, Department of Mathematics, Franklin & Marshall College Email address :zxiao@fandm.edu

1810.07020v4.Superfluid_spin_transport_in_ferro__and_antiferromagnets.pdf

Super uid spin transport in ferro- and antiferromagnets E. B. Sonin Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (Dated: March 25, 2019) This paper focuses on spin super uid transport, observation of which was recently reported in antiferromagnet Cr 2O3[Yuan et al. , Sci. Adv. 4, eaat1098 (2018)]. This paper analyzes the role of dissipation in transformation of spin current injected with incoherent magnons to a super uid spin current near the interface where spin is injected. The Gilbert damping parameter in the Landau{ Lifshitz{Gilbert theory does not describe dissipation properly, and the dissipation parameters are calculated from the Boltzmann equation for magnons scattered by defects. The two- uid theory is developed similar to the two- uid theory for super uids. This theory shows that the in uence of temperature variation in bulk on the super uid spin transport (bulk Seebeck eect) is weak at low temperatures. The scenario that the results of Yuan et al. are connected with the Seebeck eect at the interface between the spin detector and the sample is also discussed. The Landau criterion for an antiferromagnet put in a magnetic eld is derived from the spectrum of collective spin modes. The Landau instability starts in the gapped mode earlier than in the Goldstone gapless mode, in contrast to easy-plane ferromagnets where the Goldstone mode becomes unstable. The structure of the magnetic vortex in the geometry of the experiment is determined. The vortex core has the skyrmion structure with nite magnetization component normal to the magnetic eld. This magnetization creates stray magnetic elds around the exit point of the vortex line from the sample, which can be used for experimental detection of vortices. I. INTRODUCTION The concept of spin super uidity is based on the anal- ogy of the equations of magnetodynamics with the equa- tions of super uid hydrodynamics.1. The analogy led to the suggestion that in magnetically ordered media persis- tent spin currents are possible, which are able to trans- port spin on macroscopical distances without essential losses.2 The phenomenon of spin super uidity has been dis- cussed for several decades.2{15We dene the term super- uidity in its original meaning known from the times of Kamerlingh Onnes and Kapitza: transport of some phys- ical quantity (mass, charge, or spin) over macroscopical distances without essential dissipation. This requires a constant or slowly varying phase gradient at macroscopic scale with the total phase variation along the macroscopic sample equal to 2 multiplied by a very large number. Spin super uidity assumes the existence of spin current proportional to the gradient of the phase (spin super- current). In magnetically ordered media the phase is an angle of rotation in spin space around some axis (further in the paper the axis z). In contrast to the dissipative spin-diusion current proportional to the gradient of spin density, the spin supercurrent is not accompanied by dis- sipation. Spin super uidity require special topology of the order parameter space. This topology is realized at the pres- ence of the easy-plane magnetic anisotropy, which con- nes the magnetization of the ferromagnet or sublattice magnetizations of the antiferromagnet in an easy plane. In this case one may expect that the current state is sta- ble with respect to phase slips, which lead to relaxation of the supercurrent. In the phase slip event a vortex with 2phase variation around it crosses streamlines of thesupercurrent decreasing the total phase variation across streamlines by 2 . The concept of the phase slip was introduced by Anderson16for super uid4He and later was used in studying spin super uidity.2,3 Phase slips are suppressed by energetic barriers for vor- tex expansion. But these barriers disappear when phase gradients reach critical values determined by the Landau criterion. The physical meaning of the Landau criterion is straightforward: the current state becomes unstable when there are elementary excitations with negative en- ergy. So, to check the Landau criterion one must know the full spectrum of collective modes. Sometimes any presence of spin current proportional to the phase gradient is considered as a manifestation of spin super uidity.17,18However, spin current propor- tional to the spin phase gradient is ubiquitous and ex- ists in any spin wave or domain wall, also in the ground state of disordered magnetic media. In all these cases the total variation of the phase is smaller, or on the or- der of. Connecting these cases with spin super uid- ity makes this phenomenon trivial and already observed in old experiments on spin waves in the middle of the 20th Century. One may call the supercurrent produced by the total phase variation of the order or less than 2microscopical supercurrent, in contrast to persistent macroscopical supercurrents able to transport spin over macroscopical distances. The analogy with usual super uids is exact only if the spin space is invariant with respect to spin rotation around the hard axis normal to the easy plane. Then there is the conservation law for the spin component along the hard axis. In reality this invariance is bro- ken by in-plane anisotropy. But this anisotropy is usu- ally weak, because it originates from the spin-orbit in- teraction, which is relativistically small compared to thearXiv:1810.07020v4 [cond-mat.mes-hall] 22 Mar 20192 exchange interaction, i.e., inversely proportional to the speed of light.19Macroscopical spin supercurrents are still possible if the energy of supercurrents exceeds the in-plane anisotropy energy. Thus, one cannot observe macroscopical spin supercurrents not only at large cur- rents as in usual super uids, but also at small currents.2 From the time when the concept of spin super uidity (in our denition of this term) was suggested2, it was debated about whether the super uid spin current is a \real" transport current. As a response to these con- cerns, in Ref 2 a Gedanken (at that time) experiment for demonstration of reality of super uid spin transport was proposed. The spin is injected to one side of a mag- netically ordered layer of thickness dand spin accumula- tion is checked at another side. If the layer is not spin- super uid, then the spin is transported by spin diusion. The spin current and the spin density exponentially de- cay at the distance of the spin diusion length, and the density of spin accumulated at the other side decreases exponentially with growing distance d. However, if the conditions for spin super uidity are realized in the layer, then the super uid spin current decays much slower, and the accumulated spin density at the side opposite to the side where the spin is injected is inversely proportional tod+C, whereCis some constant. The interest to long-distance spin transport, especially to spin super uid transport, revived recently. Takei and Tserkovnyak7carried out a microscopic analysis of in- jection of spin to and ejection of spin out of the spin- super uid medium in an easy-plane ferromagnet justify- ing the aforementioned scheme of super uid spin trans- port. Takei et al.8extended this analysis to easy-plane antiferromagnets. Finally Yuan et al.20were able to real- ize the suggested experiment in antiferromagnetic Cr 2O3 observing spin accumulation inversely proportional to the distance from the interface where spin was injected into Cr2O3. Previously Borovik-Romanov et al.21reported evi- dence of spin super uidity in the Bphase of super uid 3He. They detected phase slips in a channel with su- per uid spin current close to its critical value. It was important evidence that persistent spin currents are pos- sible. But real long-distance transportation of spin by these currents was not demonstrated. Moreover, it is impossible to do in the nonequilibrium magnon Bose{ Einstein condensate, which was realized in the Bphase of3He super uid6and in yttrium-iron-garnet magnetic lms.22The nonequilibrium magnon Bose{Einstein con- densate requires pumping of spin in the whole bulk for its existence. In the geometry of the aforementioned spin transport experiment this would mean that spin is per- manently pumped not only by a distant injector but also all the way up the place where its accumulation is probed. Thus, the spin detector measures not only spin coming from a distant injector but also spin pumped close to the detector. Therefore, the experiment does not prove the existence of long-distance spin super uid transport. There were also reports on experimental detection ofspin super uidity in magnetically ordered solids17,18, but they addressed microscopical spin supercurrent.23As ex- plained above, \super uidity" connected with such cur- rents was well proved by numerous old experiments on spin waves and does not need new experimental conr- mations. The work of Yuan et al.20was the rst report on long-distance super uid spin transport with spin ac- cumulation decreasing with distance from the injector as expected from the theory. Long distance super uid spin transport was also recently reported in a graphene quan- tum antiferromagnet.24 The experiment on super uid spin transport20has put to rest another old dispute about the spin super uidity concept. At studying spin super uidity in the Bphase of super uid3He, it was believed4that spin super uidity is possible only if there are mobile carriers of spin and a counter ow of carriers with opposite spins transports spin. If so, then spin super uidity is impossible in insu- lators. Moreover, Shi et al.25argued that it is a critical aw of spin-current denition if it predicts spin currents in insulators. Since Cr 2O3is an insulator the experiment of Yuan et al.20rules out this presumption. Boosted by the super uid spin transport experiment20 this paper addresses some issues deserving further inves- tigation. It is especially needed because Lebrun et al.26 made an experiment in an antiferromagnetic iron oxide similar to that of Yuan et al.20and observed similar de- pendence of spin accumulation on the distance from the injector. However, Lebrun et al.26explain it not by spin transport from the distant injector but by the Seebeck eect at the detector, which is warmed by the heat ow from the injector. We shall compare these two interpre- tations in Sec. VIII. We analyzed the role of dissipation in the super uid spin transport. A widely used approach to address dis- sipation in magnetically ordered solids is the Landau{ Lifshitz{Gilbert (LLG) theory with the Gilbert damp- ing parameter. But we came to the conclusion that the Gilbert damping does not provide a proper descrip- tion of dissipation processes in easy-plane ferromagnets. The Gilbert damping is described by a single parame- ter, which scales alldissipation processes independently from whether they do violate the spin conservation law, or do not. Meanwhile, the processes violating the spin conservation law, the Bloch spin relaxation in particular, originate from spin-orbit interaction and must be rela- tivistically small as explained above. This requires the presence of a small factor in the intensity of the Bloch spin relaxation, which is absent in the Gilbert damping approach. So we determined the dissipation parameters from the Boltzmann equation for magnons scattered by defects. Dissipation is possible only in the presence of thermal magnons, and we developed the two- uid theory for easy-plane ferromagnets similar to that in super uid hydrodynamics for the clamped regime, when the gas of quasiparticles cannot freely drift without dissipation in the laboratory frame. As mentioned above, to check the Landau criterion for3 super uidity, one must calculate the spectrum of collec- tive modes and check whether some modes have nega- tive energies. The Landau critical gradient is determined by easy-plane crystal anisotropy and was known qualita- tively both for ferro- and antiferromagnets long ago.2For easy-plane ferromagnets the Landau critical gradient was recently determined quantitatively from the spin-wave spectrum in the analysis of ferromagnetic spin-1 BEC of cold atoms.15But Cr 2O3, which was investigated in the experiment,20has no crystal easy-plane anisotropy, and an \easy plane" necessary for spin super uidity is produced by an external magnetic eld. The magnetic eld should exceed the spin- op eld, above which mag- netizations of sublattices in antiferromagnet are kept in a plane normal to the magnetic eld. We analyze the magnon spectrum in the spin current states in this situ- ation. The analysis has shown that the Landau critical gradient is determined by the gapped mode, but not by the Goldstone gapless mode as in the cases of easy-plane ferromagnets. Within the two- uid theory the role of spatial temper- ature variation was investigated. This variation produces the bulk Seebeck eect. But the eect is weak because it is proportional not to the temperature gradient, but to a higher (third) spatial derivative of the temperature. The transient processes near the interface through which spin is injected were also discussed. Conversion from spin current of incoherent thermal magnons to co- herent (super uid) spin transport is among these pro- cesses. The width of the transient layer (healing length), where formation of the super uid spin current occurs, can be determined by dierent scales at dierent condi- tion. But at low temperatures it is apparently not less than the magnon mean-free-path. In reality the decay of super uid currents starts at val- ues less than the Landau critical value via phase slips produced by magnetic vortices. The dierence in the spectrum of collective modes in ferro- and antiferromag- nets leads to the dierence in the structure of magnetic vortices. In the past magnetic vortices were investi- gated mostly in ferromagnets (see Ref. 15 and references therein). The present work analyzes a vortex in an anti- ferromagnet. The vortex core has a structure of skyrmion with sublattice magnetizations deviated from the direc- tion normal to the magnetic eld. At the same time inside the core the total magnetization has a component normal to the magnetic eld. In the geometry of the Cr2O3experiment this transverse magnetization creates surface magnetic charges at the point of the exit of the vortex line from the sample. Dipole stray magnetic elds produced by these charges hopefully can be used for de- tection of magnetic vortices experimentally. Section II reminds the phenomenological model of Ref. 2 describing the spin diusion and super uid spin transport. Section III reproduces the derivation of the spectrum of the collective spin mode and the Landau criterion in a spin current state of an easy-plane ferro- magnet known before15. This is necessary for compari-son with the spectrum of the collective spin modes and the Landau criterion in a spin current state of an easy- plane antiferromagnet derived in Sec. IV. Thus, Sec. III, as well as Sec. II, do not contain new results, but were added to the paper to make it self-sucient and more readable. In Sec. V we address two- uid eects and dis- sipation parameters (spin diusion and second viscosity coecients) deriving them from the Boltzmann equation for magnons. The section also estimates the bulk See- beck eect and shows that it is weak. Section VI ana- lyzes the transient layer near the interface through which spin is injected and where the bulk super uid spin cur- rent is formed. Various scales determining the width of this layer (healing length) are discussed. In Sec. VII the skyrmion structure of the magnetic vortex in an anti- ferromagnets is investigated. The concluding Sec. VIII summarizes the results of the work and presents some numerical estimations for the antiferromagnetic Cr 2O3 investigated in the experiment. The Appendix analyzes dissipation in the LLG theory with the Gilbert damping. It is argued that this theory predicts dissipation coe- cients incompatible with the spin conservation law. II. SUPERFLUID SPIN TRANSPORT VS SPIN DIFFUSION Here we remind the simple phenomenological model of spin transport suggested in Ref. 2 (see also more recent Refs. 5, 7, and 8). The equations of magnetodynamics are dMz dt=r
JM0 z T1; (1) d' dt= M0 z +r2': (2) Hereis the magnetic susceptibility along the axis z, 'is the angle of rotation (spin phase) in the spin space around the axis z, andM0 z=MzHis a nonequilib- rium part of the magnetization density along the mag- netic eldHparallel to the axis z. The time T1is the Bloch time of the longitudinal spin relaxation. The term /r2'in Eq. (2) is an analog of the second viscosity in super uid hydrodynamic.27,28The magnetization density Mzand the magnetization current Jdier from the spin density and the spin current by sign and by the gyromag- netic factor . Nevertheless, we shall call the current J the spin current to stress its connection with spin trans- port. The total spin current J=Js+Jdconsists of the super uid spin current Js=Ar'; (3) and the spin diusion current Jd=DrMz: (4)4 JzxJLzxSpin injection Spin injectionSpin injectionMedium withoutspin superfluidityMedium withspin superfluiditymxzmz 0 a) b)Spin detection dyxzPtPtCr2O3c)H c)PtPtCr2O3Hzdxy FIG. 1. Long distance spin transport. (a) Spin injection to a spin-nonsuper uid medium. (b) Spin injection to a spin- super uid medium. (c) Geometry of the experiment by Yuan et al.20. Spin is injected from the left Pt wire and ows along the Cr 2O3lm to the right Pt wire, which serves as a detector. The arrowed dashed line shows a spin-current streamline. In contrast to (a) and (b), the spin current is directed along the same axis zas a magnetization parallel to the external magnetic eld H. The pair of the hydrodynamical variables ( Mz;') is a pair of conjugate Hamiltonian variables analogous to the pair \particle density{super uid phase" in super uid hydrodynamics.1 There are two kinds of spin transport illustrated in Fig. 1. In the absence of spin super uidity ( A= 0) there is no super uid current. Equation (2) is not relevant, and Eq. (1) describes pure spin diusion [Fig. 1(a)]. Its solu- tion, with the boundary condition that the spin current J0is injected at the interface x= 0, is J=Jd=J0ex=L d; M0 z=J0r T1 Dex=L d;(5) where Ld=p DT1 (6) is the spin-diusion length. Thus the eect of spin injec- tion exponentially decays at the scale of the spin-diusion length.However, if spin super uidity is possible ( A6= 0), the spin precession equation (2) becomes relevant. As a re- sult of it, in a stationary state the magnetization M0 z cannot vary in space (Fig. 1b) since according to Eq. (2) the gradient rM0 zis accompanied by the linear in time growth of the gradient r'. The requirement of constant in space magnetization Mzis similar to the requirement of constant in space chemical potential in super uids, or the electrochemical potential in superconductors. As a consequence of this requirement, spin diusion current is impossible in the bulk since it is simply \short-circuited" by the super uid spin current. Only in AC processes the oscillating spin injection can produce an oscillating bulk spin diusion current coexisting with an oscillating super uid spin current. In the super uid spin transport the spin current can reach the other boundary opposite to the boundary where spin is injected. We locate it at the plane x=d. As a boundary condition at x=d, one can use a phenomeno- logical relation connecting the spin current with the mag- netization: Js(d) =M0 zvd, wherevdis a phenomenologi- cal constant. This boundary condition was derived from the microscopic theory by Takei and Tserkovnyak7. To- gether with the boundary condition Js(0) =J0atx= 0 this yields the solution of Eqs. (1) and (2): M0 z=T1 d+vdT1J0; Js(x) =J0 1x d+vdT1 :(7) Thus, the spin accumulated at large distance dfrom the spin injector slowly decreases as the inverse distance 1 =d [Fig. 1(b)], in contrast to the exponential decay /ed=Ld in the spin diusion transport [Fig. 1(a)]. In Figs. 1(a) and 1(b) the spin ows along the axis x, while the magnetization and the magnetic eld are directed along the axis z. In the geometry of the experi- ment of Yuan et al.20the spin ows along the magnetiza- tion axiszparallel to the magnetic eld. This geometry is shown in Fig. 1c. The dierence between two geometries is not essential if spin-orbit coupling is ignored. In this section we chose the geometry with dierent directions of the spin current and the magnetization in order to stress the possibility of the independent choice of axes in the spin and the congurational spaces. But in Sec. VII ad- dressing a vortex in an antiferromagnet we shall switch to the geometry of the experiment because in this case the dierence between geometries is important. Without dissipation-connected terms, the phenomeno- logical theory of this section directly follows from the LLG theory. For ferromagnets the LLG equation is dM dt= [HeffM]; (8) where Heff=H M=@H @M+rj@H @rjM(9) is the eective eld determined by the functional deriva- tive of the Hamiltonian H. For a ferromagnet with uni-5 axial anisotropy the Hamiltonian is H=GM2 z 2+AriM
riMMzH: (10) HereHis an external constant magnetic eld parallel to the axisz, and the exchange constant Adetermines sti- ness with respect to deformations of the magnetization eld. In the case of easy-plane anisotropy the anisotropy parameter Gis positive and coincides with the inverse susceptibility: G= 1=. Since the absolute value Mof the magnetization is a constant, one can describe the 3D magnetization vec- torMonly by two Hamiltonian conjugate variables: the magnetization zcomponent Mzand the angle 'of rota- tion around the zaxis. Then the LLG theory yields two equations _Mz=r
Js; (11) _'= ; (12) with the Hamiltonian in new variables H=M2 z 2+AM2 ?r'2 2+AM2(rMz)2 2M2 ?MzH: (13) HereM?=p M2M2z, and the spin \chemical poten- tial" and the super uid spin current are =H Mz=@H @Mzrj@H @rjMz;Js= @H @r':(14) After substitution of explicit expressions for functional derivatives of the Hamiltonian (13) the equations become _Mz =r
(AM2 ?r'); (15) _' =Mz1 A(r')2AM2(rMz)2 M4 ? +AM2 M2 ?r2Mz+H: (16) The equations (1) and (2) without dissipation terms fol- low from Eqs. (15) and (16) after linearization with re- spect to small gradients r'and nonequilibrium magne- tizationM0 z=MzHand ignoring the dependence of the spin chemical potential onrMz. ThenA= AM2 ?, andM?is determined by its valuep M22H2 in the equilibrium. III. COLLECTIVE MODES AND THE LANDAU CRITERION IN EASY-PLANE FERROMAGNETS To check the Landau criterion one should know the spectrum of collective modes. In an easy-plane ferromag- net the collective modes (spin waves) are determined byEqs. (15) and (16) linearized with respect to weak pertur- bations of stationary states. Further the angle variable  will be introduced instead of the variable Mz=Msin. Let us consider a current state with constant gradient K=r'and constant magnetization Mz=Msin=H 1AK2: (17) To derive the spectrum of collective modes, we consider weak perturbations  and  of this state: !+ , '!'+ . Equations (15) and (16) after linearization are: _2 MzAK
r = AM cosr2; _2 MzAK
r = Mcos  1AK2
 + AM cosr2:(18) For plane waves/eik
ri!tthese equations describe the gapless Goldstone mode with the spectrum:13,15 (!+w
k)2= ~c2 sk2: (19) Here ~cs=r ~cs; (20) ~= 1A K2M2k2 M2 ?; (21) and cs= M?s A (22) is the spin-wave velocity in the ground state without any spin current. In this state the spectrum becomes !=csks 1 +AM2k2 M2 ?: (23) The velocity w= 2 MzAK; (24) can be called Doppler velocity because its eect on the mode frequency is similar to the eect of the mass ve- locity on the mode frequency in a Galilean invariant uid (Doppler eect). But our system is not Galilean invariant,13and the gradient Kis present also in the right-hand side of the dispersion relation (19). In the long-wavelength hydrodynamical limit magnons have the sound-like spectrum linear in k. Quadratic cor- rections/k2become important at kM?=MpA[see Eq. (23)]. These corrections emerge from the terms in6 the Hamiltonian, which depend on rMz. So the hydro- dynamical approach is valid at scales exceeding 0=M M?p A; (25) which can be called the coherence length, in analogy with the coherence length in the Gross{Pitaevskii theory for BEC. Also in analogy with BEC, the coherence length diverges at M?!0, i.e., at the second-order phase tran- sition from the easy-plane to the easy-axis anisotropy. The same scale determines the Landau critical gradient and the vortex core radius. Telling about hydrodynamics we bear in mind hydrodynamics of a perfect uid without dissipation. Later in this paper we shall discuss hydro- dynamics with dissipation. In this case the condition k1=0is not sucient, and an additional restriction on using hydrodynamics is determined by the mean-free path of magnons. According to the Landau criterion, the current state becomes unstable at small kwhenkis parallel to wand the frequency !becomes negative. This happens at the gradientKequal to the Landau critical gradient Kc=M?p4M23M?1pA1 0: (26) Spin super uidity becomes impossible at the phase tran- sition to the easy-axis anisotropy ( M?= 0). In the oppo- site limit of small MzMthe pseudo-Doppler eect is not important, and the Landau critical gradient Kcis de- termined from the condition that the spin-wave velocity ~csvanishes at small k: Kc=1pA= M cs: (27) Expanding the Hamiltonian (13) with respect to weak perturbations  and  up to the second order one obtains the energy of the spin wave mode per unit volume, Esw=M?!(k) p~Akjkj2; (28) wherejkj2is the squared perturbation of the angle  with the wave vector kaveraged over the wave period. In the quantum theory the energy density Eswcorre- sponds to the magnon density n(k) V=Esw ~!(k)=M?jkj2 ~ p~Ak; (29) wheren(k) is the number of magnons in the plane-wave mode with the wave vector kandVis the volume of the sample. Summing over the whole kspace, the averaged squared perturbation is h2i=X kjkj2=~ p A M?Zp ~n(k)kd3k (2)3:(30)Further we proceed within the hydrodynamical ap- proach neglecting quadratic corrections to the spectrum. There are quadratic in spin-wave amplitudes corrections to the spin super uid current and to the spin chemical potential: Jsjsw= M?A(M?h2iK+ 2Mzhri);(31) jsw=A(Mzh(r)2i+ 2M?K
hri):(32) Using Eq. (30) and the relation r =pAk k; (33) which follows from the equations of motion (18), one ob- tains: Jsjsw=2~c3 s M2 ?Z n(k) K+2 Mz csk k kd3k (2)3; (34) jsw=~c2 s M2 ?Z n(k) Mz cs+2K
k k kd3k (2)3: (35) IV. COLLECTIVE MODES AND THE LANDAU CRITERION IN ANTIFERROMAGNETS For ferromagnetic state of localized spins the deriva- tion of the LLG theory from the microscopic Heisenberg model was straightforward.29The quantum theory of the antiferromagnetic state even for the simplest case of a two-sublattice antiferromagnet, which was widely used for Cr 2O3, is more dicult. This is because the state with constant magnetizations of two sublattices is not a well dened quantum-mechanical eigenstate.29Never- theless, long time ago it was widely accepted to ignore this complication and to describe the long-wavelength dy- namics by the LLG theory for two sublattices coupled via exchange interaction:30 dMi dt= [HiMi]; (36) where the subscript i= 1;2 points out to which sublattice the magnetization Mibelongs, and Hi=H Mi=@H @Mi+rj@H @rjMi(37) is the eective eld for the ith sublattice determined by the functional derivative of the Hamiltonian H. For an isotropic antiferromagnet the Hamiltonian is H=M1
M2 +A(riM1
riM1+riM2
riM2) 2 +A12rjM1
rjM2H
(M1+M2):(38)7 In the uniform ground state without the magnetic eld H the two magnetizations are antiparallel, M2=M1, and the total magnetization M1+M2vanishes. At H6= 0 the sublattice magnetizations are canted, and in the uniform ground state the total magnetization is parallel to H: m=M1+M2=H: (39) The rst term in the Hamiltonian (38), which determines the susceptibility , originates from the exchange inter- action between spins of two sublattices. This is the sus- ceptibility normal to the staggered magnetization (anti- ferromagnetic vector) L=M1M2. Since in the LLG theory absolute values of magnetizations M1andM2are xed the susceptibility parallel to Lvanishes. In the uniform state only the uniform exchange en- ergy/1=and the Zeeman energy (the rst and the last terms) are present in the Hamiltonian, which can be rewritten as H=L2m2 4H
m=M2 +m2 2mHm;(40) whereHm= (H
m)=mis the projection of the mag- netic eld on the direction of the total magnetization m. Minimizing the Hamiltonian with respect to the absolute value of m(at it xed direction, i.e., at xed Hm) one obtains H=M2 H2 m 2=M2 H2 2+H2 L 2;(41) whereHL= (H
L)=Lis the projection of the magnetic eld on the staggered magnetization L. The rst two terms are constant, while the last term plays the role of the easy-plane anisotropy energy conning Lin the plane normal to H. ForHparallel to the axis z: Ea=H2L2 z 2L2=H2sin 2: (42)Hereis the angle between the staggered magnetization Land thexyplane (see Fig. 2). ✓<latexit sha1_base64="Gd6Qju6ECAyxaLSdKYPfRQI1y7I=">AAAB/HicbVA9SwNBEN2LXzF+RS1tDoNgFe5E0DJoYxnBfEByhL3NXLJmd+/YnRPCEX+DrdZ2Yut/sfSfuEmuMIkPBh7vzTAzL0wEN+h5305hbX1jc6u4XdrZ3ds/KB8eNU2cagYNFotYt0NqQHAFDeQooJ1ooDIU0ApHt1O/9QTa8Fg94DiBQNKB4hFnFK3U7OIQkPbKFa/qzeCuEj8nFZKj3iv/dPsxSyUoZIIa0/G9BIOMauRMwKTUTQ0klI3oADqWKirBBNns2ol7ZpW+G8XalkJ3pv6dyKg0ZixD2ykpDs2yNxX/8zopRtdBxlWSIig2XxSlwsXYnb7u9rkGhmJsCWWa21tdNqSaMrQBLWwJ5cRm4i8nsEqaF1Xfq/r3l5XaTZ5OkZyQU3JOfHJFauSO1EmDMPJIXsgreXOenXfnw/mctxacfOaYLMD5+gVm+JW+</latexit><latexit 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Angle variables and0for the case when the both magnetizations are in the plane xz('0='= 0). We introduce the pairs of angle variables i,'ideter- mining directions of the sublattice magnetizations: Mix=Mcosicos'i; Miy=Mcosisin'i; Miz=Msini:(43) The equations of motion in the angle variables are cosi_i =1 M@H @'ir@H @r'i ; cosi_'i =1 M@H @ir@H @ri : (44) In the further analysis it is convenient to use other angle variables: 0=+12 2; =12 2; '0='1+'2 2; '='1'2 2: (45) In these variables the Hamiltonian becomes H=M2 (cos 20cos2'cos 2sin2')2HM cossin0 +AM2[(1 + cos 20cos 2)r'2 0+r'2 2sin 20sin 2r'0
r'+r2 0+r2] +A12M2f(cos 2sin2'+ cos 20cos2')(r2 0r2)cos 20+ cos 2 2cos 2'(r'2 0r'2) sin 2'[sin 2(r0
r'0+r
r') + sin 20(r
r'0+r0
r')]g: (46) The polar angles for the staggered magnetization Land the canting angle 0are shown in Fig. 2 for the case when the both magnetizations are in the plane xz('0='= 0). In the uniform ground state = 0,'= 0,mz= 2Msin0=H, while the angle '0is an arbitrary con-stant. Since we consider elds Hweak compared to the exchange eld, 0is always small. In the state with con- stant current K=r'0the magnetization along the magnetic eld is mz=H 1AK2=2; (47)8 whereA=AA12. In a weakly perturbed current state small but nonzero and'appear. Also the angles 0and'0dier from their values in the stationary current state: 0!0+ , '0!'0+ . Linearization of the nonlinear equations of motion with respect to weak perturbations , , , and'yields decoupled linear equations for two pairs of variables ( ;) and (;'): _ AmzK
r =AM?r2; _ AmzK
r = 1AK2 22M?  +(A+A12cos 20) cos0Mr2;(48) _ A+mzK
r =2M?  1 +A12K2
'+A+M?r2'; _' A+mzcos0K
r' =m2 z 2M?(1 +A12K2)AK2M? AA12cos 20 cos0Mr2: (49) For plane waves/eik
ri!tEq. (48) describes the gapless Goldstone mode with the spectrum: (!+ mzAK
k)2 =c2 s 1AK2 2+(A+A12cos 20)k2 2 cos20 k2:(50) Here cs= M?s 2A (51) is the spin-wave velocity in the ground state without spin current. Apart from quadratic corrections k2to the fre- quency, the gapless mode in an antiferromagnet does not dier from that in a ferromagnet, if one replaces in all expressions for the ferromagnet AbyA=2 and the pa- rameterMby 2M. Equation (49) describes the gapped mode with the spectrum (!+ mzA+K
k)2= 1 +A12K2+A+k2 2 (1 +A12K2) 2m2 z 2c2 sK2 +2 2M2(AA12cos 20)k2  :(52)Without spin current and neglecting the term /A+k2 the spectrum is !=s 2m2z 2+c2sk2: (53) This spectrum determines a new correlation length =M Hs 2A =cs H; (54) which is connected with the easy-plane anisotropy energy (42) and determines the wave vector k= 1=at which the gap and the kdependent frequency become equal. Applying the Landau criterion to the gapless mode one obtains the critical gradientp 2=Asimilar to the value (27) obtained for a ferromagnet. But in contrast to a fer- romagnet where the susceptibility is connected with weak anisotropy energy, in an antiferromagnet the sus- ceptibilityis determined by a much larger exchange energy and is rather small. As a result, in an antiferro- magnet the gapless Goldstone mode becomes unstable at the very high value of K. But at much lower values of Kthe gapped mode becomes unstable. According to the spectrum (52), the gap in the spectrum vanishes at the critical gradient Kc=1 = H cs= mz cs: (55) V. TWO-FLUID EFFECTS AND DISSIPATION FROM THE BOLTZMANN EQUATION FOR MAGNONS Knowledge of the spectrum of collective modes allows to derive the dynamical equations at nite temperatures taking into account the presence of thermal magnons. Further we follow the procedure of the derivation of the two- uid hydrodynamics in super uids.27We address the hydrodynamical limit when all parameters ( Mz,K,T) of the system slowly vary in space and time. We shall focus on ferromagnets. The equilibrium Planck distribution of magnons in a ferromagnet with a small spin current /Kis nK=1 e~!(k)=T1n0(!0)2c2 sMz M2 ?@n0(!0) @!0K
k; (56) where!0=cskand n0(!0) =1 e~!0=T1(57) is the Planck distribution in the state without spin cur- rent. In the theory of super uidity the Plank distribution of phonons in general depends not only on density and super uid velocity (analogs of our MzandK) but also on9 the normal velocity, which characterizes a possible drift of the gas of quasiparticles with respect to the laboratory frame of coordinates. This drift is possible because of the Galilean invariance of super uids. In our case the Galilean invariance is broken by possible interaction of magnons with defects, and in the equilibrium the drift of the quasiparticle gas is impossible. The case of broken Galilean invariance, when the normal velocity vanishes, was also investigated for super uids in porous media or in very thin channels, when the Galilean invariance is broken by interaction with channel walls. It was called the clamped regime.31,32 Substituting the Planck distribution (56) into Eqs. (34) and (35) one obtains the contribution of equilibrium magnons to the spin current and the spin chemical po- tential: Jsjeq= @ @K=22T4 30 M2 ?~3csK 1 +16M2 z 3M2 ? ;(58) jeq=@ @Mz=2MzT4 30~3c3sM2 ?; (59) where =TZ ln(1e~!(k)=T)d3k (2)3: (60) is the thermodynamical potential for the magnon Bose- gas. The contribution (58) decreases the super uid spin current at xed phase gradient K, similarly to the de- crease of the mass super uid current after replacing the total mass density by the lesser super uid density. Yuan et al.20used in their experiment very thin lm at low temperature, when de Broglie wavelength of magnons exceeds lm thickness, and it is useful to give also the two- uid corrections for a two-dimensional case. Repeat- ing our calculations after replacing integralsR d3k=(2)3 by integrals WR d2k=(2)2, one obtains: Jsjeq=(3)2T3 W M2 ?~2K 1 +6M2 z M2 ? ; (61) jeq=(3)MzT3 W~2c2sM2 ?; (62) where the value of the Riemann zeta function (3) is 1.202 andWis the lm thickness. The next step in derivation of the two- uid theory at - nite temperatures is the analysis of dissipation. A widely used approach of studying dissipation in magnetically or- der systems is the LLG theory with the Gilbert damp- ing term added. However, this approach is incompatible with the spin conservation law. This law, although being approximate, plays a key role in the problem of spin su- per uidity. Therefore, we derived dissipation parameters from the Boltzmann equation for magnons postponing discussion of the LLG theory with the Gilbert damping to the Appendix.Dissipation is connected with nonequilibrium correc- tions to the magnon distribution. At low temperatures the number of magnons is small, and magnon-magnon interaction is weak. Then the main source of dissipa- tion is scattering of magnons by defects. The Boltzmann equation with the collision term in the relaxation-time approximation is _n+@! @k
rnr!
@n @k=nnK : (63) If parameters, which determine the magnon distribution functionn, vary slowly in space and time one can substi- tute the equilibrium Planck distribution nKinto the left- hand side of the Boltzmann equation (63). This yields: @n0 @!_!+@n0 @T _T+@! @k
rT =nn0 ; (64) We consider small gradients Kwhen the dierence be- tweennKandn0is not important. But weak depen- dence of!onKis important at calculation of _ !. One can see that at the constant temperature Tin any sta- tionary state the left-hand side vanishes, and there is no nonequilibrium correction to the magnon distribution. Correspondingly, there is no dissipation. This is one more illustration that stationary super uid currents do not de- cay. In nonstationary cases time derivatives are determined by the equations of motions. The equations of motion for MzandKare not sucient, and the equation of heat bal- ance is needed for nding _T. In general the heat balance equation is rather complicated since it must take into ac- count interaction of magnons with other subsystems, e. g., phonons. Instead of it we consider a simpler case, when magnons are not important in the heat balance, i.e., the temperature does not depend on magnon pro- cesses. In other words we consider the isothermal regime when _T= 0. But we allow slow temperature variation in space. The temporal variation of the frequency !emerges from slow temporal variation of MzandK, and at small K _!=@! @Mz_Mz+@! @K_K=Mz M2 ? csk_Mz+2c2 s k
_K : (65) The partial derivatives @!=@Mzand@!=@Kwere deter- mined from the spectrum (19), while the time derivatives ofMzandKwere found from the linearized equations (15) and (16) assuming that r'=Kis small and ig- noring gradients of Mzin the right-hand side of Eq. (16), which are beyond the hydrodynamical limit. Then _!=Mz M2 ?c2 s cskr
K+ 2(k
r)Mz : (66) Eventually the nonequilibrium correction to the magnon10 distribution function is n0=nn0=Mz M2 ?cscs kr
K +2(k
r)MzM2 ? MzT(k
r)T @n0 @k(67) Substituting n0into Eqs. (34) and (35) one obtains dis- sipation terms in the spin current and the spin chemical potential: Jd=D rMz1 2TM2 ? MzrT ; (68) d= r
K; (69) where D=2~c3 s 32M2 z M4 ?Z @n0 @kk4dk; =~c3 s 22M2 z M4 ?Z @n0 @kk4dk: (70) In addition to the spin diusion current, the dissipative spin current Jdcontains also the current proportional to the temperature gradient. This is the bulk Seebeck eect. Estimation of the integral in these expressions requires knowledge of possible dependence of the relax- ation time on the energy. Under the assumption that is independent from the energy, D=82 2T4M2 z 45~3c3sM2 ?; =22 2T4M2 z 15~3c3sM2 ?; (71) or for the two-dimensional case, D=16(3) 2T3M2 z 3W~2c2sM2 ?; =4(3) 2T3M2 z W~2c2sM2 ?:(72) Although in antiferromagnets the Landau critical gra- dient is connected with the gapped mode, at small phase gradients the gapless Goldstone mode has lesser energy, and at low temperatures most of magnons belong to this mode. Since the Goldstone modes in ferromagnets and antiferromagnets are similar, our estimation of dissipa- tion coecients for ferromagnets is valid also for antifer- romagnets after replacing AbyA=2 andMby 2M. The microscopic analysis of this section agrees with the following phenomenological equations similar to the hydrodynamical equations for super uids in the clamped regime: _Mz=r
Js@R @+r@R @r; (73) _'= +@R @(r
Js); (74)where the spin chemical potential and the super uid spin current, =F Mz;Js= @F @r'; (75) are determined by derivatives of the free energy F=H+ TS: (76) The spin conservation law forbids the term @R=@ in the continuity equation (73), because it is not a divergence of some current. Thus, the dissipation function is compati- ble with the spin conservation law if it depends only on the gradient of the spin chemical potential , but not on itself. This does not take place in the LLG theory with the Gilbert damping discussed in the Appendix. The analysis of this section assumed the spin conservation law and corresponded to the dissipation function R=D 2r2D 2TM2 ? Mzr
rT+ 2 AM2 ?(r
Js)2: (77) In general the dissipation function contains also the term /rT2responsible for the thermal conductivity. But it is important only for the heat balance equation, which was not considered here. If the temperature does not vary in space, then the only temperature eect is a correction to the spin chemical potential. This does not aect the basic feature of super- uid spin transport: there is no gradient of the chemical potential in a stationary current state, and all dissipation processes are not eective except for the relativistically small spin Bloch relaxation. If there is spatial variation of temperature, then the spin chemical potential also varies in space. One can nd its gradient by exclusion of r
Js from Eqs. (73) and (74): r=rMz =D 2 2AMzTr(r2T): (78) Note that the spin chemical potential gradient is propor- tional not to the rst but to the third spatial derivative of the temperature. The constant temperature gradi- ent does not produce spatial variation of the chemical potential. This is an analog of the absence of thermo- electric eects proportional to the temperature gradients in superconductors.33Naturally the eect produced by higher derivatives of the temperature is weaker than pro- duced by the rst derivative. The nonuniform correction to the spin chemical po- tential strongly depends on temperature. Assuming the T4dependence of the dissipation parameters Dandin Eq. (71) the coecient before the temperature-gradient term in Eq. (78) is proportional to T8. Now the spin dif- fusion currentDrdoes not disappear in the equa- tion (73) of continuity for the spin, but it is proportional toT12. Earlier Zhang and Zhang34used the Boltzmann equa- tion for derivation of the spin diusion coecient and11 the Bloch relaxation time in an isotropic ferromagnet in a constant magnetic eld. We derived the spin diusion and the second viscosity coecients in an easy-plane fer- romagnet with dierent spin-wave spectrum. Two- uid eects in easy-plane ferromagnets were investigated by Flebus et al.35. They solved the Boltzmann equation using the equilibrium magnon distribution function with nonzero chemical potential of magnon (do not confuse it with the spin chemical potential introduced in the present paper). In contrast, we assumed complete thermaliza- tion of the magnon distribution when the magnon chem- ical potential vanishes. The thermalization assumption is questionable in the transient layer near the interface through which spin is injected, and in this layer the ap- proach Flebus et al.35may become justied. The tran- sient layer is discussed in the next section. VI. TRANSIENT (HEALING) LAYER NEAR THE INTERFACE INJECTING SPIN Injection of spin from a medium without spin super u- idity to a medium with spin super uidity may produce not only a super uid spin current but also a spin cur- rent of incoherent magnons. But at some distance from the interface between two media, which will be called the conversion healing length, the spin current of incoherent magnons (spin diusion current) must inevitably trans- form to super uid spin current, as we shall show now. We return back to Eqs. (1) and (2) but now we neglect the relativistically small Bloch spin relaxation (the term /1=T1). In Sec. II we considered the stationary solution of the these equations with constant magnetization and absent spin diusion current. But it is not the only sta- tionary solution. Another solution is an evanescent mode M0 z/r'/ex=, where =s D A(79) is the conversion healing length. We look for superposi- tion of two solutions, which satises the condition that the injected current J0transforms to the spin diusion current, while the super uid current vanishes at x= 0: J0=DrxM0 z(0);rx'(0) = 0: (80) This superposition is M0 z(x) =M0 z+J0 Dex=;rx'(x) =J0 A(1ex=); (81) whereM0 zin the right-hand side is a constant magneti- zation far from the interface x= 0. Thus, at the length the spin diusion current Jddrops from J0to zero, while the super uid spin current grows from zero to J0 and remains at larger distances constant. As pointed out in the end of Sec. II, the phenomenolog- ical equations (1) and (2) were derived assuming that thespin chemical potential =M0 z=Hdoes not depend on gradients rMz. However, the dissipation coecients Danddecrease very sharply with temperature, and the conversion healing length eventually becomes much smaller than the scale 0[see Eq. (25)], when the depen- dence of the free energy and the spin chemical potential on the gradients rMzbecomes important. But in fact addingrMz-dependent terms into the expression for , =Mz HAM2r2Mz M2 ?; (82) does not aect the expression (79) for the healing length. The generalization of the analysis reduces to replacing of M0 zin Eqs. (1), (2), and (81) by . Transformation of the injected incoherent magnon spin current to the super uid spin current is not the only tran- sient process near the interface between media with and without spin super uidity. Even in the absence of spin current the interface may aect the equilibrium mag- netic structure. For example, the interface can induce anisotropy dierent from easy-plane anisotropy in the bulk. Then the crossover from surface to bulk anisotropy occurs at the healing length of the order of the correla- tion length 0determined by Eq. (25) in ferromagnets, or the correlation length determined by Eq. (54) in anti- ferromagnets. The similar healing length was suggested for ferromagnets by Takei and Tserkovnyak7and for an- tiferromagnets by Takei et al.8although using dierent arguments. The expression (79) for was derived within hydrody- namics with dissipation. At distances shorter than the mean-free path incoherent magnons are in the ballistic regime and cannot converge to the super uid current, since conversion is impossible without dissipation. Alto- gether this means that the real healing length at which the bulk super uid spin current state is formed cannot be less than the longest from three scales: ,0, and the magnon mean-free path cs. Apparently at low tempera- tures and weak magnetization Mzthe latter is the longest one from three scales. However, close to the phase transi- tion to the easy-axis anisotropy ( Mz=M) the coherence length0diverges and becomes the longest scale. Solving the Boltzmann equation we assumed complete thermalization of the magnon distribution. At low tem- peratures when magnon-magnon interaction is weak the length at which thermalization occurs essentially exceeds the mean-free path on defects. It could be that the heal- ing length would grow up to the thermalization length. This requires a further analysis. VII. MAGNETIC VORTEX IN AN EASY-PLANE ANTIFERROMAGNET Let us consider structure of an axisymmetric vortex in an antiferromagnet with one quantum of circulation of the angle'0of rotation around the vortex axis. Now we consider the geometry of the experiment20when the12 PtPtzxyCr2O3H
























a) b) FIG. 3. Precession of magnetization maround the direction of the magnetic eld Halong the path around the vortex axis. (a) The geometry of the experiment20with the magnetic eld (the axis z) in the plane of the Cr 2O3lm. The vortex axis is normal to the lm (the axis y). (b) Precession of the magnetization mis shown in the plane xz(the plane of the lm). The path around the vortex axis (dashed lines) is inside the vortex core where the total magnetization is not parallel toH( 6= 0). magnetic eld H(the axisz) is in the lm plane. The vortex axis is the axis ynormal to the lm plane (Fig, 3a). The azimuthal component of the angle '0gradient is r'0=1 r: (83) At the same time '= 0 and0is small. Then the Hamil- tonian (46) transforms to H=2M2 2 02HM cos0+AM2cos2 r2+r2 : (84) Minimization with respect to small 0yields 0=Hcos 2M; (85) and nally the Hamiltonian is H=H2cos2 2+AM2cos2 r2+r2 :(86)The Euler{Lagrange equation for this Hamiltonian de- scribes the vortex structure in polar coordinates: d2 dr2+1 rd drsin 2 21 21 r2 = 0; (87) where the correlation length is given by Eq. (54) and determines the size of the vortex core. The vortex core has a structure of a skyrmion, in which the total weak magnetization deviates from the direction of the magnetic eld H(6= 0). The component of magnetization transverse to the magnetic eld is m?= Hsin 2 2: (88) The transverse magnetization creates stray magnetic elds at the exit of the vortex line from the sample. Fig- ure 3 shows variation of the magnetization inside the core along the path around the vortex axis parallel to the axis y. Along the path the magnetization mrevolves around the direction of the magnetic eld forming a cone. The precession in space creates an oscillating ycomponent of magnetization my=m?(r) sin, whereis the az- imuthal angle at the circular path around the vortex line. This produces surface magnetic charges 4 myat the exit of the vortex to the boundary separating the sample from the vacuum. These charges generate the curl-free stray eldh=r . At distances from the vortex exit point much larger that the core radius the stray eld is a dipole eld with the scalar potential (R) =H 2(R
n) R3Z1 0sin 2(r)r2dr = 1:2H3(R
n) R3=1:2c3 s 3H2(R
n) R3: (89) HereR(x;y;z ) is the position vector with the origin in the vortex exit point and nis a unit vector in the plane xzalong which the surface charge is maximal ( ==2). In our model the direction of nis arbitrary, but it will be xed by spin-orbit interaction or crystal magnetic anisotropy violating invariance with respect to rotations around the axis z. These interactions were ignored in our model. In principle, the stray eld can be used for detec- tion of vortices nucleated at spin currents approaching the critical value. VIII. DISCUSSION AND SUMMARY The paper analyzes the long-distance super uid spin transport. The super uid spin transport does not require a gradient of the spin chemical potential (as the electron supercurrent in superconductors does not require a gra- dient of the electrochemical potential). As result of it, mechanisms of dissipation are suppressed except for weak Bloch spin relaxation. Other dissipation mechanisms af- fect the spin transport only at the transient (healing)13 layer close to the interface through which spin is injected, or in nonstationary processes. The paper calculates the Landau critical spin phase gradient in a two-sublattice antiferromagnet when the easy-plane topology of the magnetic order parameter is provided not by crystal magnetic anisotropy but by an external magnetic eld. This was the case realized in the experiment by Yuan et al.20. For this goal it was necessary to derive the spectrum of collective modes (spin waves) in spin current states. The Landau instability destroying spin super uidity sets on not in the Goldstone gapless mode as in easy-plane ferromagnets but in the gapped mode, despite that at small spin currents the latter has energy larger than the Goldstone mode. The paper analyzes dissipation processes determining dissipation parameters (spin diusion and second viscos- ity coecients) by solving the Boltzmann equation for magnons scattered by defects. The two- uid theory sim- ilar to the super uid two- uid hydrodynamics was sug- gested. It is argued that the LLG theory with the Gilbert damping parameter is not able to properly describe dissi- pation in easy-plane magnetic insulators. Describing the whole dissipation by a single Gilbert parameter one can- not dierentiate between strong processes connected with high exchange energy (e.g., spin diusion) and weak pro- cesses connected with spin-orbit interaction (Bloch spin relaxation), which violate the spin conservation law. The formation of the super uid spin current in the transient (healing) layer near the interface through which spin is injected was investigated. The width of this layer (healing length) is determined by processes of dissipation, and at low temperatures can reach the scale of relevant mean-free paths of magnons including those at which the magnon distribution is thermalized. The structure of the magnetic vortex in the geometry of the experiment on Cr 2O3is investigated. In the vortex core there is a magnetization along the vortex line, which is normal to the magnetic eld. This magnetization pro- duces magnetic charges at the exit of the vortex line from the sample. The magnetic charges create a stray dipole magnetic eld, which probably can be used for detection of vortices. Within the developed two- uid theory the paper ad- dresses the role of the temperature variation in space on the super uid spin transport. This is important because in the experiment of Yuan et al.20the spin is created in the Pt injector by heating (the Seebeck eect). Thus the spin current to the detector is inevitably accompa- nied by heat ow. The temperature variation produces the bulk Seebeck eect, which is estimated to be rather weak at low temperatures. However, it was argued26that probably Yuan et al.20detected a signal not from spin coming from the injector but from spin produced by the Seebeck eect at the interface between the heated anti- ferromagnet and the Pt detector. Such eect has already been observed for antiferromagnet Cr 2O3.36If true, then Yuan et al.20observed not long-distance spin transport but long-distance heat transport. It is not supported bythe fact that Yuan et al. observed a threshold for super- uid spin transport at low intensity of injection, when ac- cording to the theory5violation of the approximate spin conservation law becomes essential. Investigation of su- per uid spin transport at low-intensity injection is more dicult both for theory and experiment. But the exis- tence of the threshold is supported by extrapolation of the detected signals from high-intensity to low-intensity injection. According to the experiment, the signal at the detector is not simply proportional to the squared elec- tric current j2responsible for the Joule heating in the injector, but to j2+a. The oset ais evidence of the threshold, in the analogy with the oset of IVcurves in the mixed state of type II superconductors determining the critical current for vortex deepening. With all that said, the heat-transport interpretation cannot be ruled out and deserves further investigation. According to this interpretation, one can see the signal observed by Yuan et al.20at the detector even if the Pt injector is replaced by a heater, which produces the same heat but no spin. An experimental check of this prediction would conrm or reject the heat-transport interpretation. Let us make some numerical estimations for Cr 2O3us- ing the formulas of the present paper. It follows from neutron scattering data37that the spin-wave velocity is cs= 8105cm/sec. According to Foner38, the magne- tization of sublattices is M= 590 G and the magnetic susceptibility is = 1:2104. Then the total magne- tizationmz=Hin the magnetic eld H= 9 T used in the experiment is about 10 G, and the canting an- gle0=mz=2M0:01 is small as was assumed in our analysis. The correlation length (54), which determines vortex core radius, is about 0:5106cm. The stray magnetic eld produced by magnetic charges at the exit of the vortex line from the sample is 10( 3=R3) G, where Ris the distance from the vortex exit point. The task to detect such elds does not look easy, but it is hopefully possible with modern experimental techniques. ACKNOWLEDGMENTS I thank Eugene Golovenchits, Wei Han, Mathias Kl aui, Romain Lebrun, Allures Qaiumzadeh, Victoria Sanina, So Takei, and Yaroslav Tserkovnyak for fruitful discus- sions and comments. Appendix: Dissipation in the LLG theory For ferromagnets the LLG equation taking into ac- count dissipation is dM dt= [HeffM] + M MdM dt ; (A.1) whereis the dimensionless Gilbert damping parameter. For smallthis equation is identical to the equation with14 the Landau{Lifshitz damping term: 1 dM dt= MH M + M M MH M : (A.2) Transforming the vector LLG equation to the equations for two Hamiltonian conjugate variables, the zcompo- nentMzof magnetization and the angle 'of rotation around the zaxis, one obtains Eqs. (73) and (74) without the term r(@R=@r) and with the dissipation function R= M2 ? 2M2+M 2M2 ?(r
Js)2; (A.3) which depends on the spin chemical potential itself, but not on its gradient. Meanwhile, according to the two- uid theory of Sec. V, the r-dependent term in the dissipation function was responsible for the spin-diusion term in the continuity equation for Mz. Indeed, at deriva- tion of the continuity equation (1) from the LLG theory under the assumption that M0 z==Mz=Hthe spin diusion term /Ddoes not appear. The term does appear only if in the dissipation function (A.3) is determined by the more general expression (82) taking into account the dependence on rMz. Then one obtains Eqs. (1) and (2) with the equal spin diusion and spin second viscosity coecients D== MA; (A.4) and the inverse Bloch relaxation time 1 T1= M2 ? M: (A.5) The outcome looks bizarre. The spin diusion emerges from the-dependent term in the dissipation function, which is incompatible with the spin conservation law, asif the spin diusion is forbidden by the spin conserva- tion law. Evidently this conclusion is physically incor- rect. Moreover, in the analogy of magnetodynamics and super uid hydrodynamics the magnetization Mzcorre- sponds to the uid density. In hydrodynamics the uid density gradients are usually not taken into account in the Hamiltonian and in the chemical potential since they become important only at small scales beyond the hydro- dynamical approach. This does not rule out the diusion process. Similarly, one should expect that it is possible to ignore the magnetization gradients in the spin chemi- cal potential either. It is strange that the spin diusion becomes impossible in the hydrodynamical limit. According to the Noether theorem the total magnetiza- tion along the axis zis conserved if the Hamiltonian is in- variant with respect to rotations around the axis zin the spin space. The Landau{Lifshitz theory of magnetism19 is based on the idea that the spin-orbit interaction, which breaks rotational symmetry in the spin space and there- fore violates the spin conservation law, is relativistically small compared to the exchange interaction because the former is inversely proportional to the speed of light. So, although the spin conservation law is not exact, it is a good approximation (see Sec. I). Then the spin Bloch relaxation term/1=T1, which violates the spin conser- vation law, must be proportional to a small parameter inversely proportional to the speed of light and cannot be determined by the same Gilbert parameter as other dissipation terms, which do not violate the spin conser- vation law The insuciency of the LLG theory for description of dissipation was discussed before, but mostly at higher temperatures. It was suggested to replace of the LLG equation by the Landau{Lifshitz{Bloch equation, in which the Bloch longitudinal spin relaxation is present explicitly (see, e.g., Ref. 39 and references to earlier works therein). Our analysis shows that the problem exists also at low temperatures. 1B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898 (1969). 2E. B. Sonin, Zh. Eksp. Teor. Fiz. 74, 2097 (1978), [Sov. Phys.{JETP, 47, 1091 (1978)]. 3E. B. Sonin, Usp. Fiz. Nauk 137, 267 (1982), [Sov. Phys.{ Usp., 25, 409 (1982)]. 4Y. Bunkov, in Progress of Low Temperature Physics , Vol. 14, edited by W. P. Halperin (Elsevier, 1995) p. 68. 5E. B. Sonin, Adv. Phys. 59, 181 (2010). 6Y. M. Bunkov and G. E. Volovik, \Novel super uids," (Oxford University Press, 2013) Chap. IV, pp. 253{311. 7S. Takei and Y. Tserkovnyak, Phys. Rev. Lett. 112, 227201 (2014). 8S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak, Phys. Rev. B 90, 094408 (2014). 9H. Chen and A. H. MacDonald, in Universal themes of Bose{Einstein condensation , edited by N. Proukakis, D. Snoke, and P. Littlewood (Cambridge University Press,2017) Chap. 27, pp. 525{548, arXiv:1604.02429. 10C. Sun, T. Nattermann, and V. L. Pokrovsky, Phys. Rev. Lett. 116, 257205 (2016). 11E. B. Sonin, Phys. Rev. B 95, 144432 (2017). 12J. Armaitis and R. A. Duine, Phys. Rev. A 95, 053607 (2017). 13E. Iacocca, T. J. Silva, and M. A. Hoefer, Phys. Rev. Lett. 118, 017203 (2017). 14A. Qaiumzadeh, H. Skarsv ag, C. Holmqvist, and A. Brataas, Phys. Rev. Lett. 118, 137201 (2017). 15E. B. Sonin, Phys. Rev. B 97, 224517 (2018). 16P. W. Anderson, in Quantum Fluids , edited by D. F. Brewer (North{Holand, Amsterdam, 1966) pp. 146{171. 17Y. M. Bunkov, E. M. Alakshin, R. R. Gazizulin, A. V. Klochkov, V. V. Kuzmin, V. S. L'vov, and M. S. Tagirov, Phys. Rev. Lett. 108, 177002 (2012). 18D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L'vov, and15 B. Hillebrands, Nat. Phys. 12, 1057 (2016). 19L. D. Landau and E. M. Lifshitz, Electrodynamics of con- tinuum media (Pergamon Press, 1984). 20W. Yuan, Q. Zhu, T. Su, Y. Yao, W. Xing, Y. Chen, Y. Ma, X. Lin, J. Shi, R. Shindou, X. C. Xie, and W. Han, Sci. Adv. 4, eaat1098 (2018). 21A. S. Borovik-Romanov, Y. Bunkov, V. V. Dmitriev, and Y. Mukharskii, Pis'ma Zh. Eksp. Teor. Fiz. 45, 98 (1987), [JETP Lett. 45, 124{128 (1987)]. 22S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443, 430 (2006). 23E. B. Sonin, J. Phys. Conf. Ser. 969, 012116 (2018). 24P. Stepanov, S. Che, D. Shcherbakov, J. Yang, R. Chen, K. Thilahar, G. Voigt, M. W. Bockrath, D. Smirnov, K. Watanabe, T. Taniguchi, R. K. Lake, Y. Barlas, A. H. MacDonald, and C. N. Lau, Nat. Phys. 14, 907 (2018). 25J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett. 96, 076604 (2006). 26R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Bal- drati, J. Cramer, A. Brataas, R. A. 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1804.09242v1.Generalisation_of_Gilbert_damping_and_magnetic_inertia_parameter_as_a_series_of_higher_order_relativistic_terms.pdf

Generalisation of Gilbert damping and magnetic inertia parameter as a series of higher-order relativistic terms Ritwik Mondalz, Marco Berritta and Peter M. Oppeneer Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-751 20 Uppsala, Sweden E-mail: ritwik.mondal@physics.uu.se Abstract. The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion remains as the cornerstone of contemporary magnetisation dynamics studies, wherein the Gilbert damping parameter has been attributed to rst-order relativistic eects. To include magnetic inertial eects the LLG equation has previously been extended with a supplemental inertia term and the arising inertial dynamics has been related to second-order relativistic eects. Here we start from the relativistic Dirac equation and, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin Hamiltonian that contains relativistic correction terms to any higher order. Using the Heisenberg equation of spin motion we derive general relativistic expressions for the tensorial Gilbert damping and magnetic inertia parameters, and show that these ten- sors can be expressed as series of higher-order relativistic correction terms. We further show that, in the case of a harmonic external driving eld, these series can be summed and we provide closed analytical expressions for the Gilbert and inertial parameters that are functions of the frequency of the driving eld. 1. Introduction Spin dynamics in magnetic systems has often been described by the phenomenological Landau-Lifshitz (LL) equation of motion of the following form [1] @M @t= MHeM[MHe]; (1) where is the gyromagnetic ratio, Heis the eective magnetic eld, and is an isotropic damping parameter. The rst term describes the precession of the local, classical magnetisation vector M(r;t) around the eective eld He. The second term describes the magnetisation relaxation such that the magnetisation vector relaxes to the direction of the eective eld until nally it is aligned with the eective eld. To include zPresent address: Department of Physics, University of Konstanz, D -78457 Konstanz, GermanyarXiv:1804.09242v1 [cond-mat.other] 3 Apr 20182 large damping, the relaxation term in the LL equation was reformulated by Gilbert [2, 3] to give the Landau-Lifshitz-Gilbert (LLG) equation, @M @t= MHe+M@M @t; (2) whereis the Gilbert damping constant. Note that both damping parameters and are here scalars, which corresponds to the assumption of an isotropic medium. Both the LL and LLG equations preserve the length of the magnetisation during the dynamics and are mathematically equivalent (see, e.g. [4]). Recently, there have also been attempts MHeff Precession NutationDamping Figure 1. Sketch of extended LLG magnetisation dynamics. The green arrow denotes the classical magnetisation vector which precesses around an eective eld. The red solid and dotted lines depict the precession and damping. The yellow path signies the nutation, or inertial damping, of the magnetisation vector. to investigate the magnetic inertial dynamics which is essentially an extension to the LLG equation with an additional term [5{7]. Phenomenologically this additional term of magnetic inertial dynamics, MI@2M=@t2, can be seen as a torque due to second-order time derivative of the magnetisation [8{11]. The essence of the terms in the extended LLG equation is described pictorially in Fig. 1. Note that in the LLG dynamics the magnetisation is described as a classical vector eld and not as a quantum spin vector. In their original work, Landau and Lifshitz attributed the damping constant to relativistic origins [1]; later on, it has been more specically attributed to spin-orbit coupling [12{15]. In the last few decades, several explanations have been proposed towards the origin of damping mechanisms, e.g., the breathing Fermi surface model [16, 17], torque-torque correlation model [18], scattering theory formulation [19], eective eld theories [20] etc. On the other hand, the origin of magnetic inertia is less discussed in the literature, although it's application to ultrafast spin dynamics and switching could potentially be rich [9]. To account for the magnetic inertia, the breathing Fermi surface model has been extended [11, 21] and the inertia parameter has been associated with the magnetic susceptibility [22]. However, the microscopic origins of both Gilbert3 damping and magnetic inertia are still under debate and pose a fundamental question that requires to be further investigated. In two recent works [23, 24], we have shown that both quantities are of relativistic origin. In particular, we derived the Gilbert damping dynamics from the relativistic spin-orbit coupling and showed that the damping parameter is not a scalar quantity but rather a tensor that involves two main contributions: electronic and magnetic ones [23]. The electronic contribution is calculated as an electronic states' expectation value of the product of dierent components of position and momentum operators; however, the magnetic contribution is given by the imaginary part of the susceptibility tensor. In an another work, we have derived the magnetic inertial dynamics from a higher-order (1 =c4) spin-orbit coupling and showed that the corresponding parameter is also a tensor which depends on the real part of the susceptibility [24]. Both these investigations used a semirelativistic expansion of the Dirac Hamiltonian employing the Foldy-Wouthuysen transformation to obtain an extended Pauli Hamiltonian including the relativistic corrections [25, 26]. The thus-obtained semirelativistic Hamiltonian was then used to calculate the magnetisation dynamics, especially for the derivation of the LLG equation and magnetic inertial dynamics. In this article we use an extended approach towards a derivation of the generalisation of those two (Gilbert damping and magnetic inertia) parameters from the relativistic Dirac Hamiltonian, developing a series to fully include the occurring higher-order relativistic terms. To this end we start from the Dirac Hamiltonian in the presence of an external electromagnetic eld and derive a semirelativistic expansion of it. By doing so, we consider the direct eld-spin coupling terms and show that these terms can be written as a series of higher-order relativistic contributions. Using the latter Hamiltonian, we derive the corresponding spin dynamics. Our results show that the Gilbert damping parameter and inertia parameter can be expressed as a convergent series of higher-order relativistic terms and we derive closed expressions for both quantities. At the lowest order, we nd exactly the same tensorial quantities that have been found in earlier works. 2. Relativistic Hamiltonian Formulation To describe a relativistic particle, we start with a Dirac particle [27] inside a material, and, in the presence of an external eld, for which one can write the Dirac equation asi~@ (r;t) @t=H (r;t) for a Dirac bi-spinor . Adopting furthermore the relativistic density functional theory (DFT) framework we write the corresponding Hamiltonian as [23{25] H=c
(peA) + ( 1)mc2+V 1 =O+ ( 1)mc2+E; (3) whereVis the eective unpolarised Kohn-Sham potential created by the ion-ion, ion- electron and electron-electron interactions. Generally, to describe magnetic systems, an4 additional spin-polarised energy (exchange energy) term is required. However, we have treated eects of the exchange eld previously, and since it doesn't contribute to the damping terms we do not consider it explicitly here (for details of the calculations involving the exchange potential, see Ref. [23, 25]). The eect of the external electromagnetic eld has been accounted through the vector potential, A(r;t),cdenes the speed of light, mis particle's mass and 1is the 44 unit matrix. andare the Dirac matrices which have the form = 0 0! ;  = 10 01! ; where= (x;y;z) are the Pauli spin matrix vectors and 1is 22 unit matrix. Note that the Dirac matrices form the diagonal and o-diagonal matrix elements of the Hamiltonian in Eq. (3). For example, the o-diagonal elements can be denoted as O=c
(peA), and the diagonal matrix elements can be written as E=V 1. In the nonrelativistic limit, the Dirac Hamiltonian equals the Pauli Hamiltonian, see e.g. [28]. In this respect, one has to consider that the Dirac bi-spinor can be written as (r;t) = (r;t) (r;t)! ; where the upper and lowercomponents have to be considered as \large" and \small" components, respectively. This nonrelativistic limit is only valid for the case when the particle's momentum is much smaller than the rest mass energy, otherwise it gives an unsatisfactory result [26]. Therefore, the issue of separating the wave functions of particles from those of antiparticles is not clear for any given momentum. This is mainly because the o-diagonal Hamiltonian elements link the particle and antiparticle. The Foldy-Wouthuysen (FW) transformation [29] has been a very successful attempt to nd a representation where the o-diagonal elements have been reduced in every step of the transformation. Thereafter, neglecting the higher-order o-diagonal elements, one nds the correct Hamiltonian that describes the particles eciently. The FW transformation is an unitary transformation obtained by suitably choosing the FW operator [29], UFW=i 2mc2O: (4) The minus sign in front of the operator is because of the property that andO anticommute with each other. With the FW operator, the FW transformation of the wave function adopts the form 0(r;t) =eiUFW (r;t) such that the probability density remains the same, j j2=j 0j2. In this way, the time-dependent FW transformed Hamiltonian can be expressed as [26, 28, 30] HFW=eiUFW Hi~@ @t eiUFW+i~@ @t: (5)5 According to the Baker-Campbell-Hausdor formula, the above transformed Hamilto- nian can be written as a series of commutators, and the nally transformed Hamiltonian reads HFW=H+i UFW;Hi~@ @t +i2 2! UFW; UFW;Hi~@ @t +i3 3! UFW; UFW; UFW;Hi~@ @t +:::: : (6) In general, for a time-independent FW transformation, one has to work with@UFW @t= 0. However, this is only valid if the odd operator does not contain any time dependency. In our case, a time-dependent transformation is needed as the vector potential is notably time-varying. In this regard, we notice that the even operators and the term i~@=@t transform in a similar way. Therefore, we dene a term Fsuch thatF=Ei~@=@t. The main theme of the FW transformation is to make the odd terms smaller in every step of the transformation. After a fourth transformation and neglecting the higher order terms, the Hamiltonian with only the even terms can be shown to have the form as [26, 30{33] H000 FW= ( 1)mc2+O2 2mc2O4 8m3c6+O6 16m5c10 +E1 8m2c4[O;[O;F]]  8m3c6[O;F]2+3 64m4c8 O2;[O;[O;F]] +5 128m4c8 O2; O2;F :(7) Here, for any two operators AandBthe commutator is dened as [ A;B] and the anticommutator as fA;Bg. As already pointed out, the original FW transformation can only produce correct and expected higher-order terms up to rst order i.e., 1 =c4 [26, 30, 33]. In fact, in their original work Foldy and Wouthuysen derived only the terms up to 1 =c4, i.e., only the terms in the rst line of Eq. (7), however, notably with the exception of the fourth term [29]. The higher-order terms in the original FW transformation are of doubtful value [32, 34, 35]. Therefore, the Hamiltonian in Eq. (7) is not trustable and corrections are needed to achieve the expected higher-order terms. The main problem with the original FW transformation is that the unitary operators in two preceding transformations do not commute with each other. For example, for the exponential operators eiUFWandeiU0 FW, the commutator [ UFW;U0 FW]6= 0. Moreover, as the unitary operators are odd, this commutator produces even terms that have not been considered in the original FW transformation [26, 30, 33]. Taking into account those terms, the correction of the FW transformation generates the Hamiltonian as [33] Hcorr: FW= ( 1)mc2+O2 2mc2O4 8m3c6+O6 16m5c10 +E1 8m2c4[O;[O;F]] + 16m3c6fO;[[O;F];F]g+3 64m4c8 O2;[O;[O;F]] +1 128m4c8 O2; O2;F 1 32m4c8[O;[[[O;F];F];F]]: (8)6 Note the dierence between two Hamiltonians in Eq. (7) and Eq. (8) that are observed in the second and consequent lines in both the equations, however, the terms in the rst line are the same. Eq. (8) provides the correct higher-order terms of the FW transformation. In this regard, we mention that an another approach towards the correct FW transformation has been employed by Eriksen; this is a single step approach that produces the expected FW transformed higher-order terms [34]. Once the transformed Hamiltonian has been obtained as a function of odd and even terms, the nal form is achieved by substituting the correct form of odd terms Oand even termsEin the expression of Eq. (8) and calculating term by term. Since we perform here the time-dependent FW transformation, we note that the commutator [O;F] can be evaluated as [ O;F] =i~@O=@t. Therefore, following the denition of the odd operator, the time-varying elds are taken into account through this term. We evaluate each of the terms in Eq. (8) separately and obtain that the particles can be described by the following extended Pauli Hamiltonian [24, 26, 36] Hcorr: FW=(peA)2 2m+Ve~ 2m
B(peA)4 8m3c2+(peA)6 16m5c4 e~ 2m2B2 2mc2+e~ 4m2c2( (peA)2 2m;
B) e~2 8m2c2r
Etote~ 8m2c2
[Etot(peA)(peA)Etot] e~2 16m3c4 (peA);@Etot @t ie~2 16m3c4
@Etot @t(peA) + (peA)@Etot @t +3e~ 64m4c4n (peA)2e~
B;~r
Etot+
[Etot(peA)(peA)Etot]o +e~4 32m4c6r
@2Etot @t2+e~3 32m4c6
@2Etot @t2(peA)(peA)@2Etot @t2 : (9) The elds in the last Hamiltonian (9) are dened as B=rA, the external magnetic eld,Etot=Eint+Eextare the electric elds where Eint=1 erVis the internal eld that exists even without any perturbation and Eext=@A @tis the external eld (only the temporal part is retained here because of the Coulomb gauge). It is clear that as the internal eld is time-independent, it does not contribute to the fourth and sixth lines of Eq. (9). However, the external eld does contribute to the above terms wherever it appears in the Hamiltonian. The above-derived Hamiltonian can be split in two parts: (1) a spin-independent Hamiltonian and (2) a spin-dependent Hamiltonian that involves the Pauli spin matrices. The spin-dependent Hamiltonian, furthermore, has two types of coupling terms. The direct eld-spin coupling terms are those which directly couples the elds with the magnetic moments e.g., the third term in the rst line, the second term in the third line of Eq. (9) etc. On the other hand, there are relativistic terms that do not directly couple the spins to the electromagnetic eld - indirect eld-spin coupling terms. These7 terms include e.g., the second term of the second line, the fth line of Eq. (9) etc. The direct eld-spin interaction terms are most important because these govern the directly manipulation of the spins in a system with an electromagnetic eld. For the external electric eld, these terms can be written together as a function of electric and magnetic eld. These terms are taken into account and discussed in the next section. The indirect coupling terms are often not taken into consideration and not included in the discussion (see Ref. [36, 37] for details). In this context, we reiterate that our current approach of deriving relativistic terms does not include the exchange and correlation eect. A similar FW transformed Hamiltonian has previously been derived, however, with a general Kohn-Sham exchange eld [23, 25, 26]. As mentioned before, in this article we do not intend to include the exchange-correlation eect, while mostly focussing on the magnetic relaxation and magnetic inertial dynamics. 2.1. The spin Hamiltonian The aim of this work is to formulate the spin dynamics on the basis of the Hamiltonian in Eq. (9). The direct eld-spin interaction terms can be written together as electric or magnetic contributions. These two contributions can be expressed as a series up to an order of 1=m5[36] HS magnetic =e mS
" B+1 2X n=1;2;3;41 2i!cn@nB @tn# +O1 m6 ; (10) HS electric =e mS
" 1 2mc2X n=0;2i 2!cn@nE @tn(peA)# +O1 m6 ; (11) where the Compton wavelength and pulsation have been expressed by the usual denitions c=h=mc and!c= 2c=cwith Plank's constant h. We also have used the spin angular momentum operator as S= (~=2). Note that we have dropped the notion of total electric eld because the the involved elds ( B,E,A) are external only, the internal elds are considered as time-independent. The involved terms in the above two spin-dependent Hamiltonians can readily be explained. The rst term in the magnetic contribution in Eq. (10) explains the Zeeman coupling of spins to the external magnetic eld. The rest of the terms in both the Hamiltonians in Eqs. (11) and (10) represent the spin-orbit coupling and its higher-order corrections. We note that these two spin Hamiltonians are individually not Hermitian, however, it can be shown that together they form a Hermitian Hamiltonian [38]. As these Hamiltonians describe a semirelativistic Dirac particle, it is possible to derive from them the spin dynamics of a single Dirac particle [24]. The eect of the indirect eld-spin terms is not yet well understood, but they could become important too in magnetism [36, 37], however, those terms are not of our interest here. The electric Hamiltonian can be written in terms of magnetic contributions with the choice of a gauge A=Br=2. The justication of the gauge lies in the fact8 that the magnetic eld inside the system being studied is uniform [26]. The transverse electric eld in the Hamiltonian (10) can be written as E=1 2 r@B @t : (12) Replacing this expression in the electric spin Hamiltonian in Eq. (11), one can obtain a generalised expression of the total spin-dependent Hamiltonian as HS(t) =e mS
h B+1 21X n=1;2;:::1 2i!cn@nB @tn +1 4mc21X n=0;2;:::i 2!cn r@n+1B @tn+1 (peA)i : (13) It is important to stress that the above spin-Hamiltonian is a generalisation of the two Hamiltonians in Eqs. (10) and (11). We have already evaluated the Hamiltonian forms forn= 1;2;3;4 and assume that the higher-order terms will have the same form [36]. This Hamiltonian consists of the direct eld-spin interaction terms that are linear and/or quadratic in the elds. In the following we consider only the linear interaction terms, that is we neglect the eAterm in Eq. (13). Here, we mention that the quadratic terms could provide an explanation towards the previously unknown origin of spin-photon coupling or optical spin-orbit torque and angular magneto-electric coupling [38{40]. The linear direct eld-spin Hamiltonian can then be recast as HS(t) =e mS
h B+1 21X n=1;2;:::1 2i!cn@nB @tn +1 4mc21X n=0;2;:::i 2!cn@n+1B @tn+1(r
p)r@n+1B @tn+1
pi : (14) This is nal form of the Hamiltonian and we are interested to describe to evaluate its contribution to the spin dynamics. 3. Spin dynamics Once we have the explicit form of the spin Hamiltonian in Eq. (14), we can proceed to derive the corresponding classical magnetisation dynamics. Following similar procedures of previous work [23, 24], and introducing a magnetisation element M(r;t), the magnetisation dynamics can be calculated by the following equation of motion @M @t=X jgB 1 i~D Sj;HS(t)E ; (15) whereBis the Bohr magneton, gis the Land e g-factor that takes a value 2 for electron spins and is a suitably chosen volume element. Having the spin Hamiltonian in Eq.9 (14), we evaluate the corresponding commutators. As the spin Hamiltonian involves the magnetic elds, one can classify the magnetisation dynamics into two situations: (a) the system is driven by a harmonic eld, (b) the system is driven by a non-harmonic eld. However, in the below we continue the derivation of magnetisation dynamics with the harmonic driven elds. The magnetisation dynamics driven by the non-harmonic elds has been discussed in the context of Gilbert damping and inertial dynamics where it was shown that an additional torque contribution (the eld-derivative torque) is expected to play a crucial role [23, 24, 26]. The magnetisation dynamics due to the very rst term of the Hamiltonian in Eq. (14) is derived as [24] @M(1) @t= MB; (16) with the gyromagnetic ratio =gjej=2m. Here the commutators between two spin operators have been evaluated using [ Sj;Sk] =i~Sljkl, wherejklis the Levi-Civita tensor. This dynamics actually produces the precession of magnetisation vector around an eective eld. To get the usual form of Landau-Lifshitz precessional dynamics, one has to use a linear relationship of magnetisation and magnetic eld as B=0(M+H). With the latter relation, the precessional dynamics becomes 0MH, where 0= 0 denes the eective gyromagnetic ratio. We point out that the there are relativistic contributions to the precession dynamics as well, e.g., from the spin-orbit coupling due to the time-independent eld Eint[23]. Moreover, the contributions to the magnetisation precession due to exchange eld appear here, but are not explicitly considered in this article as they are not in the focus of the current investigations (see Ref. [23] for details). The rest of the terms in the spin Hamiltonian in Eq. (14) is of much importance because they involve the time-variation of the magnetic induction. As it has been shown in an earlier work [23] that for the external elds and specically the terms with n= 1 in the second terms and n= 0 in the third terms of Eq. (14), these terms together are Hermitian. These terms contribute to the magnetisation dynamics as the Gilbert relaxation within the LLG equation of motion, @M(2) @t=M A
@M @t ; (17) where the Gilbert damping parameter Ahas been derived to be a tensor that has mainly two contributions: electronic and magnetic. The damping parameter Ahas the form [23, 24] Aij=e0 8m2c2X `;k hripk+pkriihr`p`+p`r`iik  1+1
kj; (18) where 1is the 33 unit matrix and is the magnetic susceptibility tensor that can be introduced only if the system is driven by a eld which is single harmonic [26]. Note that the electronic contributions to the Gilbert damping parameter are given by the10 expectation value hripkiand the magnetic contributions by the susceptibility. We also mention that the tensorial Gilbert damping tensor has been shown to contain a scalar, isotropic Heisenberg-like contribution, an anisotropic Ising-like tensorial contribution and a chiral Dzyaloshinskii-Moriya-like contribution [23]. In an another work, we took into account the terms with n= 2 in the second term of Eq. (14) and it has been shown that those containing the second-order time variation of the magnetic induction result in the magnetic inertial dynamics. Note that these terms provide a contribution to the higher-order relativistic eects. The corresponding magnetisation dynamics can be written as [24] @M(3) @t=M C
@M @t+D
@2M @t2 ; (19) with a higher-order Gilbert damping tensor Cijand inertia parameter Dijthat have the following expressions Cij= 0~2 8m2c4@ @t( 1+1)ijandDij= 0~2 8m2c4( 1+1)ij. We note that Eq. (19) contains two fundamentally dierent dynamics { the rst term on the right-hand side has the exact form of Gilbert damping dynamics whereas the second term has the form of magnetic inertial dynamics [24]. The main aim of this article is to formulate a general magnetisation dynamics equation and an extension of the traditional LLG equation to include higher-order relativistic eects. The calculated magnetisation dynamics due to the second and third terms of Eq. (14) can be expressed as @M @t=e mMh1 21X n=0;1;:::1 2i!cn+1@n+1B @tn+1 +1 4mc21X n=0;2;:::i 2!cn@n+1B @tn+1hr
piD r@n+1B @tn+1
pEi : (20) Note the dierence in the summation of rst terms from the Hamiltonian in Eq. (14). To obtain explicit expressions for the Gilbert damping dynamics, we employ a general linear relationship between magnetisation and magnetic induction, B=0(H+M). The time-derivative of the magnetic induction can then be replaced by magnetisation and magnetic susceptibility. For the n-th order time-derivative of the magnetic induction we nd @nB @tn=0@nH @tn+@nM @tn : (21) Note that this equation is valid for the case when the magnetisation is time-dependent. Substituting this expression into the Eq. (20), one can derive the general LLG equation and its extensions. Moreover, as we work out the derivation in the case of harmonic driving elds, the dierential susceptibility can be introduced as =@M=@H. The rst term ( n-th derivative of the magnetic eld) can consequently be written by the11 following Leibniz formula as @nH @tn=n1X k=0(n1)! k!(nk1)!@nk1(1) @tnk1
@k @tk@M @t ; (22) where the magnetic susceptibility 1is a time-dependent tensorial quantity and harmonic. Using this relation, the rst term and second terms in Eq. (20) assume the form @M @t rst=e0 2mM1X n=0;1;:::1 2i!cn+1nX k=0n! k!(nk)!@nk( 1+1) @tnk
@k @tk@M @t ; (23) @M @t second=e0 4m2c2M 1X n=0;2;:::1 2i!cnnX k=0n! k!(nk)!h@nk( 1+1) @tnk
@k @tk@M @t hr
pi D r@nk( 1+1) @tnk
@k @tk@M @t
pEi :(24) These two equations already provide a generalisation of the higher-order magnetisation dynamics including the Gilbert damping (i.e., the terms with k= 0) and the inertial dynamics (the terms with k= 1) and so on. 4. Discussion 4.1. Gilbert damping parameter It is obvious that, as Gilbert damping dynamics involves the rst-order time derivative of the magnetisation and a torque due to it, kmust take the value k= 0 in the equations (23) and (24). Therefore, the Gilbert damping dynamics can be achieved from the following equations: @M @t rst=e0 2mM1X n=0;1;:::1 2i!cn+1@n( 1+1) @tn
@M @t; (25) @M @t second=e0 4m2c2M1X n=0;2;:::1 2i!cnh@n( 1+1) @tn
@M @t hr
pi D r@n( 1+1) @tn
@M @t
pEi : (26) Note that these equations can be written in the usual form of Gilbert damping as M G
@M @t
, where the Gilbert damping parameter Gis notably a tensor [2, 23]. The12 general expression for the tensor can be given by a series of higher-order relativistic terms as follows Gij=e0 2m1X n=0;1;:::1 2i!cn+1@n( 1+1)ij @tn +e0 4m2c21X n=0;2;:::1 2i!cnh@n( 1+1)ij @tn(hrlplihrlpii)i : (27) Here we have used the Einstein summation convention on the index l. Note that there are two series: the rst series runs over even and odd numbers ( n= 0;1;2;3;


), however, the second series runs only over the even numbers ( n= 0;2;4;


). Eq. (27) represents a general relativistic expression for the Gilbert damping tensor, given as a series of higher-order terms. This equation is one of the central results of this article. It is important to observe that this expression provides the correct Gilbert tensor at the lowest relativistic order, i.e., putting n= 0 the expression for the tensor is found to be exactly the same as Eq. (18). The analytic summation of the above series of higher-order relativistic contributions can be carried out when the susceptibility depends on the frequency of the harmonic driving eld. This is in general true for ferromagnets where a dierential susceptibility is introduced because there exists a spontaneous magnetisation in ferromagnets even without application of a harmonic external eld. However, if the system is driven by a nonharmonic eld, the introduction of the susceptibility is not valid anymore. In general the magnetic susceptibility is a function of wave vector and frequency in reciprocal space, i.e.,=(q;!). Therefore, for the single harmonic applied eld, we use 1/ei!tand then-th order derivative will follow @n=@tn(1)/(i!)n1. With these arguments, one can express the damping parameter of Eq. (27) as (see Appendix A for detailed calculations) Gij=e0 4m2c2~ i+hrlplihrlpii ( 1+1)ij +e0 4m2c2" (2!!c+!2)~ i+!2(hrlplihrlpii) 4!2 c!2# 1 ij: (28) Here, the rst term in the last expression is exactly the same as the one that has been derived in our earlier investigation [23]. As the expression of the expectation value hripjiis imaginary, the real Gilbert damping parameter will be given by the imaginary part of the susceptibility tensor. This holds consistently for the higher-order terms as well. The second term in Eq. (28) stems essentially from an innite series which contain higher-order relativistic contributions to the Gilbert damping parameter. As !cscales with c, these higher-order terms will scale with c4or more and thus their contributions will be smaller than the rst term. Note that the higher-order terms will diverge when != 2!c1021sec1, which means that the theory breaks down at the limit!!2!c. In this limit, the original FW transformation is not dened any more because the particles and antiparticles cannot be separated at this energy limit.13 4.2. Magnetic inertia parameter Magnetic inertial dynamics, in contrast, involves a torque due to the second-order time- derivative of the magnetisation. In this case, kmust adopt the value k= 1 in the afore-derived two equations (23) and (24). However, if k= 1, the constraint nk0 dictates that n1. Therefore, the magnetic inertial dynamics can be described with the following equations: @M @t rst=e0 2mM1X n=1;2;:::1 2i!cn+1n! (n1)!@n1( 1+1) @tn1
@2M @t2; (29) @M @t second=e0 4m2c2M1X n=2;4;:::1 2i!cnn! (n1)!h@n1( 1+1) @tn1
@2M @t2 hr
pi D r@nk( 1+1) @tnk
@2M @t2
pEi : (30) Similar to the Gilbert damping dynamics, these dynamical terms can be expressed asM I
@2M @t2 which is the magnetic inertial dynamics [8]. The corresponding parameter has the following expression Iij=e0 2m1X n=1;2;:::1 2i!cn+1n! (n1)!@n1( 1+1)ij @tn1 +e0 4m2c21X n=2;4;:::1 2i!cnn! (n1)!h@n1( 1+1)ij @tn1(hrlplihripli)i : (31) Note that as ncannot adopt the value n= 0, the starting values of nare dierent in the two terms. Importantly, if n= 1 we recover the expression for the lowest order magnetic inertia parameter Dij, as given in the equation (19) [24]. Using similar arguments as in the case of the generalised Gilbert damping parameter, when we consider a single harmonic eld as driving eld, the inertia parameter can be rewritten as follows (see Appendix A for detailed calculations) Iij=e0~2 8m3c4( 1+1)ije0~2 8m3c4!2+ 4!!c (2!c!)2 1 ij +e0 8m3c4~ i(hrlplihripli)16!!3 c (4!2 c!2)2 1 ij: (32) The rst term here is exactly the same as the one that was obtained in our earlier investigation [24]. However, there are now two extra terms which depend on the frequency of the driving eld and that vanish for !!0. Again, in the limit !!2!c, these two terms diverge and hence this expression is not valid anymore. The inertia parameter will consistently be given by the real part of the susceptibility.14 5. Summary We have developed a generalised LLG equation of motion starting from fundamental quantum relativistic theory. Our approach leads to higher-order relativistic correction terms in the equation of spin dynamics of Landau and Lifshitz. To achieve this, we have started from the foundational Dirac equation under the presence of an electromagnetic eld (e.g., external driving elds or THz excitations) and have employed the FW transformation to separate out the particles from the antiparticles in the Dirac equation. In this way, we derive an extended Pauli Hamiltonian which eciently describes the interactions between the quantum spin-half particles and the applied eld. The thus- derived direct eld-spin interaction Hamiltonian can be generalised for any higher-order relativistic corrections and has been expressed as a series. To derive the dynamical equation, we have used this generalised spin Hamiltonian to calculate the corresponding spin dynamics using the Heisenberg equation of motion. The obtained spin dynamical equation provides a generalisation of the phenomenological LLG equation of motion and moreover, puts the LLG equation on a rigorous foundational footing. The equation includes all the torque terms of higher-order time-derivatives of the magnetisation (apart from the Gilbert damping and magnetic inertial dynamics). Specically, however, we have focussed on deriving an analytic expression for the generalised Gilbert damping and for the magnetic inertial parameter. Our results show that both these parameters can be expressed as a series of higher-order relativistic contributions and that they are tensors. These series can be summed up for the case of a harmonic driving eld, leading to closed analytic expressions. We have further shown that the imaginary part of the susceptibility contributes to the Gilbert damping parameter while the real part contributes to the magnetic inertia parameter. Lastly, with respect to the applicability limits of the derived expressions we have pointed out that when the frequency of the driving eld becomes comparable to the Compton pulsation, our theory will not be valid anymore because of the spontaneous particle-antiparticle pair-production. 6. Acknowledgments We thank P-A. Hervieux for valuable discussions. This work has been supported by the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation (Contract No. 2015.0060), the European Union's Horizon2020 Research and Innovation Programme under grant agreement No. 737709 (FEMTOTERABYTE, http://www.physics.gu.se/femtoterabyte).15 Appendix A. Detailed calculations of the parameters for a harmonic eld In the following we provide the calculational details of the summation towards the results given in Eqs. (28) and (32). Appendix A.1. Gilbert damping parameter Eq. (27) can be expanded as follows Gij=e0 2m1 2i!c( 1+1)ij+e0 4m2c2(hrlplihrlpii) ( 1+1)ij +e0 2m1X n=1;2;:::1 2i!cn+1 (i!)n1 ij+e0 4m2c21X n=2;4;:::1 2i!cn (hrlplihrlpii) (i!)n1 ij =e0 2m1 2i!c( 1+1)ij+e0 4m2c2(hrlplihrlpii) ( 1+1)ij +e0 2m1 2i!c1X n=1;2;:::! 2!cn 1 ij+e0 4m2c21X n=2;4;:::! 2!cn (hrlplihrlpii)1 ij =e0 4m2c2~ i+hrlplihrlpii ( 1+1)ij +e0 4m2c2" ~ i1X n=1;2;:::! 2!cn + (hrlplihrlpii)1X n=2;4;:::! 2!cn# 1 ij =e0 4m2c2~ i+hrlplihrlpii ( 1+1)ij +e0 4m2c2~ i! 2!c!+ (hrlplihrlpii)!2 4!2 c!2 1 ij =e0 4m2c2~ i+hrlplihrlpii ( 1+1)ij +e0 4m2c2" (2!!c+!2)~ i+!2(hrlplihrlpii) 4!2 c!2# 1 ij: (A.1) We have used the fact that! !c<1 and the summation formula 1 +x+x2+x3+:::=1 1x;1<x< 1: (A.2)REFERENCES 16 Appendix A.2. Magnetic inertia parameter Eq. (31) can be expanded as follows Iij=e0 2m1 2i!c2 ( 1+1)ij+e0 2m1X n=2;3;:::1 2i!cn+1n! (n1)!@n1( 1+1)ij @tn1 +e0 4m2c21X n=2;4;:::1 2i!cnn! (n1)!h@n1( 1+1)ij @tn1(hrlplihripli)i =e0 2m1 2i!c2 ( 1+1)ij+1X n=2;3;:::1 2i!cn+1n! (n1)!(i!)n11 ij +e0 4m2c21X n=2;4;:::1 2i!cnn! (n1)!(hrlplihripli) (i!)n11 ij =e0~2 8m3c4( 1+1)ij+e0 2m1 2i!c21X n=2;3;:::! 2!cn1n! (n1)!1 ij +e0 4m2c21 2i!c1X n=2;4;:::! 2!cn1n! (n1)!(hrlplihripli)1 ij =e0~2 8m3c4( 1+1)ije0~2 8m3c41X n=2;3;:::! 2!cn1n! (n1)!1 ij +e0 8m3c4~ i1X n=2;4;:::! 2!cn1n! (n1)!(hrlplihripli)1 ij =e0~2 8m3c4( 1+1)ije0~2 8m3c41 ij" 2! 2!c + 3! 2!c2 + 4! 2!c3 +:::# +e0 8m3c4~ i(hrlplihripli)1 ij" 2! 2!c + 4! 2!c3 + 6! 2!c5 +:::# =e0~2 8m3c4( 1+1)ije0~2 8m3c4!2+ 4!!c (2!c!)2 1 ij +e0 8m3c4~ i(hrlplihripli)16!!3 c (4!2 c!2)2 1 ij: (A.3) Here we have used the formula 1 + 2x+ 3x2+ 4x3+ 5x4+:::=1 (1x)2;1<x< 1: (A.4) References [1] Landau L D and Lifshitz E M 1935 Phys. Z. Sowjetunion 8101{114 [2] Gilbert T L 2004 IEEE Trans. Magn. 403443{3449 [3] Gilbert T L 1956 Formulation, foundations and applications of the phenomenologi- cal theory of ferromagnetism Ph.D. thesis Illinois Institute of Technology, ChicagoREFERENCES 17 [4] Lakshmanan M 2011 Phil. Trans. Roy. Soc. London A 3691280{1300 [5] Wegrowe J E and Ciornei M C 2012 Am. J. Phys. 80607{611 URL http: //dx.doi.org/10.1119/1.4709188 [6] Olive E, Lansac Y, Meyer M, Hayoun M and Wegrowe J E 2015 J. Appl. Phys. 117 213904 URL http://dx.doi.org/10.1063/1.4921908 [7] Wegrowe J E and Olive E 2016 J. Phys.: Condens. Matter 28106001 URL http://stacks.iop.org/0953-8984/28/i=10/a=106001 [8] Ciornei M C, Rub  J M and Wegrowe J E 2011 Phys. Rev. B 83020410 URL http://link.aps.org/doi/10.1103/PhysRevB.83.020410 [9] Kimel A V, Ivanov B A, Pisarev R V, Usachev P A, Kirilyuk A and Rasing T 2009 Nat. Phys. 5727{731 URL http://dx.doi.org/10.1038/nphys1369 [10] Bhattacharjee S, Nordstr om L and Fransson J 2012 Phys. Rev. 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1412.4032v1.Spin_waves_in_micro_structured_yttrium_iron_garnet_nanometer_thick_films.pdf

arXiv:1412.4032v1 [cond-mat.mes-hall] 12 Dec 2014Spin waves in micro-structured yttrium iron garnet nanomet er-thick films Matthias B. Jungfleisch,1,a)Wei Zhang,1Wanjun Jiang,1Houchen Chang,2Joseph Sklenar,3Stephen M. Wu,1 John E. Pearson,1Anand Bhattacharya,1John B. Ketterson,3Mingzhong Wu,2and Axel Hoffmann1 1)Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 2)Department of Physics, Colorado State University, Fort Col lins, Colorado 80523, USA 3)Department of Physics and Astronomy, Northwestern Univers ity, Evanston, Illinois 60208, USA (Dated: 1 May 2018) We investigated the spin-wave propagation in a micro-structured y ttrium iron garnet waveguide of 40 nm thickness. Utilizing spatially-resolved Brillouin light scattering microsc opy, an exponential decay of the spin- wave amplitude of (10 .06±0.83)µm was observed. This leads to an estimated Gilbert damping constant ofα= (8.79±0.73)×10−4, which is larger than damping values obtained through ferromagnet ic resonance measurements in unstructured films. The theoretically calculated s patial interference of waveguide modes was compared to the spin-wave pattern observed experimentally b y means of Brillouin light scattering spec- troscopy. I. INTRODUCTION Magnonics is an emerging field of magnetism study- ing the spin dynamics in micro- and nanostructured devices aiming for the development of new spintron- ics applications.1–3Up to now, ferromagnetic metals (for example, Permalloy and Heusler alloys) have been widely used for the investigation of magnetization dy- namicson the nanoscale.4–10However, the Gilbert damp- ing of Permalloy is two orders of magnitude higher than that of ferrimagnetic insulator yttrium iron garnet (YIG, Y3Fe5O12). Recent progress in the growth of YIG films allows for the fabrication of low-damping nanometer- thick YIG films,11–14which are well-suited for patterning of micro-structured YIG devices. This enables investiga- tions of spin-wave propagation in plain YIG microstruc- tures of sub-100 nm thicknesses which are a step forward for future insulator-based magnonics applications. Inthiswork,weexperimentallydemonstratespin-wave propagation in a micro-structured YIG waveguide of 40 nm thickness and 4 µm width. By utilizing spatially- resolvedBrillouinlightscattering(BLS) microscopy4–7,11 the exponential decay length of spin waves is deter- mined. The corresponding damping parameter of the micro-structured YIG is estimated and compared to that determined from ferromagnetic resonance (FMR) mea- surements. Furthermore, we show that different spin- wave modes quantized in the direction perpendicular to the waveguide lead to a spatial interference pattern. We compare the experimental results to the theoretically ex- pected spatial interference of the waveguide modes. a)Electronic mail: jungfleisch@anl.govII. EXPERIMENT Figure1shows a schematic illustration of the sam- ple layout. The YIG film of 40 nm thickness was de- posited by magnetron sputtering on single crystal pol- ished gadolinium gallium garnet (GGG, Gd 3Ga4O12) substrates of 500 µm thickness with (111) orientation under high-purity argon atmosphere. The film was sub- sequently annealed in-situ at 800◦C for 4 hours under an oxygenatmosphereof1.12Torr. Themagneticproperties ofthe unstructured film werecharacterizedby FMR: The peak-to-peak linewidth µ0∆Has a function of the exci- tation frequency fis depicted in Fig. 2(a). The Gilbert damping parameter αFMRcan be obtained from FMR measurements using15 √ 3µ0∆H=2αFMR |γ|f+µ0∆H0, (1) FIG. 1. (Color online) Schematic illustration of the sample layout. The 4 µm wide yttrium iron garnet waveguide is magnetized transversally by the bias magnetic field H. Spin waves are excited by a shortened coplanar waveguide and the spin-wave intensity is detected by means of spatially-reso lved Brillouin light scattering microscopy. Colorbar indicate s spin- wave intensity.2 whereµ0is the vacuum permeability, γis the gyromag- netic ratio, fis the resonance frequency and µ0∆H0 is the inhomogeneous linewidth broadening. We find a damping parameter of αFMR= (2.77±0.49)×10−4[fit shown as a red solid line in Fig. 2(a)]. The resonance field,µ0H, as a function of the excitation frequency fis shown in Fig. 2(b). A fit to16 f=µ0|γ| 2π/radicalbig H(H+Meff) (2) yields an effective magnetization Meff= (122 ± 0.30) kA/m [solid line in Fig. 2(b)]. In a subsequent fab- rication process, YIG waveguides of 4 µm width were patterned by photo-lithography and ion milling with an Ar plasma at 600 V for 5 min. In a last step, a shortened coplanar waveguide (CPW) made of Ti/Au (3 nm/150 nm) is patterned on top ofthe YIG waveguide (see Fig. 1). The shortened end of the CPW has a width of 5µm. The Oersted field of an alternating microwave signal applied to the CPW exerts a torque on the mag- netic moments in the YIG and forces them to precess. The bias magnetic field is applied perpendicular to the shortaxisofthe waveguide(Fig. 1) providingefficient ex- citation of Damon-Eshbach spin waves. The microwave powerPMW= 1 mW is sufficiently small to avoid pos- sible perturbations of spin-wave propagation caused by nonlinearities. III. DISCUSSIONS In order to detect spin-wave propagation in the YIG waveguide spatially-resolved BLS microscopy with a res- olution of 250 nm is employed. To characterize the prop- agating spin waves, the BLS intensity was recorded at differentdistancesfromtheantenna. Aspatially-resolved BLS intensity map is shown in Fig. 3(b) at an exemplary excitation frequency of f= 4.19 GHz. Spin waves are excited near the antenna and propagate towards the op- posite end of the waveguide. To further analyze the data and to minimize the influence of multi-mode propagation in the YIG stripe (this will be discussed below), the BLS intensity is integrated over the width of the waveguide. The correspondingBLS intensity as a function of the dis- tance from the antenna is illustrated in Fig. 3(c). The decayofthespin-waveamplitudecanbedescribedby:7,11 I(z) =I0e−2z λ+b, (3) wherezis the distance from the antenna, λis the de- cay length of the spin-wave amplitude and bis an offset. From Fig. 3(b), it is apparent that the data-points fol- low an exponential behavior. A fit according to Eq. ( 3) yields the decay length λ. For an excitation frequency off= 4.19 GHz we find λ= (10.06±0.83)µm. This value is larger than decay lengths reported for Permalloy (<6µm, see Ref. 8, 21, and 22), but it is smaller thanFIG. 2. (Color online) (a) Ferromagnetic resonance peak-to - peaklinewidth µ0∆Has afunctionoftheresonance frequency fof the unstructured 40 nm YIG film. The red solid line represents a fit to Eq. (1). A Gilbert damping parameter α= (2.77±0.49)×10−4is determined. (b) Ferromagnetic resonance field µ0Has a function of f. Error bars are smaller than the data symbols. the largest decay length found for the Heusler-compound Co2Mn0.6Fe0.4Si (8.7 – 16.7µm, see Ref. 7). Pirro et al. reported a decay length of 31 µm in thicker YIG waveg- uides(100nm) grownbyliquidphaseepitaxy(LPE)with a 9 nm thick Pt capping layer.11In order to understand this discrepancy between our and their results, one has to take into account two facts: (1) State-of-the-art LPE fabrication technology can not be employed to grow film thicknesses below ∼100 nm. To date, sputtering offers an alternative approach to grow sub-100 nm thick YIG films with a sufficient quality.23(2) Taking into account the spin-wave group velocity vg=∂ω/∂kand the spin- wave lifetime τ, the theoretically expected decay length λcan be calculated from λ=vg·τ. The groupvelocity vg can be derived directly from the dispersion relation (see Fig.4). A thinner YIG film has flatter dispersion and a smallervg. Consequently, the expected decay length is smaller for thinner YIG samples and so it is natural that the decay length reported here is shorter than the one found in Ref. 11 for 100 nm thick YIG waveguides. We estimate the groupvelocity from the spin-wavedis- persion to be vg= 0.35−0.40µm/ns. Using our exper- imentally found decay length, the spin-wave lifetime is determined to be τ= 29 ns. We can use the BLS-data to determine the corresponding Gilbert damping param- eterαBLS. In case of Damon-Eshbach spin waves, the damping is given by243 FIG. 3. (Color online) (a) Calculated spatial interference pattern of the first two odd waveguide modes ( n= 1 and n= 3. (b) Spatially-resolved BLS intensity map at an ex- citation frequency f= 4.19 GHz, applied microwave power PMW= 1 mW, biasing magnetic field µ0∆H= 83 mT. The numbers 1 – 4 highlight the main features of the interference pattern. (c) Corresponding BLS intensity integrated over t he entire width of the YIG waveguide. An exponential decay of the spin-wave amplitude λ= (10.06±0.83)µm is found. αBLS=1 τ(γµ0Meff 2+2πf)−1. (4) A Gilbert damping parameter of the micro-structured YIG waveguide obtained by BLS characterization is found to be αBLS= (8.79±0.73)×10−4, which is a factor of 3 times larger than that determined by FMR in the unstructuredfilm [ αFMR= (2.77±0.49)×10−4]. This difference mightbe attributed to the micro-structuringof the YIG waveguide by Ar ion beam etching. The etch- ing might enhance the roughness of the edges of the YIG waveguides and the resist processing could have an in- fluence on the surface quality17,18which could possibly lead to an enhancement of the two-magnon scattering process.19It would be desirable to perform FMR mea- surements on the YIG waveguide. However, since the structured bar is very small, the FMR signal is van- ishingly small which makes it difficult to determine the Gilbert damping in this way. While the discussion above only considered the BLS- intensity integrated over the waveguide width, we will focus now on the spatial interference pattern shown in Fig.3(b). The spin-wave intensity map can be under- stood by taking into account the dispersion relation of magnetostatic spin waves in an in-plane magnetized fer- romagnetic thin film (see Fig. 4). Due to the lateral con- finement, thewavevectorisquantizedacrossthewidthof the YIG waveguide, ky=nπ/w, wheren∈N. The wave vectorkzalong the long axis of the waveguide ( z-axis)is assumed to be non-quantized. We follow the approach presented in Ref. 8. The dynamic magnetization is as- sumed to be pinned at the edges of the waveguide which can be considered by introducing an effective width of the waveguide.8,20 Figure4shows the calculated dispersion relations of different spin-wave modes quantized across the width of the strip. The dashed line represents a fixed excitation frequency. At a particular frequency different spin-waves modes with different wave vectors kzare excited simulta- neously. Thisleadstothe occurrenceofspatiallyperiodic interference patterns. In the present excitation config- uration, only modes with an odd quantization number ncan be excited ( ndetermines the number of maxima across the width of the waveguide). Since the intensity of the dynamic magnetization of these modes decreases with increasing nas 1/n2, we only consider the first two odd modes n= 1 and n= 3. According to Ref. 8 the spatial distribution of the dy- namic magnetization of the n-th mode can be expressed as mn(y,z)∝sin(nπ wy)cos(kn zz−2πft+φn),(5) wherefis the excitation frequency, kn zis the longitudinal wave vector of the n-th spin-wave mode and φnis the phase.25The spin-waveintensity distribution Inof then- th mode can be derived by averaging mn(y,z)2over one oscillation period 1 /f. The entire interference pattern can be obtained from the same procedure using the sum m1(y,z) +1 3m3(y,z). The factor 1/3 accounts for the lower excitation efficiency of the n= 3 mode. Thus, the intensity is given by8 IΣ(y,z)∝sin(π wy)2+1 9sin(3π wy)2 +2 3sin(π wy)sin(3π wy)cos(∆kzz+∆φ),(6) where ∆kz=k3 z−k1 zand ∆φ=φ3−φ1. This pattern re- peatsperiodically. Thephaseshift φshiftsthe entirepat- FIG. 4. (Color online) Dispersion relations the first five waveguide modes of a transversally magnetized YIG stripe.26 Only modes with a odd quantization number ncan be excited (solid lines). fdenotes the excitation frequency.4 tern along the z-direction and the wave-vector difference ∆kz= 0.97 rad/µm can be calculated from the disper- sion relation (Fig. 4). The calculated spatial interference pattern is depicted in Fig. 3(a) using ∆ φ= 0 and taking into account for the exponential decay of the spin-wave amplitude by multiplying Eq. ( 6) withe−2z/λusing the experimentallydetermined λ= 10.03µm. As is apparent from Fig. 3(a) and (b) a qualitative agreement between calculation and experiment is found. (The numbers 1 – 4 highlight the main features of the interference pattern in experiment and calculation.) The small difference in Fig.3(a) and (b) can be explained by considering the fact that in the calculation a spin-wave propagation at an angle of exactly 90◦(Damon-Eshbach configuration) with respect to the antenna/externalmagnetic field is as- sumed. However, in experiment small misalignments of the external magnetic field might lead to a small asym- metry in the interference pattern. IV. CONCLUSION In summary, we demonstrated spin-wave excitation and propagation in micro-fabricated pure YIG wave- guides of 40 nm thickness. BLS-characterization re- vealed a decay length of the spin-wave amplitude of 10µm leading to an estimated Gilbert damping pa- rameter of αBLS= (8.79±0.73)×10−4. This value is a factor 3 larger than the one determined for the unstructured YIG film by means of FMR techniques [αFMR= (2.77±0.49)×10−4]. The difference might be attributed to micro-structuring using ion beam etching. 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1907.04577v2.The_superior_role_of_the_Gilbert_damping_on_the_signal_to_noise_ratio_in_heat_assisted_magnetic_recording.pdf

The superior role of the Gilbert damping on the signal-to-noise ratio in heat-assisted magnetic recording O. Muthsam,1,a)F. Slanovc,1C. Vogler,1and D. Suess1 University of Vienna, Physics of Functional Materials, Boltzmanngasse 5, 1090 Vienna, Austria (Dated: 25 September 2019) In magnetic recording the signal-to-noise ratio (SNR) is a good indicator for the quality of written bits. However, a priori it is not clear which parameters have the strongest in uence on the SNR. In this work, we investigate the role of the Gilbert damping on the SNR. Grains consisting of FePt like hard magnetic material with two dierent grain sizes d= 5 nm and d= 7 nm are considered and simulations of heat-assisted magnetic recording (HAMR) are performed with the atomistic simulation program VAMPIRE. The simulations display that the SNR saturates for damping constants larger or equal than 0.1. Additionally, we can show that the Gilbert damping together with the bit length have a major eect on the SNR whereas other write head and material parameters only have a minor relevance on the SNR. I. INTRODUCTION The next generation recording technology to increase the areal storage density of hard drives beyond 1.5 Tb/in2 is heat-assisted magnetic recording (HAMR)1{6. Higher areal storage densities (ADs) require smaller recording grains. These grains need to have high anisotropy to be thermally stable. HAMR uses a heat pulse to locally enhance the temperature of the high anisotropy record- ing medium beyond the Curie temperature. Due to the heating, the coercivity of the grain drops and it can be written with the available head elds. After the grain is written, the medium is cooled and the information is safely stored. A good indicator for the quality of the written bits is the so-called signal-to-noise ratio (SNR) which gives the power of the signal over the power of the noise7. To achieve high areal storage densities, record- ing materials that show good magnetic properties even at small grain sizes and thus yield high SNR values are needed. However, a priori it is not clear which parame- ters have the strongest in uence on the SNR. In this work, we investigate the eect of a varying damp- ing constant on the SNR. HAMR simulations with the atomistic simulation program VAMPIRE8are performed for cylindrical recording grains with two dierent diame- tersd= 5 nm and d= 7 nm and a height h= 8 nm. The material parameters of FePt like hard magnetic record- ing media according to the Advanced Storage Technol- ogy Consortium (ASTC)9are used. Damping constants between= 0:01 and= 0:5 are considered. Addition- ally, we present an equation to include the in uence of the bit length to the SNR. With this we can explain a SNR decrease of about 8.25 dB for 5- nm grains, which results when changing the material and writing parame- ters in the HAMR simulations from those used in former simulations10{12to those according to the Advanced Stor- age Technology Consortium9, with the damping constant and the bit length only. The structure of this paper is as follows: In Section II, the HAMR model is introduced and it is explained how a)Electronic mail: olivia.muthsam@univie.ac.atthe SNR is determined. In Section III, the results are presented and in Section IV they are discussed. II. HAMR MODEL Cylindrical recording grains with height h= 8 nm and diametersd= 5 nm and d= 7 nm are considered. One grain can be interpreted as one grain of a state-of-the- art granular recording medium. A simple cubic crystal structure is used. The exchange interaction Jijand the eective lattice parameter aare adjusted so that the sim- ulations lead to the experimentally obtained saturation magnetization and Curie temperature13,14. In the simu- lations, only nearest neighbor exchange interactions be- tween the atoms are included. A continuous laser pulse with Gaussian shape and the full width at half maximum (FWHM) of 60 nm is assumed in the simulations. The temperature prole of the heat pulse is given by T(x;y;t ) = (TwriteTmin)ex2+y2 22+Tmin (1) =Tpeak(y)
ex2 22+Tmin (2) with =FWHMp 8 ln(2)(3) and Tpeak(y) = (TwriteTmin)ey2 22: (4) v= 15 m/s is the speed of the write head. xandylabel the down-track and the o-track position of the grain, respectively. In our simulations both the down-track po- sitionxand the o-track position yare variable. The ambient and thus minimum temperature of all simula- tions isTmin= 300 K. The applied eld is modeled as a trapezoidal eld with a eld duration of 0.57 ns and a eld rise and decay time of 0.1 ns, resulting in a bit length of 10.2 nm. The eld strength is assumed to be +0 :8 T and0:8 T inz-direction. Initially, the magnetization of each grain points in + z-direction. The trapezoidal eldarXiv:1907.04577v2 [physics.app-ph] 24 Sep 20192 tries to switch the magnetization of the grain from + z- direction toz-direction. At the end of every simulation, it is evaluated if the bit has switched or not. The material and write head parameters according to the Advanced Storage Technology Consortium9are shown Table I. A. Determination of SNR To calculate the signal-to-noise ratio, the read-back signal of a written bit pattern has to be determined. To write the bit pattern and get the read-back signal from it, the following procedure is used. First, a switching prob- ability phase diagram is needed for the writing process of the bit pattern. Since it is very time consuming to com- pute a switching probability phase diagram with atom- istic or micromagnetic simulations, an analytical model developed by Slanovc et al15is used in this work. The model uses eight input parameters (the maximum switch- ing probability Pmax, the down-track jitter down;the o- track jitter o;the transition curvature c, the bit length b, the half maximum temperature F50, the position p2of the phase diagram in Tpeakdirection and the position p3 of the phase diagram in down-track direction) to deter- mine a switching probability phase diagram. Slanovc et alshowed that the maximum switching probability Pmax and the down-track jitter down are the input parameters with the strongest in uence on the SNR. Note, that the bit lengthbalso has a strong in uence on the SNR. In the further course of this work, an equation to include the bit length to the SNR calculations is shown. Thus, the bit length can be assumed constant during the SNR determination. The transition curvature cdid not show strong in uence on the SNR for the used reader model and the o-track jitter ois neglectable since the reader width is with 30 :13 nm smaller than the track width with 44:34 nm and thus does not sense the o-track jitter. p2 andp3only shift the bit pattern and can thus be xed for comparability. For this reason, it is reasonable to x the input parameters, except for the maximum switch- ing probability Pmaxand the down-track jitter down. The xed input parameters are determined by a least square t from a switching probability phase diagram computed with a coarse-grained Landau-Lifshitz-Bloch (LLB) model16for pure hard magnetic grains with mate- rial parameters given in Table I. The tting parameters are summarized in Table II for grain diameters d= 5 nm andd= 7 nm. Further, it is necessary to compute the down-track jitter down and the maximal switching probability Pmaxfor the considered set of material and write head parameters, see Table I. In the simulations, the switching probability of a recording grain at various down-track positions xat a peak temperature Tpeak=Tc+ 60 K is calculated with the atomistic simulations program VAMPIRE8, yielding a down-track probability function P(x). To get the down- track jitter and the maximum switching probability, the switching probability curve is tted with a Gaussian cu-mulative distribution function ;2=1 2(1 + erf(xp 22))
Pmax (5) with erf(x) =2pZx 0e2d; (6) where the mean value , the standard deviation and the mean maximum switching probability Pmax2[0;1] are the tting parameters. The standard deviation , which determines the steepness of the transition function, is a measure for the transition jitter and thus for the achievable maximum areal grain density of a recording medium. The tting parameter Pmaxis a measure for the average switching probability at the bit center. Note, that the calculated jitter values down only consider the down-track contribution of the write jitter. The so-called aparameter is given by a=q 2 down+2g (7) wheregis a grain-size-dependent jitter contribution17. The write jitter can then be calculated by writear S W(8) whereWis the reader width and S=D+Bis the grain size, i.e. the sum of the grain diameter Dand the non- magnetic boundary B15,18. For eachdown andPmaxcombination a switching prob- ability phase diagram is computed with the analytical model. With the resulting phase diagram, the writing process of a certain bit pattern is simulated on granu- lar recording medium15. Here, the switching probabil- ity of the grain is set according to its position in the phase diagram. The writing process is repeated for 50 dierent randomly initialized granular media. Finally, the read-back signal is determined with a reader model where the reader width is 30.13 nm and the reader res- olution in down-track direction is 13 :26 nm. The SNR can then be computed from the read-back signal with the help of a SNR calculator provided by SEAGATE19. The resulting SNR value is given in dB (SNR dB). In the following, the SNR dBis simply called SNR unless it is explicitly noted dierent. III. RESULTS A. SNR Dependency on Damping First, the in uence of the damping constant on the SNR is investigated in more detail. The damping con- stant is varied from = 0:01 to= 0:5 for two dierent grain sizesd= 5 nm and d= 7 nm. All other parameters are taken from Table I. The bit length in the simula- tions is 10:2 nm and the track width is 44 :34 nm. The down-track jitter curves are computed at Tpeak= 760 K and tted with eq. (5). In Figure 1, the SNR over the3 Curie temp. TC[K]DampingUniaxial anisotropy ku[J/link]Jij[J/link] s[B]v[m/s]eld duration (fd) [ns]FWHM [nm] 693.5 0.02 9:12410236:7210211.6 15 0.57 60 TABLE I. Material and write head parameters of a FePt like hard magnetic granular recording medium accoring to the Advanced Storage Technology Consortium. grain size 5 nm 7 nm o[K] 22.5 14.4 Pmax 0.995 0.997 F50[K] 602 628 b[nm] 10.2 10.2 c[104nm/K2]3.88 4.89 p2[K] 839 830 p3[nm] 27.5 25.8 TABLE II. Reference parameters that are evaluated via least square t of the simulated phase diagrams for grain sizes 5 nm and 7 nm. Details of the parameters can be found in15. damping constants for both grain sizes is visible. Note that the SNR is proportional to the number of grains, meaning that the number of grains per bit has to be kept constant to determine a nearly constant SNR20. How- ever, since the dimensions of the granular media used for the writing and reading process are xed, less grains form one bit ford= 7 nm. Thus, the SNR values for the larger grain size are smaller than for the small grains. The results show that changing the damping constant from= 0:01 to= 0:02 already increases the SNR by 3.66 dB for 5 nm-grains. For d= 7 nm, the SNR gain is 1.65 dB. For 5 nm-grains, damping constants 0:1 lead to the best results with a total improvement of 6 dB com- pared to= 0:01. Surprisingly, enhancing the damping constant beyond 0 :1 does not show any further improve- ment, the SNR saturates. This behavior is the same for the 7 nm-grains. However, the total betterment of the SNR is only 2.24 dB for the larger grains. The SNR sat- uration results from the fact that Pmax= 1 for0:1. Simultaneously, the down-track jitter down varies only marginally for 0:1 (see Table III) such that it does not alter the SNR. The correlation between the SNR and the maximum switching probability Pmaxis shown in Fig- ure 2. It shows that the tted SNR curve reproduces the data very well. By further studying the switching dynamics of a 5 nm- grain, one can show that the assumed pulse duration of the heat pulse and the applied eld strength are crucial for the saturation of the SNR. In Figure 3, it is displayed how the duration of the heat pulse in uences the maxi- mum switching probability and with it the SNR. During the duration of the heat pulse the eld is considered to constantly point in zdirection. The results demon- strate that Pmaxdoes not saturate for small pulse du- rations. If longer pulse durations 0:5 ns are assumed, aPmaxsaturation can be seen. A similar eect can be seen for a change of the eld strength when the pulse duration is assumed to be 0 :5 ns (see Figure 4). For a small head eld with a strength of 0 :5 T,Pmaxshows no saturation whereas it does for larger head elds. From FIG. 1. Resulting SNR for various damping constants for grains with two dierent diameters d= 5 nm and d= 7 nm. FIG. 2. SNR and Pmaxdepending on the damping constant for grain size with a diameter of 5 nm. the simulations with varying duration of the heat pulse and eld strength, it can also be seen that the SNR can be improved for smaller damping constants if the dura- tion of the heat pulse is increased due to a smaller head velocity or the eld strength are enhanced. B. SNR Dependency on bit length The in uence of the bit length on the SNR was al- ready studied by Slanovc et al15. In this work, the fol-4 5 nm 7 nm down[nm]Pmaxdown[nm]Pmax 0.01 2.0 0.917 1.13 0.955 0.02 0.9475 0.974 0.83 0.99 0.05 0.7 0.989 0.549 1.0 0.1 0.688 1.0 0.442 1.0 0.3 0.495 1.0 0.48 1.0 0.5 0.64 1.0 0.636 1.0 TABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for pure hard magnetic material with dierent damping constants . FIG. 3. Maximum switching probability Pmaxover damping for dierent pulse lengths of the heat pulse. A eld strength of0:8 T for grains with diameter d= 5 nm is assumed. lowing calculation is important. For the SNR calcula- tions a bit length b1= 10:2 nm is assumed since this is the bit length resulting from the ASTC parameters. The track width in the simulations is again 44 :34 nm. How- ever, the bit length can change due to a variation of the write head parameters (eld duration and head velocity). Therefore, the bit length for the former parameters10{12 is 22 nm. To write a bit pattern with larger bit lengths (b >12 nm) the simulations of new granular media are required. This is computationally very expensive. Thus, a dierent approach is needed to qualitatively investi- gate the in uence of the bit length. For the SNR with SNR dB= 10 log10(SNR), there holds18 SNR/b a2T50 bW S (9) with the bit length band the read-back pulse width T50 which is proportional to the reader resolution in down- track direction. The ratio T50=bis called user bit density and is usually kept constant18. Further, the reader width Wand the grain size Sare constant. Since the aim is to qualitatively describe the SNR for a bit length b2from SNR calculations with a bit length b1;the a-parameter a is also assumed to be constant. The SNR dBfor a dierent bit lengthb2can then be calculated by FIG. 4. Maximum switching probability Pmaxover damping for dierent eld strengths. The durations of the heat pulse of 0:5 ns for grains with diameter d= 5 nm is assumed. SNR dB(b2)SNR dB(b1) = 10 log10(SNR(b2))10 log10(SNR(b1)) = 10 log10(b2 2)10 log10(b2 1) = 20 log10(b2 b1) (10) since all other parameters are the same for both bit lengths. Thus, one can compute the SNR dBvalue for a varied bit length b2via the SNR dBof the bit length b1 by SNR dB(b2) = SNR dB(b1) + 20 log10(b2 b1): (11) The curve achieved by eq. (11) with b1= 10:2 nm agrees qualitatively very well with the SNR(bit length) data from Slanovc et al15. It is thus reasonable to use this equation to include the bit length to the SNR. C. Combination of damping and bit length5 Curie temp. TC[K]DampingUniaxial anisotropy ku[J/link]Jij[J/link] s[B]v[m/s]eld duration (fd) [ns]FWHM [nm] 536.6 0.1 9:1210235:1710211.7 20 1.0 20 TABLE IV. Material and write head parameters of a FePt like hard magnetic granular recording medium that were used in former works10{12. Parameter set diameter [nm] Tpeak[K]bit length [nm] Pmaxdown [nm] SNR [dB] ASTC 5 760 10.2 0.974 0.95 17.51 Parameters of former works10{12 5 600 22 0.984 0.384 25.76 ASTC 7 760 10.2 0.99 0.83 15.35 Parameters of former works10{12 7 600 22 1.0 0.44 22.75 TABLE V. Resulting Pmax; down and SNR values for the simulations with ASTC parameters and those used in former simulations. The simulations with write head and material parameters according to the ASTC are compared to simulations with parameters used in former works10{12. Main dierences to the currently used parameters are the bit length, the damping constant, the height of the grain, the exchange interaction, the atomistic spin moment, the full width at half maximum, the head velocity and the eld duration. These former parameters are summarized in Table IV. Comparing the SNR values of both parameter sets shows that ford= 5 nm the SNR is about 8.25 dB larger for the former used parameters than for the ASTC parameters and ford= 7 nm it is7:4 dB larger. The question is if the damping and bit length variation can fully explain this deviation. Increasing the damping constant from = 0:02 to= 0:1, yields about +2 :25 dB ford= 5 nm and +0 :72 dB for d= 7 nm. Additionally, with the calculations from Sec- tion III B, one can show that by changing the bit length fromb1= 10:2 nm tob2= 22 nm gives SNR dB(b2) = SNR dB(b1) + 6:85 dB: (12) Combined, this shows that the dierence in the SNR can be attributed entirely to the damping and the bit length enhancement. Moreover, simulations where the other material and write head parameters are changed one by one conrm this ndings. The other write head and material parameters that are changed in the simu- lations have only minor relevance on the SNR compared to the damping constant and the bit length. IV. CONCLUSION To conclude, we investigated how the damping con- stant aects the SNR. The damping constant was varied between= 0:01 and= 0:5 for two dierent grain sizes d= 5 nm and d= 7 nm and the SNR was determined. In practice, the damping constant of FePt might be in- creased by enhancing the Pt concentration21,22. Another option would be to use a high/low Tcbilayer structure23 and increase the damping of the soft magnetic layer by doping with transition metals24{28. An interesting nd- ing of the study is the enormous SNR improvement of6 dB that can be achieved for 5 nm-grains when enhanc- ing the damping constant from = 0:01 to= 0:1 and beyond. It is reasonable that the SNR improves with larger damping. This results from the oscillatory behavior of the magnetization for small damping dur- ing switching. In fact, smaller damping facilitates the rst switching but with larger damping it is more likely that the grain will switch stably during the cooling of the thermal pulse29. This leads to a smaller switching time distribution for larger damping constants and in the fur- ther course to higher SNR values. However, an increase of the duration of the heat pulse due to a smaller head velocity or an increase of the eld strength can improve the SNR even for smaller damping constant. Furthermore, the results display a SNR saturation for damping constants 0:1. This SNR saturation can be explained with the saturation of the maximum switching probability and the only marginal change of the down- track jitter for 0:1. Indeed, one can check that for shorter pulse widths and smaller eld strength, the be- havior is dierent and the SNR does not saturate. In this case, the SNR rises for increasing damping constants. Summarizing, the SNR saturation for a varying damping constant depends strongly on the used eld strength and the duration of the heat pulse. The qualitative behavior for 7 nm-grains is the same. In- terestingly, the SNR change for a varying damping con- stant is not as signicant as for grains with d= 5 nm. This results from the higher maximum switching proba- bility and the smaller down-track jitter down for 7 nm- grains even for small damping constants. This is as expected since larger grain sizes lead to an elevated maximum switching probability11and smaller transition jitter7compared to smaller grain sizes. This limits the possible increase of the recording performance in terms ofPmaxanddown and thus the possible SNR gain. Ad- ditionally, the SNR saturation value is smaller for 7 nm- grains since one bit consists of fewer grains. The overall goal was to explain the decrease of the SNR by about 8:25 dB and 7 :4 dB ford= 5 nm and d= 7 nm, respectively, when changing from recording parameters used in former simulations10{12to the new ASTC pa- rameter. Indeed, together with the bit length variation, the SNR variation could be fully attributed to the damp-6 ing enhancement. The other changed parameters like the atomistic spin moment, the system height, the exchange interaction and the full width at half maximum have only a minor relevance compared to the in uence of the damp- ingand the bit length. In fact, the variation of the bit length gave the largest SNR change. However, since an increase of the bit length is not realistic in recording devices, the variation of the material parameters, especially the increase of the damp- ing constant, is a more promising way to improve the SNR. V. ACKNOWLEDGEMENTS The authors would like to thank the Vienna Sci- ence and Technology Fund (WWTF) under grant No. MA14-044, the Advanced Storage Technology Consor- tium (ASTC), and the Austrian Science Fund (FWF) under grant No. I2214-N20 for nancial support. The computational results presented have been achieved us- ing the Vienna Scientic Cluster (VSC). 1L. Burns Jr Leslie and others. Magnetic recording system . Google Patents, December 1959. 2G. W. Lewicki and others. Thermomagnetic recording and magneto-optic playback system . Google Patents, December 1971. 3Mark H. Kryder, Edward C. Gage, Terry W. McDaniel, William A. Challener, Robert E. Rottmayer, Ganping Ju, Yiao- Tee Hsia, and M. 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2307.05865v1.Asymptotic_behavior_of_solutions_to_the_Cauchy_problem_for_1_D_p_system_with_space_dependent_damping.pdf

arXiv:2307.05865v1 [math.AP] 12 Jul 2023Asymptotic behavior of solutions to the Cauchy problem for 1 -D p-system with space dependent damping Akitaka Matsumura and Kenji Nishihara∗ Osaka University and Waseda University, Japan Abstract We consider the Cauchy problem for one-dimensional p-system with damping of space- dependent coefficient. This system models the compressible flow thr ough porous media in the Lagrangeancoordinate. Ourconcernisanasymptoticbehaviorof solutions,whichisexpected to be the diffusion wave based on the Darcy law. In fact, in the constan t coefficient case Hsiao and Liu [3] showed the asymptotic behavior under suitable smallness cond itions for the first time. After this work, there are many literatures, but there are few wo rks in the space-dependent damping case, as far as we know. In this paper we treat this space- dependent case, as a first step when the coefficient is around some positive constant. 1 Introduction We consider the Cauchy problem for the p-system with space-dependent damping (1.1) vt−ux= 0,(t,x)∈R+×R, ut+p(v)x=−αu(α=α(x)), (v,u)(0,x) = (v0,u0)(x)→(v±,u±), x→ ±∞,(v±>0), which models the 1-D compressible flow through porous media i n the Lagrangean coordinate, whereu=u(t,x) is the velocity of the flow at time tand position x,v(>0) is the specific volume, p(>0) is the pressure with p′(v)<0, and the coefficient α=α(x)>0. Our interest is in the large time behavior of the solution ( v,u) to (1.1), which is expected to be the diffusion wave (¯ v,¯u) to (1.2)/braceleftBigg ¯vt−¯ux= 0, p(¯v)x=−α¯u, by the Darcy law. In fact, when αis a constant, Hsiao and Liu [3] showed its asymptotic behavi or to (¯v,¯u) under some smallness conditions for the first time. The conv ergence rates of ( v−¯v,u−¯u) were improved by the second author [9]. After these, there ar e many works including the case α=α(t). We cite only [4, 5, 6, 8, 10, 13, 14, 15]. See the references t herein, too. However, as far as we know, there are few results on the asymptotics towar d diffusion wave in space-dependent damping case. Concerning the blow-up results, there are som e works(see Chen et al. [1], Sugiyama [12] and reference therein). ∗E-mail addresses: akitaka@muh.biglobe.ne.jp (A. Matsumu ra), kenji@waseda.jp (K. Nishihara) 1In this paper we consider the case of x-dependent coefficient α=α(x). The Darcy law says that the velocity of flow in porous media is proportional to th e pressure gradient, and so it may be reasonable that the coefficient αdepends on the position x. By the situation of media, αwill be assumed variously. However, as a first step we treat the simpl est case (1.3) α=α(x)→α, x→ ±∞ for some constant α>0. When (1.4) v+=v−(andu+/ne}ationslash=u−in general) , the asymptotic profile ¯ vof (1.2) is expected to be a variant of the Gauss function. Whi le, when (1.5) v+/ne}ationslash=v−, ¯vbe an approximate similarity solution, and the treatment be comes complicated. In both cases, the problems are reformulated to the Cauchy pr oblems for the quasilinear wave equations with damping of the coefficient α(x), whose results are directly applied to the original ones. Those details are stated in the next section. As for the results obtained in this paper we be- lieve that there are still plenty of rooms for improvement. U nder various assumptions on α=α(x) orα(t,x) many discussions are expected in the future. The content of this paper is as follows. In Section 2 our probl em is reformulated in each case of (1.4) and (1.5) to the Cauchy problem for a quasilinear wav e equation of second order with damping. The case around the constant state of vis treated in Subsection 2.1, while the different end states case is mentioned in Subsection 2.2. In each subse ction, the theorem on the reformulated one will be stated clear, so that our goal for (1.1) in each cas e is obtained as a corollary. For those proofs, a standard energy method is applied, for which a seri es of a priori estimates are necessary. The case around the constant state (1.4) is shown in Section 3 . In the final section the convergence to the similarity solution in case of (1.5) will be treated. Notations . For the function spaces, Lp=Lp(R)(1≤p≤ ∞) is a usual Lebesgue space with norm |f|p= (/integraldisplay R|f(x)|pdx)1/p(1≤p <∞),|f|∞= sup R|f(x)|. The integral domain Ris often abbreviated when it is clear. For any integer l≥0,Hl=Hl(R) denotes the usual l-th order Sobolev space with norm /bardblf/bardblHl=/bardblf/bardbll= (l/summationdisplay j=0/bardbl∂j xf/bardbl2)1/2. Whenl= 0andp= 2, weoftenusethenotation /bardblf/bardbl=/bardblf/bardbl0=|f|2. Forbrevity, /bardblf,g,···/bardbl2 Hk×Hl×··· =/bardblf/bardbl2 Hk+/bardblg/bardbl2 Hl+···=/bardblf/bardbl2 k+/bardblg/bardbl2 l+···. The set of k-times continuously differentiable functions inRwith compact support is denoted by Ck 0(R). The space Ck([0,T];X) is a set of k-times differentiable functions on [0 ,T] to the Hilbert space X. Also, by Corcwe denote a generic positive constant independent of the data and time t, whose value may change in line to line. 22 Reformulation of the problem and results In this section we heuristically reformulate the problem (1 .1) to the quasi-linear wave equation with damping, and state the results on the reformulated one, whic h derive our main results on (1.1). Rewrite (1.1) as (2.1) vt−ux= 0, ut+p(v)x+αu= 0, (v,u)(0,x) = (v0,u0)(x)→(v±,u±), x→ ±∞, under the assumption (1.3). By the Darcy law, the asymptotic profile (¯v,¯u) is expected to be given by (1.2) as in the previous papers. However, in the case thatαis depending on xwe meet difficulties to directly construct (¯ v,¯u) by (2.2) ¯ vt−/parenleftBig−p(¯v)x α/parenrightBig x= 0 and ¯ u=−p(¯v)x α. In this paper, to avoid the difficulties, we shall introduce a s impler profile in each case of (1.4) and (1.5). 2.1 Reformulation of the problem around the constant state Start to discuss the simpler case (1.4). Since both (1.3) and (1.4) are assumed, we here adopt an asymptotic profile ( V,U) by (2.1.1)/braceleftBigg Vt−Ux= 0, p′(v)Vx+αU= 0, V(t,±∞) =v(:=v+=v−), or (2.1.2) Vt−µVxx= 0, V(t,±∞) =v(µ:=|p′(v)| α), U=|p′(v)| αVx=µVx, so that, as an exact solution, (2.1.3) V(t,x) =v+δ0√ 4πµ(1+t)e−x2 4µ(1+t), U(t,x) =−2πδ0x (4πµ(1+t))3/2e−x2 4µ(1+t), whereδ0is some constant, determined later. Then ( V,U) satisfies (2.1.4)/braceleftBigg Vt−Ux= 0, Ut+p(V)x+αU=Ut+(α−α)U+(p(V)−p′(v)V)x. Here, we note that ( V,U)→(v,0) asx→ ±∞. 3To arrange the condition u(0,x) =u0(x)→u±(x→ ±∞), we introduce a correction function (ˆv,ˆu). By (2.1) 2(second equation of (2.1)) we can expect u(t,x)∼e−αtu±asx→ ±∞and so we determine (ˆ v,ˆu) as (2.1.5) ˆvt−ˆux= 0, ˆut+αˆu= 0, (ˆv,ˆu)(t,x)→(0,e−αtu±), x→ ±∞. Thus we define (ˆ v,ˆu) by (2.1.6) ˆu(t,x) =e−αt(u−+(u+−u−)/integraltextx −∞m0(y)dy) =:e−αtM0(x), ˆv(t,x) =∂ ∂x(e−αt −αM0(x)) =e−αt −α{(u+−u−)m0(x)−α′(x)M0(x)(1 α(x)+t)}, wherem0∈C∞ 0(R) with/integraltext∞ −∞m0(y)dy= 1. In the result, (2.1.5) holds with (2.1.7)/integraldisplay∞ −∞ˆv(0,x)dx=/integraldisplay∞ −∞∂ ∂x(1 −αM0(x))dx=1 −α(u+−u−). Combining (2.1), (2.1.4), (2.1.5), we have (2.1.8) (v−V−ˆv)t−(u−U−ˆu)x= 0, (u−U−ˆu)t+(p(v)−p(V))x+α(u−U−ˆu) =−{Ut+(α−α)U+(p(V)−p′(v)V)x}. Sinceu−U−ˆu→0 (x→ ±∞) is expected, /integraltext∞ −∞(v−V−ˆv)(t,x)dx=/integraltext∞ −∞(v0(x)−V(0,x)−ˆv(0,x))dx =/integraltext∞ −∞{(v0(x)−v)−(V(0,x)−v)}dx+1 α(u+−u−) =/integraltext∞ −∞(v0(x)−v)dx−δ0+1 α(u+−u−). Therefore, we choose δ0by (2.1.9) δ0=/integraldisplay∞ −∞(v0(x)−v)dx+1 α(u+−u−), so that/integraltext∞ −∞(v−V−ˆv)(t,x)dx= 0 for any t≥0. Thus, if we define φby (2.1.10) φ(t,x) =/integraldisplayx −∞(v−V−ˆv)(t,y)dy, then we may expect φ(t,·)∈H1, and it holds that φx=v−V−ˆv. Then, by (2.1.8) 1, it also holds φt=u−U−ˆu, and (2.1.8) 2can be written as (2.1.11)φtt+(p(V+ ˆv+φx)−p(V))x+αφt =−{Ut+(α−α)U+(p(V)−p′(v)V)x}. 4Modify the second term in (2.1.11) to get the following refor mulated problem: (2.1.12) φtt+(p(V+ ˆv+φx)−p(V+ ˆv))x+αφt =−{Ut+(α−α)U+(p(V)−p′(v)V)x+(p(V+ ˆv)−p(V))x}=:F=:4/summationdisplay i=1Fi, φ(0,x) =φ0(x) :=/integraltextx −∞(v0(y)−V(0,y)−ˆv(0,y))dy, φt(0,x) =φ1(x) :=u0(x)−U(0,x)−ˆu(0,x). Our aim is to show ( φx,φt)(t) =o(t−1/4,t−3/4)inL2-sense and, if possible, o(t−1/2,t−1) in L∞-sense as t→ ∞, so that ( V,U+ˆu) is an asymptotic profile of ( v,u), since (ˆ v,ˆu) decays rapidly. Thus, ournextjobis to showtheexistence and asymptotic beh aviorof theuniqueglobal-in-time solution φto (2.1.12). We assume (2.1.13) p∈C3(R+), p′(v)<0 (v >0), α∈C2(R), α−α∈L1∩L2,|x|1/2(α−α)∈L2, (1+|x|)(|αx|+|αxx|)∈L2andα≥α0>0 (α0: constant) , and (2.1.14)v0−v∈L1,/integraldisplayx −∞(v0−v)(y)dy∈L2, v0−v∈H2andu0−u±∈L2(R±),u0x∈H1, so (φ0,φ1)∈H3×H2. Under these assumptions there exists a unique local-in-ti me solution φin ∩2 i=0Ci([0,t0];H3−i) for some t0>0(for the proof, see Matsumura [7]). Main part in the proof fo r the global-in-time solution and its asymptotic behavior is the following a priori estimates. Proposition 2.1.1 (A priori estimate) Assume (2.1.13) and (2.1.14). Then there exist positive constants ε0andC0such that, if δ1:=|v0−v|1+|u+−u−| ≤ε0and ifφ∈ ∩2 i=0Ci([0,T];H3−i) is a solution to (2.1.12) for some T >0, which satisfies sup 0≤t≤T{/bardbl(φ,φt)(t)/bardblH3×H2+(1+t)/bardbl(φtx,φtxx)(t)/bardbl} ≤ε0, then it holds (2.1.15)(1+t)2/bardbl(φtt,φt,φxx)(t)/bardbl2 H1×H2×H1+(1+t)/bardblφx(t)/bardbl2+/bardblφ(t)/bardbl2 +/integraltextt 0{(1+τ)2/bardbl(φtt,φtx,φtxx)(τ)/bardbl2+(1+τ)/bardbl(φt,φxx)(τ)/bardbl2+/bardblφx(τ)/bardbl2}dτ ≤C0(/bardblφ0,φ1/bardbl2 H3×H2+δ1),0≤t≤T. Remark 2.1.1 Note that none of smallness assumptions on αis not made. Combining the local existence and a priori estimates of solu tion, we have theorem for (2.1.12). Theorem 2.1.1 Under the assumptions of Proposition 2.1.1, unique global- in-time solution φ∈ ∩2 i=0Ci([0,∞);H3−i)to (2.1.12) exists and satisfies the decay properties(2.1.1 5) for0≤t <∞. 5Once we have Theorem 2.1.1, if we newly define ( v,u) by (v,u) = (V+ˆv+φx,U+ˆu+φt), then we can have a unique solution ( v,u) to (2.1), satisfying ( v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2). Thus, from Theorem 2.1.1 with |φx(t)|∞≤C/bardblφx(t)/bardbl1/2/bardblφxx(t)/bardbl1/2≤C(1+t)−3/4, we have main theorem in case of (1.4). Theorem 2.1.2 (Around the constant state) Suppose (2.1.13) and (2.1.14). If /bardblφ0,φ1/bardbl2 H3×H2 +δ1is suitably small, then the solution (v,u)to (1.1) such that (v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2) uniquely exists and it holds that, as t→ ∞, (2.1.16)/braceleftBigg (v−V)(t,x) =O(t−1/2), (u−U−ˆu)(t,x) =O(t−1),inL2-sense, and (2.1.17) v(t,x) =V(t,x)+O(t−3/4)inL∞-sense. Since/bardbl(V,U)(t)/bardbl ∼(t−1/4,t−3/4) and/bardbl(ˆv,ˆu)(t)/bardblH1×L∞=O(e−ct), (V,U+ ˆu) is an asymptotic profile of ( v,u) inL2-framework, and Vis so ofvinL∞-framework. 2.2 Reformulation around the different end states and result s Similar to the last subsection, as an asymptotic profile we ad opt the similarity solution ( V,U), which is defined by (2.2.1)/braceleftBigg Vt−Ux= 0, p(V)x+αU= 0, V(t,±∞) =v±(v+/ne}ationslash=v−), or (2.2.2)/braceleftBiggVt−(µ(V)Vx)x= 0, V(t,±∞) =v±, U=−1 αp(V)x=−1 αp′(V)Vx=:µ(V)Vx. As a special solution, (2.2.2) has a similarity solution of t he form (2.2.3) V(t,x) =˜V(ξ), ξ= (x−x0)/√ 1+t. wherex0is any shift constant(cf. [2]). We denote ˜Vstill byV. Lemma 2.2.1 The similarity solution Vto (2.2.2) 1satisfies the following: (2.2.4) |V(t,x)−v−| ≤Cδ0e−c(x−x0)2 1+t,(t,x)∈R+×R−, |V(t,x)−v+| ≤Cδ0e−c(x−x0)2 1+t,(t,x)∈R+×R+, |Vx(t,x)| ≤Cδ0(1+t)−1/2e−c(x−x0)2 1+t,(t,x)∈R+×R, |Vxx(t,x)| ≤Cδ0(1+t)−1e−c(x−x0)2 1+t,(t,x)∈R+×R, |Vxxx(t,x)| ≤Cδ0(1+t)−3/2e−c(x−x0)2 1+t,(t,x)∈R+×R, whereδ0=|v+−v−|. 6SinceV=V(ξ)→v±(ξ→ ±∞),U=µ(V)Vx→0 asx→ ±∞. On the other hand, u(0,x) =u0(x)→u±(x→ ±∞), and so we must introduce the correction function (ˆ v,ˆu), samely as in the preceding subsection. We expect u(t,x)∼e−αtu±asx→ ±∞by (2.1) 2, and so we determine (ˆ v,ˆu) as (2.2.5) ˆvt−ˆux= 0, ˆut+αˆu= 0, (ˆv,ˆu)(t,x)→(0,e−αtu±), x→ ±∞. Thus, we define (ˆ v,ˆu) by (2.2.6) ˆu(t,x) =e−αt(u−+(u+−u−)/integraltextx −∞m0(y)dy) =:e−αtM0(x), ˆv=∂ ∂x(e−αt −αM0(x)) =e−αt −α{(u+−u−)m0(x)−α′(x)M0(x)(1 α(x)+t)}, wherem0∈C∞ 0(R) with/integraltext∞ −∞m0(y)dy= 1. By (2.1), (2.2.1) and (2.2.5), (2.2.7)/braceleftBigg (v−V−ˆv)t−(u−U−ˆu)x= 0, (u−U−ˆu)t+(p(v)−p(V))x+α(u−U−ˆu) =−{Ut+(α−α)U}. Sinceu−U−ˆu→0 is expected as x→ ±∞, integrating (2.2.7) 1over [0,t]×R, we have /integraldisplay∞ −∞(v−V−ˆv)(t,x)dx=/integraldisplay∞ −∞(v0(x)−V(x−x0))dx+1 α(u+−u−). For a given v0(x),/integraltext∞ −∞(v0(x)−V(x−x0))dx→ ±∞or→ ∓∞(x0→ ±∞), and so we can choose x0as (2.2.8)/integraldisplay∞ −∞(v0(x)−V(x−x0))dx=−1 α(u+−u−), (setx0= 0 without loss of generality) so that, for any t≥0, /integraldisplay∞ −∞(v−V−ˆv)(t,x)dx= 0. Thus, if we define φby (2.2.9) φ(t,x) =/integraldisplayx −∞(v(t,y)−V(y√1+t)−ˆv(t,y))dy, then (φx,φt) = (v−V−ˆv,u−U−ˆu) and φtt+(p(V+ ˆv+φx)−p(V))x+αφt=−{Ut+(α−α)U} or, as in the previous subsection, (2.2.10)φtt+(p(V+ ˆv+φx)−p(V+ ˆv))x+αφt =−{Ut+(α−α)U+(p(V+ ˆv)−p(V))x}=:G=:3/summationdisplay i=1Gi, 7which is our reformulated problem in the present case. Forma l reformulation is almost similar to the case around the constant state. However, since V(±∞) =v±, v+/ne}ationslash=v−, the energy estimates of φare rather different from those in case of v+=v−. Our aim is to derive /bardbl(φx,φt)(t)/bardbl=o(1,t−1/4) and, if possible, |(φx,φt)(t)|∞=o(1,t−1/2) ast→ ∞, so that ( V,U+ ˆu) becomes an asymptotic profileof( v,u) inL2andL∞sense, respectively, since(ˆ v,ˆu) decays rapidly. Henceitis importantto show the existence of global-in-time solution φto (2.2.10), combining the existence of local-in-time solution and the a priori estimates with decay rates. For the given data (2.2.11) ( φ,φt)(0) = (φ0,φ1)∈H3×H2, there exists a unique local-in-time solution φ∈ ∩2 i=0Ci([0,t0];H3−i) to (2.2.10) (for the proof, see Matsumura [7]). As usual, if we can have a priori estimates /bardbl(φ,φt)(t)/bardbl2 H3×H2≤C(/bardblφ0,φ1/bardbl2 H3×H2+δ1), δ1=|v+−v−|+|u+−u−| for the solution φ∈ ∩2 i=0Ci([0,T];H3−i) with some decay properties, then continuation arguments are well done and global-in-time solution φis obtained, which may satisfies desired decay rates. But, in this case it seems to be hopeless. In fact, to get the bo undedness of /bardblφ(t)/bardbl, multiplying (2.2.10) by φand integrating it over [0 ,t]×R, we have /integraldisplay (αφ2+φtφ)dx+/integraldisplayt 0/integraldisplay (p(ˆV)−p(ˆV+φx))φxdxdτ≤C(/bardblφ0,φ1/bardbl2+/integraldisplayt 0/bardblφt/bardbl2dτ+/integraldisplayt 0/integraldisplay Gφdxdτ ), or /bardblφ(t)/bardbl2+/integraldisplayt 0/bardblφx(τ)/bardbl2dτ≤C(/bardblφ0,φ1/bardbl2+/integraldisplayt 0/bardblφt(τ)/bardbl2dτ+/integraldisplayt 0|/integraldisplay Gφdx|dτ). whereˆV=V+ ˆv. For an example, in the final term we need to estimate G2as /integraldisplayt 0|/integraldisplay (α−α)Uφdx|dτ≤/integraldisplayt 0|U|∞|φ|∞|α−α|1dτ≤Cδ0/integraldisplayt 0(1+τ)−1/2/bardblφ(τ)/bardbl1/2/bardblφx(τ)/bardbl1/2dτ, and to cover this by good terms, which seems to be impossible, even if the other terms can be well-evaluated. However, fortunately there is not φitself in the nonlinearity of (2.2.10), and so, for the conti n- uation arguments the uniform estimate in [0 ,T] of/bardbl(φx,φt)(t)/bardbl2is necessary, while growing up of /bardblφ(t)/bardblmay be permitted. Thus, our a priori estimate is the following. Proposition 2.2.1 (A priori estimate) Assume that /braceleftBigg p∈C3(R+), p′(v)<0(v >0),andα∈C1(R), α−α∈L1∩L2,αx∈L2 withα≥α0>0(α0:constant). Then there exist positive constants ε0andC0such that, if δ1:=|v+−v−|+|u+−u−| ≤ε0and if φ∈ ∩2 i=0Ci([0,T];H3−i)is a solution to (2.2.10)-(2.2.11) for some T >0, which satisfies sup 0≤t≤T{(1+t)−γ/2/bardblφ(t)/bardbl+/bardbl(φt,φx)(t)/bardbl2+(1+t)1/2/bardbl(φtxx,φxxx)(t)/bardbl} ≤ε0, 8then it holds that (2.2.12)(1+t)−γ/bardblφ(t)/bardbl2+(1+t)1−γ/bardblφx(t)/bardbl2+(1+t)/bardbl(φt,φtt,φxx)(t)/bardbl2 H2×H1×H1 +/integraltextt 0{(1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+(1+τ)1−γ/bardblφt(τ)/bardbl2 +(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2+/bardbl(φttx,φtxx)(τ)/bardbl2}dτ ≤C0(/bardblφ0,φ1/bardbl2 H3×H2+δ1),0≤t≤T. Proposition 2.2.1 yields the following theorem, together w ith the local existence theorem. Theorem 2.2.1 Under the assumption in Proposition 2.2.1, if both /bardblφ0,φ1/bardblH3×H2andδ1= |v+−v−|+|u+−u−|are sufficiently small, then there exists a unique global-in- time solution φ∈ ∩2 i=0Ci([0,∞);H3−i)to (2.2.10)-(2.2.11), which satisfies decay property (2.2. 12) for0≤t < ∞. As same as in the preceding subsection, once we have Theorem 2 .2.1, if we define ( v,u) by (v,u) = (V+ ˆv+φx,U+ ˆu+φt), then we can have a unique solution ( v,u) of (2.1) satisfying (v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2). Also, since /bardblφx(t)/bardbl ≤C(1+t)−(1−γ)/2,|φx(t)|∞≤C/bardblφx(t)/bardbl1/2/bardblφxx(t)/bardbl1/2≤C(1+t)−(2−γ)/2 and/bardblφt(t)/bardbl ≤C(1+t)−1/2, the similarity solution ( V,U+ ˆu) is an asymptotic profile of ( v,u) in L2-sense, and Vis so ofvinL∞-sense. Thus, our main theorem is the following. Theorem 2.2.2 (Main Theorem) Under the assumption in Proposition 2.2.1, suppose that v0−V∈L1,/integraldisplay· −∞(v0(y)−V(0,y))dy∈L2,(v0−V(0,·),u0−U(0,·)−ˆu(0,·))∈H2×H2. If both/bardbl/integraltext· −∞(v0(y)−V(0,y))dy/bardbl+/bardblv0−V(0,·),u0−U(0,·)−ˆu(0,·)/bardblH2×H2andδ1=|v+− v−|+|u+−u−|are suitably small, then there exists a unique solution (v,u)to (1.1) satisfying (v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2), which satisfies, as t→ ∞, (v−V,u−U−ˆu)(t,x) =O(t−(1−γ)/2,t−1/2)inL2-sense, and (v−V)(t,x) =O(t−(2−γ)/2)inL∞-sense with1/2< γ <1. 3 A priori estimate around the constant state We prove Proposition 2.1.1 in a series of several steps. To do that, let φbe a smooth solution in ∩2 i=0Ci([0,T];H3−i) for some T >0 to (2.1.12) and rewrite (2.1.12) as (3.1) φtt+(p(ˆV+φx)−p(ˆV))x+αφt=F, whereˆV:=V+ ˆvand (3.2) F=−{Ut+(α−α)U+(p(V)−p′(v)V)x+(p(V+ ˆv)−p(V))x}=:4/summationdisplay i=1Fi. 9Also, denote the a priori assumption as δ= sup 0≤t≤T{/bardbl(φ,φt)(t)/bardblH3×H2+(1+t)/bardbl(φtx,φtxx)(t)/bardbl}(≤1) together with δ1=|v0−v|1+|u+−u−| ≥ |δ0|+(1+1 α)|u+−u−| and I0=/bardblφ0,φ1/bardbl2 H3×H2+δ1(≤1), whereδ0is defined in (2.1.9). Step 1.First we multiply (3.1) by φtand integrate it over R: (3.3)d dt/integraltext1 2φ2 tdx+/integraltext αφ2 tdx+/integraltext (p(ˆV)−p(ˆV+φx))φtxdx ≤/integraltext |φtF|dx≤ε/integraltext φ2 tdx+Cε/integraltext F2dx for small constant ε >0. The integral domain Rand variable tare often abbreviated. The 3-rd term is estimated as /integraltext (p(ˆV)−p(ˆV+φx))φtxdx =/integraltext (/integraltextφx 0(p(ˆV)−p(ˆV+s))ds)tdx+/integraltext (p(ˆV+φx)−p(ˆV)−p′(V)φx)ˆVtdx ≥d dt/integraltext (/integraltextφx 0(p(ˆV)−p(ˆV+s))ds)dx−C(|Vt|∞+|ˆvt|∞)/integraltext φ2 xdx ≥d dt/integraltext (/integraltextφx 0(p(ˆV)−p(ˆV+s))ds)dx−Cδ1δ(1+t)−3/2. The estimates of Fare as follow, including those necessary. Lemma 3.1 (Estimates of F)It holds that, for t∈R+ (3.4) /bardblF(t)/bardbl2≤Cδ1(1+t)−5/2, /bardblFx(t)/bardbl2≤Cδ1(1+t)−3, /bardblFt(t)/bardbl2≤Cδ1(1+t)−9/2, /bardblFxt(t)/bardbl2≤Cδ1(1+t)−5. The proof is given in Appendix. Thus, integrating (3.3) over [0 ,t], we have (3.5) /bardbl(φt,φx)(t)/bardbl2+/integraldisplayt 0/bardblφt(τ)/bardbl2dτ≤C/bardblφ0x,φ1/bardbl2+C(δ1+δδ1)≤CI0. Next, multiplying (3.1) by φ, we have d dt/integraltext (α 2φ2+φφt)dx−/integraltext φ2 tdx+/integraltext (p(ˆV)−p(ˆV+φx))φxdx ≤/integraltext |φF|dx≤sup 0≤t≤T/bardblφ/bardbl/bardblF/bardbl ≤Cδ1(1+t)−5/4, 10and hence, using (3.5), (3.6)/bardblφ(t)/bardbl2+/integraltextt 0/bardblφx(τ)/bardbl2dτ≤C(/bardblφ0,φ0x,φ1/bardbl2+/bardblφt(t)/bardbl2+/integraltextt 0/bardblφt(τ)/bardbl2dτ)+Cδ1 ≤CI0. Using (3.6), we can multiply (3.3) by 1+ tand get (3.7) /bardblφ(t)/bardbl2+(1+t)/bardbl(φt,φx)(t)/bardbl2+/integraldisplayt 0(/bardblφx(τ)/bardbl2+(1+τ)/bardblφt(τ)/bardbl2)dτ≤CI0. We repeat the similar procedure to Step 1 for higher derivati ves ofφ. Step 2.Differentiate (3.1) in t: (3.8) φttt+αφtt+(p′(ˆV+φx)φxt+(p′(ˆV+φx)−p′(ˆV))ˆVt)x=Ft. Multiplying (3.8) by φttand integrating it over R, we have (3.9)d dt/integraltext (1 2φ2 tt+−p′(ˆV+φx) 2φ2 xt)dx+/integraltext αφ2 ttdx ≤Cδ1{(1+t)−4/bardblφx/bardbl2+(1+t)−3/bardblφxx/bardbl2+(1+t)−9/2}+C(δ+δ1)/bardblφxt/bardbl2, because of (3.4) 3, where the third term of (3.8) is estimated as /integraltext (−p′(ˆV+φx)φxtφxtt+{(p′(ˆV+φx)−p′(ˆV))ˆVxt+p′′(ˆV+φx)ˆVtφxx +(p′′(ˆV+φx)−p′′(ˆV))ˆVxˆVt}φtt)dx ≥d dt/integraltext−p′(ˆV+φx) 2φ2 xtdx+/integraltextp′′(ˆV+φx) 2(ˆVt+φxt)φ2 xtdx −ε/integraltext φ2 ttdx−Cε/integraltext (ˆV2 xtφ2 x+ˆV2 tφ2 xx+ˆV2 xˆV2 tφ2 x)dx (0< ε≪1) and |/integraldisplayp′′(ˆV+φx) 2(ˆVt+φxt)φ2 xtdx| ≤C(δ1(1+t)−3/2+δ)/bardblφxt/bardbl2≤C(δ1+δ)/bardblφxt/bardbl2 by the a priori assumption. Also, the product of (3.8) and φtyields (3.10)d dt/integraltext (α 2φ2 t+φtφtt)dx+c 2/integraltext φ2 xtdx ≤/integraltext φ2 ttdx+Cδ1{(1+t)−9/4/bardblφt/bardbl2+(1+t)−3/bardblφx/bardbl2}, by|/integraltext φtFtdx| ≤C/bardblφt/bardbl/bardblFt/bardbl ≤Cδ1(1+t)−9/4/bardblφt/bardbl. By calculating (3.9)+ λ·(3.10) (0 < λ≪1), (3.11)d dt/integraltext {(1 2φ2 tt+λφttφt+λα 2φ2 t)+−p′(ˆV+φx) 2φ2 xt}dx+/integraltext (α 2φ2 tt+λc 2φ2 tx)dx ≤C(δ+δ1(1+t)−1)/bardblφxt/bardbl2 +Cδ1{(1+t)−3/bardblφx/bardbl2+(1+t)−9/4/bardblφt/bardbl2+(1+t)−9/2+(1+t)−3/bardblφxx/bardbl2}. 11Whenδ+δ1≪1, integrating (3.11), (1+ t)·(3.11) and (1+ t)2·(3.11) over [0 ,t], yields (3.12) (1+ t)/bardbl(φtt,φtx,φt)(t)/bardbl2+/integraldisplayt 0(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2dτ≤CI0 and (3.13) (1+ t)2/bardbl(φtt,φtx,φt)(t)/bardbl2+/integraldisplayt 0(1+τ)2/bardbl(φtt,φtx)(τ)/bardbl2dτ≤CI0+Cδ1/integraldisplayt 0/bardblφxx(τ)/bardbl2dτ. Step 3.In a similar fashion to x-derivative of (3.1), we have (3.14)d dt/integraltext (1 2φ2 tx+−p′(ˆV+φx) 2φ2 xx)dx+c/integraltext φ2 txdx ≤Cδ1{(1+t)−3/bardblφx/bardbl2+(1+t)−3}+C/bardblφt/bardbl2+Cδ/bardblφxx/bardbl2. Here, since we have differentiate (3.1) in x, the additional term αxφtcomes out and |/integraldisplay αxφtφtxdx| ≤ε/bardblφtx/bardbl2+Cε/bardblαx/bardbl/bardblαxx/bardbl·/bardblφt/bardbl2. Though we omit the estimates of the other terms, integrating (3.14) and (1 + t)·(3.14) over [0 ,t] and using (3.7), we have (3.15) (1+ t)/bardbl(φtx,φxx)(t)/bardbl2+/integraldisplayt 0(1+τ)/bardblφtx(τ)/bardbl2dτ≤CI0+Cδ/integraldisplayt 0/bardblφxx(τ)/bardbl2dτ. In (3.13) and (3.15), the final terms are not estimated yet. Step 4.Multiplying the variant of (3.1) (3.1)′φtt+αφt+p′(ˆV+φx)φxx=F+(p′(ˆV)−p′(ˆV+φx))ˆVx by−φxxand integrating it over R, we obtain /integraldisplay φ2 xxdx≤ −d dt/integraldisplay φxφxtdx+/integraldisplay φ2 xtdx+C/integraldisplay φ2 tdx+Cδ1{(1+t)−2/bardblφx/bardbl2+(1+t)−5/2} and, also integrating over [0 ,t], (3.16)/integraldisplayt 0/bardblφxx(τ)/bardbl2dτ≤CI0+/bardbl(φt,φxt)(t)/bardbl2+/integraldisplayt 0/bardbl(φt,φxt)(τ)/bardbl2dτ≤CI0 by (3.12). Applying (3.16) to (3.13) and (3.15) with δ≪1, we get (3.17)(1+t)2/bardbl(φtt,φtx,φt)(t)/bardbl2+(1+t)/bardbl(φx,φxx)(t)/bardbl2+/bardblφ(t)/bardbl2 +/integraltextt 0{(1+τ)2/bardbl(φtt,φtx)(τ)/bardbl2+(1+τ)/bardblφt(τ)/bardbl2+/bardbl(φxx,φx)(τ)/bardbl2}dτ≤CI0. By (3.17), integration of (1 + t)2(−φxx)·(3.1)′inRand (1 + t)(−φxx)·(3.1)′over [0,t]×Ralso yield (3.18) (1+ t)2/bardblφxx(t)/bardbl2+/integraldisplayt 0(1+τ)/bardblφxx(τ)/bardbl2dτ≤CI0. 12Step 5. Though the details are omitted, the combination of∂2 ∂t∂x(3.1)·φttx,∂ ∂t(3.1)·(−φxxt) and ∂ ∂x(3.1)·(−φxxx) yields (3.19) (1+ t)2/bardbl(φttx,φtxx,φxxx)(t)/bardbl2+/integraldisplayt 0{(1+τ)2/bardblφtxx(τ)/bardbl2+(1+τ)/bardblφxxx(τ)/bardbl2}dτ≤CI0. Thus, we have obtained (2.17) by (3.17) - (3.19) and complete d the proof of Proposition 2.1. 4 Proof of Proposition 2.2.1 in different end states case Letφbe a smooth solution in ∩2 i=0Ci([0,T];H3−i) for some T >0 to the reformulation problem (2.2.10)-(2.2.11), which is re-written as (4.1)/braceleftBigg φtt+(p(ˆV+φx)−p(ˆV))x+αφt=G, (φ,φt)(0) = (φ0,φ1)∈H3×H2, whereˆV=V+ ˆvand (4.2) G=3/summationdisplay i=1Gi=−{Ut+(α−α)U+(p(V+ ˆv)−p(V))x}. To get the a priori estimate (2.2.12), denote the a priori ass umption as δ= sup 0≤t≤T{(1+t)−γ/2/bardblφ(t)/bardbl+/bardbl(φt,φx)(t)/bardbl2+(1+t)1/2/bardbl(φtxx,φxxx)(t)/bardbl}(≤1), together with I0=/bardblφ0,φ1/bardbl2 H3×H2+δ1(δ1=|v+−v−|+|u+−u−|). Note that the notations δandδ1in Sections 3 and 4 are slightly different from each other. Same as in the preceding section, the proof is given in a serie s of several steps. Step 1. Multiply (4.1) by φtand integrate it over R: (4.3)d dt/integraldisplay1 2φ2 tdx+/integraldisplay αφ2 tdx−/integraldisplay (p(ˆV+φx)−p(ˆV))φxtdx=/integraldisplay φtGdx and (3-rd term) =d dt/integraltext /integraltextφx 0(p(ˆV)−p(ˆV+s))dsdx+/integraltext (p(ˆV+φx)−p(ˆV)−p′(ˆV)φx)ˆVtdx ≥d dt/integraltext /integraltextφx 0(p(ˆV)−p(ˆV+s))dsdx−Cδ1(1+t)−1/integraltext φ2 xdx. Hence, integrating (4.3) over [0 ,t], we have 1 2/bardblφt(t)/bardbl2+/integraltext /integraltextφx 0(p(ˆV)−p(ˆV+s))dsdx+/integraltextt 0/integraltext αφ2 tdxdτ ≤C/bardblφ0,φ1/bardbl2 H1×L2+Cδ1/integraltextt 0(1+τ)−1/bardblφx(τ)/bardbl2dx+/integraltextt 0/integraltext φtGdxdt, 13or (4.4)/bardbl(φt,φx)(t)/bardbl2+/integraldisplayt 0/bardblφt(τ)/bardbl2dτ≤CI0+Cδ1/integraldisplayt 0(1+τ)−1/bardblφx(τ)/bardbl2dx+C/integraldisplayt 0/integraldisplay φtGdxdt. Next, multiply (4.1) by (1+ t)−γφ(1/2< γ <1) and integrate it over [0 ,t]×Rto get (4.5)(1+t)−γ/integraltext (α 2φ2+φφt)dx+γ/integraltextt 0(1+τ)−γ−1/integraltext (φ2+φφt)dxdτ+c/integraltextt 0(1+τ)−γ/integraltext φ2 xdxdτ ≤CI0+/integraltextt 0(1+τ)−γ/integraltext φ2 tdxdτ+C/integraltextt 0/integraltext (1+τ)−γφGdxdτ. Because /integraldisplay α(1+t)−γφφtdx=d dt(1+t)−γ/integraldisplayα 2φ2dx+αγ 2(1+t)−γ−1/integraldisplay φ2dx, /integraldisplay (1+t)−γφφttdx=d dt(1+t)−γ/integraldisplay φφtdx+γ(1+t)−γ−1/integraldisplay φφtdx−(1+t)−γ/integraldisplay φ2 tdx, and −/integraldisplay (1+t)−γ(p(ˆV+φx)−p(ˆV))φxdx≥c(1+t)−γ/integraldisplay φ2 xdx. Adding (4.4) to λ·(4.5) (0< λ≪1, in particular, Cδ1λ≤c), we have (4.6)(1+t)−γ/bardblφ(t)/bardbl2+/bardbl(φt,φx)(t)/bardbl2 +/integraltextt 0((1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+/bardblφt(τ)/bardbl2)dτ ≤CI0+C/integraltextt 0(/integraltext φtGdx+(1+τ)−γ/integraltext φGdx)dτ. Itisnecessarytoestimatethelasttermcarefully. Estimat eeachterm, usingtheaprioriassumption (2.2.13) with δ0=|v+−v−|: (i)/integraltext φtG1dx≤Cδ0/integraltext φ2 tdx+Cδ0/integraltext (1+τ)−3e−2cx2 1+τdx≤Cδ0(/bardblφt(τ)/bardbl2+(1+τ)−5/2) by|Ut| ≤C(|Vx||Vt|+|Vxt|, (ii)/integraltext φtG2dx=d dt/integraltext (α−α)φU dx+/integraltext (α−α)φUtdx with |/integraltext (α−α)φU dx| ≤C|U|∞/bardblφ/bardbl/bardblα−α/bardbl ≤Cδ0δ(1+t)−(1−γ)/2≤Cδ0, |/integraltext (α−α)φUtdx| ≤ |φ|∞|Ut|∞|α−α|1≤Cδ0(1+τ)−3/2/bardblφ/bardbl1/2/bardblφx/bardbl1/2 ≤Cδ0(1+τ)(2γ−5)/4·(1+τ)−(γ+1)/4/bardblφ/bardbl1/2·(1+τ)−γ/4/bardblφx/bardbl1/2 ≤Cδ0{(1+τ)−(5−2γ)/2+(1+τ)−(γ+1)/bardblφ/bardbl2+(1+τ)−γ/bardblφx/bardbl2}, (iii)|(1+τ)−γ/integraltext φG1dx| ≤Cδ0/integraltext |φ|(1+τ)−γ−3/2e−cx2 1+τdx ≤Cδ0((1+τ)−γ/bardblφ/bardbl2)1/2(/integraltext (1+τ)−γ−3e−2cx2 1+τdx)1/2≤Cδ0δ(1+τ)−(2γ+5)/4, (iv)|(1+τ)−γ/integraltext φ(α−α)U dx| ≤(1+τ)−γ|φ|∞|α−α|1|U|∞ ≤Cδ0(1+τ)−γ−1/2/bardblφ/bardbl1/2/bardblφx/bardbl1/2 =Cδ0(1+τ)−(2γ+1)/4·(1+τ)−(γ+1)/4/bardblφ/bardbl1/2·(1+τ)−γ/4/bardblφx/bardbl1/2 ≤Cδ0{(1+τ)−γ−1/2+(1+τ)−γ−1/bardblφ/bardbl2+(1+τ)−γ/bardblφx/bardbl2}, 14and (v) since |G3| ≤C|u+−u−|e−ct,/integraltext (φtG3+(1+τ)−γφG3)dxis well estimated. Thus, the final term in (4.6) is absorbed in the left-hand side ifδ1=δ0+|u+−u−|is small, and we have (4.7)(1+t)−γ/bardblφ(t)/bardbl2+/bardbl(φt,φx)(t)/bardbl2 +/integraltextt 0((1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+/bardblφt(τ)/bardbl2)dτ≤CI0. Step 2.We want to have decay properties of /bardbl(φt,φx)(t)/bardbland modify Step 1. In fact, multiplying (4.1) by (1+ t)1−γφtand integrating it over [0 ,t]×R, we have (4.8)(1+t)−γ/bardblφ(t)/bardbl2+(1+t)1−γ/bardbl(φt,φx)(t)/bardbl2 +/integraltextt 0((1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+(1+τ)1−γ/bardblφt(τ)/bardbl2)dτ ≤CI0, together with (4.6). Note that 1 /2< γ <1, and so ( φt,φx)(t) decays in L2-sense. Let us derive (4.8). Multiplying (4.3) by (1+ t)1−γyields d dt/integraltext1 2(1+t)1−γφ2 tdx−/integraltext1−γ 2(1+t)−γφ2 tdx+/integraltext α(1+t)1−γφ2 tdx +d dt/integraltext (1+t)1−γ/integraltextφx 0(p(ˆV)−p(ˆV+s))dsdx−(1−γ)/integraltext (1+t)−γ/integraltextφx 0(p(ˆV)−p(ˆV+s))dsdx +(1+t)1−γ/integraltext (p(ˆV+φx)−p(ˆV)−p′(ˆV)φx)ˆVtdx = (1+t)1−γ/integraltext φtGdx, and hence d dt(1+t)1−γ{/integraltext1 2φ2 tdx+/integraltext /integraltextφx 0(p(ˆV)−p(ˆV+s))dsdx}+α0(1+t)1−γ/integraltext φ2 tdx ≤C(1+t)−γ/bardbl(φt,φx)(t)/bardbl2+(1+t)1−γCδ1(1+t)−1/bardblφx(t)/bardbl2+C(1+t)1−γ/integraltext φtGdx. Here, the final term is estimated as follows and absorbed in th e left-hand side: (vi) (1+ t)1−γ/integraltext φtG1dx≤(1+t)1−γ/bardblφt|/bardblUt/bardbl ≤Cδ0{/bardblφt(t)/bardbl2+(1+t)−γ−3/2} (vii)(1+t)1−γ/integraltext φtG2dx=d dt(1+t)1−γ/integraltext (α−α)φU dx −(1−γ)(1+t)−γ/integraltext (α−α)φU dx−(1+t)1−γ/integraltext (α−α)φUtdx, with (1+t)1−γ/integraltext (α−α)φU dx≤(1+t)1−γ|φ|∞|U|∞|α−α|1 ≤Cδ0(1+t)−γ+1/2/bardblφ/bardbl1/2/bardblφx/bardbl1/2 =Cδ0(1+t)−(2γ−1)/4·((1+t)(1−γ)/2/bardblφx/bardbl)1/2·((1+t)−γ/2/bardblφ/bardbl)1/2 ≤Cδ0{(1+t)−γ/bardblφ(t)/bardbl2+(1+t)1−γ/bardblφx(t)/bardbl2+(1+t)−(2γ−1)/2} and the same estimates as (iii), (iv). (viii) For G3, same as (v). 15Therefore, using (4.7), we have (4.8). We further need to evaluate higher derivatives of φ. Step 3.Differentiate (4.1) in t: (4.9) φttt+αφtt+(p′(ˆV+φx)φxt+(p′(ˆV+φx)−p′(ˆV))ˆVt)x=Gt. We multiply (4.9) by φt,φttand (1+ t)φt, (1+t)φtt. First, estimate/integraltext (4.9)·φtdx: (4.10)d dt/integraltext (φttφt+α 2φ2 t)dx+/integraltext |p′(ˆV+φx)|φ2 xtdx −/integraltext (p′(ˆV+φx)−p′(ˆV))ˆVtφtxdx=/integraltext φtGtdx. Since|ˆVt|∞≤Cδ1(1+t)−1≤Cδ1(1+t)−γ, |3-rd term | ≤Cδ1/bardblφtx(t)/bardbl2+Cδ1(1+t)−γ/bardblφx(t)/bardbl2. It is easy to show (4.11) /bardblGt/bardbl ≤Cδ1(1+t)−3/2 and|/integraltext φtGtdx| ≤Cδ1(1+t)1−γ/bardblφt(t)/bardbl2+Cδ1(1+t)−(4−γ), so that (4.12)/integraldisplayt 0/bardblφtx(τ)/bardbl2dτ≤CI0+C(/bardblφtt(t)/bardbl2+/integraldisplayt 0/bardblφtt(τ)/bardbl2dτ) by (4.8). Secondly, multiplying (4.9) by φtt, (4.13)d dt/integraltext1 2φ2 ttdx+/integraltext αφ2 ttdx−/integraltext p′(ˆV+φx)φxtφxttdx +/integraltext ((p′(ˆV+φx)−p′(ˆV))ˆVt)xφttdx=/integraltext φttGtdx, and (3-rd term) =d dt/integraltext|p′(ˆV+φx)| 2φ2 xtdx+/integraltextp′′(ˆV+φx) 2(ˆVt+φxt)φ2 xtdx ≥d dt/integraltext|p′(ˆV+φx)| 2φ2 xtdx−C(δ1+δ)/bardblφxt(t)/bardbl2, |4-th term | ≤Cδ1/bardblφtt(t)/bardbl2+Cδ1/integraltext {(p′(ˆV+φx)−p′(ˆV))ˆVxt +p′′(ˆV+φx)φxxˆVt+(p′′(ˆV+φx)−p′′(ˆV))ˆVtˆVx}2dx ≤Cδ1/bardblφtt(t)/bardbl2+Cδ1{(1+t)−3/bardblφx(t)/bardbl2+(1+t)−2/bardblφxx(t)/bardbl2}. Here, by (4.1) φxx= (−p′(ˆV+φx))−1(φtt+αφt+(p′(ˆV+φx)−p′(ˆV))ˆVx−G) and hence (4.14) (1+ t)−2/bardblφxx(t)/bardbl2≤C(1+t)−2(/bardbl(φt,φtt)(t)/bardbl2+/bardblG/bardbl2)+Cδ1(1+t)−3/bardblφx(t)/bardbl2. Since (4.15) /bardblG/bardbl2≤C(/bardblUt/bardbl2+|U|2 ∞/bardblα−α/bardbl2+|u+−u−|e−ct)≤Cδ1(1+t)−1 16and also |/integraldisplay φttGtdx| ≤ε/bardblφtt(t)/bardbl2+Cε/integraldisplay G2 tdx≤ε/bardblφtt(t)/bardbl2+Cδ1(1+t)−3, all bad terms are absorbed, and integration of (4.13) over [0 ,t] yields (4.16) /bardbl(φtt,φtx)(t)/bardbl2+/integraldisplayt 0/bardblφtt(τ)/bardbl2dτ≤CI0+C(δ1+δ)/integraldisplayt 0/bardblφxt(τ)/bardbl2dτ. Inserting (4.12) to (4.16), we have (4.17) /bardbl(φtt,φtx)(t)/bardbl2+/integraldisplayt 0/bardbl(φtt,φtx(τ)/bardbl2dτ≤CI0, provided that δ1+δis suitably small. Thirdly, based on (4.17), calculate (1+ t)·(4.10): (4.18)d dt(1+t)/integraltext (φttφt+α 2φ2 t)dx+(1+t)/integraltext |p′(ˆV+φx)|φ2 xtdx ≤/integraltext (φttφt+α 2φ2 t)dx+(1+t)/integraltext (φ2 tt+(p′(ˆV+φx)−p′(ˆV))ˆVtφtx)dx +(1+t)/integraltext φtGtdx ≤C/integraltext φ2 tdx+(1+t)/integraltext (φ2 tt+εφ2 tx)dx +Cε(1+t)|ˆV|2 ∞/integraltext φ2 xdx+C(1+t)1−γ/bardblφt(t)/bardbl2+C(1+t)1+γ/bardblGt/bardbl2 and (1+t)/integraldisplay φttφtdx≤ε(1+t)/integraldisplay φ2 tdx+Cε(1+t)/integraldisplay φ2 ttdx. Hence, integrating (4.18) over [0 ,t] and using (4.8), (4.11), we have (4.19)(1+t)/bardblφt(t)/bardbl2+/integraltextt 0(1+τ)/bardblφxt(τ)/bardbl2dτ ≤CI0+C(1+t)/bardblφtt(t)/bardbl2+C/integraltextt 0(1+τ)/bardblφtt(τ)/bardbl2dτ. Finally, multiplication of (4.13) by 1+ tand integration over [0 ,t] yield (1+t)/bardbl(φtt,φtx)(t)/bardbl2+/integraltextt 0(1+τ)/bardblφtt(τ)/bardbl2dτ ≤C/bardbl(φtt,φtx)(t)/bardbl2+C(δ1+δ)/integraltextt 0(1+τ)/bardblφtx(τ)/bardbl2dτ +Cδ1/integraltextt 0((1+τ)/bardblφtt(τ)/bardbl2+(1+τ)−2/bardblφx(τ)/bardbl2+(1+τ)−1/bardblφxx(τ)/bardbl2)dτ. By (4.14), /bardblφxx(t)/bardbl2≤C/bardbl(φt,φtt)(t)/bardbl2+Cδ1(1+t)−1/bardblφx(t)/bardbl2+Cδ1(1+t)−1. Therefore, by (4.17) we have (4.20)(1+t)/bardbl(φtt,φtx)(t)/bardbl2+/integraltextt 0(1+τ)/bardblφtt(τ)/bardbl2dτ ≤CI0+C(δ1+δ)/integraltextt 0(1+τ)/bardblφtx(τ)/bardbl2dτ. Substituting (4.20) into (4.19), we obtain (4.21) (1+ t)/bardbl(φt,φtt,φtx)(t)/bardbl2+/integraldisplayt 0(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2dτ≤CI0, 17provided that δ1+δis small. Additionally, we have (4.22) (1+ t)/bardblφxx(t)/bardbl2≤CI0, because, by (4.14) and (4.15) with (4.21), (1+t)/bardblφxx(t)/bardbl2≤C(1+t)/bardbl(φt,φtt)(t)/bardbl2+Cδ1/bardblφx(t)/bardbl2+Cδ1. Step 4.In the final step we estimate the third order derivatives of φ. To do so, differentiate (4.1) inxandx,t: (4.23)φttx+αxφt+αφtx+(p′(ˆV+φx)φxx)x =−((p′(ˆV+φx)−p′(ˆV))x+Gx, and (4.24)φtttx+αxφtt+αφttx+(p′(ˆV+φx)φtxx)x =−(p′′(ˆV+φx)(ˆVt+φtx)φxx)x−((p′(ˆV+φx))ˆVx)tx+Gtx =:H+Gtx=:h1+h2+Gtx. First, multiplying (4.24) by φttxand integrating it over R, we have d dt/integraltext (1 2φ2 ttx+|p′(ˆV+φx)| 2φ2 txx)dx+/integraltext αφ2 ttxdx ≤ε/integraltext φ2 ttxdx+Cε/integraltext φ2 ttdx+C|ˆVt+φtx|∞/integraltext φ2 txxdx+Cε/integraltext (H2+G2 tx)dx, which derives (4.25)d dt/integraltext (1 2φ2 ttx+|p′(ˆV+φx)| 2φ2 txx)dx+α0/integraltext φ2 ttxdx ≤C{(δ1+δ)/integraltext φ2 txxdx+/integraltext φ2 ttdx+/bardblH/bardbl2+/bardblGtx/bardbl2}, since (4.26) |ˆVt+φtx|∞≤C(δ1+/radicalbig I0δ)(1+t)−1≤C(δ1+δ)(1+t)−1 by (2.2.13), (4.21) and ε≪1. Next, multiplying (4.24) by φtx, similarly as above, we have (4.27)d dt/integraltext (φttxφtx+α 2φ2 tx)dx−/integraltext φ2 ttxdx+/integraltext |p′(ˆV+φx)|φ2 txxdx ≤C|αx|∞(/bardblφtt/bardbl2+/bardblφtx/bardbl2)+C(/bardblφtx/bardbl2+/bardblH/bardbl2+/bardblGtx/bardbl2) ≤C{/bardbl(φtt,φtx)(t)/bardbl2+/bardblH/bardbl2+/bardblGtx/bardbl2}. For small λ >0, add (4.25) to λ·(4.27), and then (4.28)d dt/integraltext {(1 2φ2 ttx+λφttxφtx+αλ 2φ2 tx)+|p′(ˆV+φx)| 2φ2 txx}dx +/integraltext (α0 2φ2 ttx+λ 2|p′(ˆV+φx)|φ2 txx)dx ≤C(/bardbl(φtt,φtx)(t)/bardbl2+/bardblH/bardbl2+/bardblGtx/bardbl2) provided that δ1+δ≤λ 2|p′(ˆV+φx)|. Here,/bardblH/bardbl2is estimated as the following. 18Lemma 4.1 (4.29)/bardblH/bardbl2≤C{(δ1+δ)(1+t)−1/bardbl(φttx,φtxx,φtx,φt)(t)/bardbl2 +(δ1+δ)(1+t)−2/bardblφxx(t)/bardbl2+δ1(1+t)−3/bardblφx(t)/bardbl2+δ1(1+t)−2}. Proof of Lemma 4.1 . First, in (4.24) h1=−p′′(ˆV+φx)(ˆVt+φtx)φxxx−p′′(ˆV+φx)(ˆVtx+φtxx)φxx −p′′′(ˆV+φx)(ˆVx+φxx)(ˆVt+φtx)φxx =:h11+h12+h13, with /bardblh11/bardbl2≤C(δ1+δ)(1+t)−1/bardblφxxx/bardbl2by (4.26) /bardblh12/bardbl2≤C(|ˆVtx|2 ∞/bardblφxx/bardbl2+/bardblφxx/bardbl/bardblφxxx/bardbl/bardblφtxx/bardbl2) ≤Cδ1(1+t)−3/bardblφxx/bardbl2+(I0+δ)(1+t)−1/bardblφtxx/bardbl2, /bardblh13/bardbl2≤C(|ˆVx|2 ∞+/bardblφxx/bardbl/bardblφxxx/bardbl)(|ˆVt|2 ∞+/bardblφtx/bardbl/bardblφtxx/bardbl)/bardblφxx/bardbl2 ≤C(δ1+δ)(1+t)−2/bardblφxx/bardbl2. For the estimate of /bardblφxxx/bardbl2in/bardblh11/bardbl2we back to (4.23): −p′(ˆV+φx)φxxx=p′′(ˆV+φx)(ˆVx+φxx)φxx+((p′(ˆV+φx)−p′(ˆV))ˆVx)x +φttx+αxφt+αφtx−Gx, and hence (4.30)/bardblφxxx/bardbl2≤C{(|ˆVx|2 ∞+/bardblφxx/bardbl/bardblφxxx/bardbl)/bardblφxx/bardbl2+(|ˆVxx|2 ∞+|ˆVx|4 ∞)/bardblφx/bardbl2 +|ˆVx|2 ∞/bardblφxx/bardbl2+/bardblφttx,φtx,φt/bardbl2+/bardblGx/bardbl2} ≤C{(I0+δ)(1+t)−1/bardblφxx/bardbl2+δ1(1+t)−2/bardblφx/bardbl2 +/bardblφttx,φtx,φt/bardbl2+δ1(1+t)−1}. Here,/bardblGx/bardbl2≤Cδ1(1+t)−1is easily seen. In a similar fashion to the above, we have /bardblh2/bardbl2≤Cδ1{(1+t)−1/bardblφtxx/bardbl2+(1+t)−2/bardblφtx,φxx/bardbl2+(1+t)−4/bardblφx/bardbl2}. Combining /bardblh1/bardbl2,/bardblh2/bardbl2with (4.30), we get (4.29) and complete the proof of Lemma 4.1 .q.e.d. We also note that (4.31) /bardblGtx/bardbl2≤Cδ1(1+t)−3. We now return back to (4.28). Take δ1+δas sufficiently small, then the term φtxxandφttxin (4.29) are absorbed into the left hand side, and integration of (4.28) over [0 ,t] yields (4.32) /bardbl(φttx,φtxx,φtx)(t)/bardbl2+/integraldisplayt 0/bardbl(φttx,φtxx)(τ)/bardbl2dτ≤CI0, because of (4.8), (4.21)-(4.22). 19To get further decay rate, we want to multiply (4.28) by 1+ t, but, if so, the final term in (4.29) is not integrable in t. So, we here use the technique found in Nishikawa [11], that i s, multiply (4.28) by (1+t)1+ν(ν >0,notν <0), so that, by (4.32), (4.33)(1+t)1+ν/bardbl(φttx,φtxx,φtx)(t)/bardbl2+/integraltextt 0(1+τ)1+ν/bardbl(φttx,φtxx)(τ)/bardbl2dτ ≤C{(1+ν)/integraltextt 0(1+τ)ν/bardbl(φttx,φtxx,φtx)(τ)/bardbl2dτ+/integraltextt 0(1+τ)1+ν/bardbl(φtt,φtx)(τ)/bardbl2dτ +/integraltextt 0(1+τ)1+ν/bardblH/bardbl2dτ}+Cδ1 ν(1+t)ν. Divide (4.33) by (1+ t)ν, and use1+τ 1+t≤1 and (4.32) just obtained, then we get desired estimate (4.34)(1+t)/bardbl(φttx,φtxx,φtx)(t)/bardbl2+/integraltextt 0/bardbl(φttx,φtxx)(τ)/bardbl2dτ ≤C{/integraltextt 0/bardbl(φttx,φtxx,φtx)(τ)/bardbl2dτ+/integraltextt 0(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2dτ+δ1} ≤CI0. Here, note that, although(1+τ)1+ν (1+t)νcomes out in the second term in (4.34), it only holds that (1+τ)1+ν (1+t)ν≥0(0≤τ≤t). Additionally, multiplying (4.30) by 1+ t, we get (4.35) (1+ t)/bardblφxxx(t)/bardbl2≤CI0. Thus, we have obtained the estimate (2.2.12) and completed t he proof of Proposition 2.2.1. Appendix. We prove Lemma 3.1. By (A1) V−v=δ0/radicalbig 4πµ(1+t)e−x2 4µ(1+t), (A2) U=Vx=−2πδ0x (4πµ(1+t))3/2e−x2 4µ(1+t), and (2.1.6), we easily know (A3)|∂k x(V−v)|p≤Cδ0(1+t)−1 2(1−1 p)−k 2, |∂k xU|p=|∂k xVx|p≤Cδ0(1+t)−1 2(1−1 p)−k+1 2 and (A4) |∂k xˆv|p+|∂l tˆv|p≤C|u+−u−|e−ct≤Cδ1e−ct(0< c < α 0). Rewrite (3.2): (3.2) F=−{Ut+(α−α)U+(p(V)−p′(v)V)x+(p(V+ ˆv)−p(V))x}=:4/summationdisplay i=1Fi. 20By (A3)-(A4), /bardblF1(t)/bardbl2≤ /bardblUt/bardbl2=/bardblVxxx/bardbl2≤Cδ0(1+t)−7/2, /bardblF3(t)/bardbl2≤/integraltext |p′(V)−p′(v)|2|Vx|2dx≤C|Vx|2 ∞/bardblV−v/bardbl2≤Cδ0(1+t)−5/2, /bardblF4(t)/bardbl2≤Cδ1e−ct. ForF2, by (A2) /bardblF2(t)/bardbl2≤/integraltext |α−α|2|Vx|2dx≤Cδ0/integraltext|α−α|2|x| (1+t)3/2dx·|Vx|∞ ≤Cδ0(1+t)−5/2if|x|1/2(α−α)∈L2. Hence we have /bardblF(t)/bardbl2≤Cδ1(1+t)−5/2. Next, /bardblF1x(t)/bardbl2=/bardblUtx/bardbl2≤Cδ0(1+t)−9/2, /bardblF2x(t)/bardbl2≤C/integraltext (|αxU|2+|α−α|2U2 x)dx ≤Cδ0(1+t)−3(/bardblx·αx/bardbl2+/bardblα−α/bardbl2)≤Cδ0(1+t)−3, /bardblF3x(t)/bardbl2≤/integraltext (p′′(V)V2 x+(p′(V)−p′(v))Vxx)2dx≤Cδ0(1+t)−7/2, /bardblF4x(t)/bardbl2≤Cδ1e−ct, and hence /bardblFx(t)/bardbl2≤Cδ1(1+t)−3. ForFt, /bardblF1t(t)/bardbl2=/bardblUtt/bardbl2=/bardblVxtt/bardbl2≤Cδ0(1+t)−11/2, /bardblF3t(t)/bardbl2=/bardbl(p′(V)−p′(v))Vtx+p′′(V)VxVt/bardbl2 ≤Cδ0((1+t)−1/2−4+(1+t)−3/2−3) =Cδ0(1+t)−9/2, /bardblF4t(t)/bardbl2≤Cδ1e−ct. ForF2t, /bardblF2t(t)/bardbl2=/bardbl(α−α)Ut/bardbl2≤ /bardbl(α−α)Vxxx/bardbl2≤Cδ0(1+t)−5/bardblx·(α−α)/bardbl2 ≤Cδ0(1+t)−5ifx·(α−α)∈L2, because of Vxxx=Cδ0x·((1+t)−5/2+Cx2(1+t)−7/2)e−x2 4µ(1+t). Hence/bardblFt(t)/bardbl2≤Cδ1(1+t)−9/2. The estimate of /bardblFtx(t)/bardbl2is done samely as above and omitted. Acknowledgment .The authors would like to thank Professor Ming Mei who shortl y visited Tokyo Institute of Technology in 2022-23. Through discussi ons with him they had a motive to consider the present problem. References [1] S. Chen, H. Li, J. Li, M. Mei and K. Zhang, Global and blow-u p solutions for compressible Eulerequationswithtime-dependentdamping, J.Differentia l Equations268(2020), 5035-5077. 21[2] C.J. van Duyn and L.A. Peletier, A class of similarity sol utions of the nonlinear diffusion equation, Nonlinear Anal. TMA 1 (1977), 223-233. [3] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math Ph ys. 143 (1992), 599-605. [4] F.M. Huang, P. Marcati and R.H. Pan, Convergence to the Ba renblatt solution for the com- pressible Euler equation with damping and vacuum, Arch. Rat ion. Mech. Anal. 176 (2005), 599-605. [5] S.JiandM. Mei, Optimaldecay rates ofthecompressibleE ulerequations withtime-dependent damping in Rn: (I) Under-damping case, J. Nonlinear Sci. 33 (2023), 1-47. [6] P. Marcati, M. Mei and B. Rubino, Optimal convergence rat es to diffusion waves for solutions of the hyperbolic conservation laws with damping. J. Math. F luid Mech.7 (2005)m 224-240. [7] A. Matsumura, Global existence and asymptotics of the so lutions of the second-order quasilin- ear hyperbolic equations with the first-order dissipation, Publ. RIMS, Kyoto Univ. 13 (1977), 349-379. [8] M. Mei, Best asymptotic profile for hyperbolic p-system w ith damping, SIAM J Math. Anal. 42 (2010), 1-23. [9] K. Nishihara, Convergence rates to nonlinear diffusion wa ves for solutions of system of hyper- bolic conservation laws with damping, J. Differential Equati ons 131 (1996), 171-188. [10] K. Nishihara, W.-K. Wang and T. Yang, Lp-convergence rates to nonlinear diffusion waves for p-system with damping, J. Differential Equations 131 (2000), 1 91-218. [11] M. Nishikawa, Convergence rate to the traveling wave fo r viscous conservation laws, Funkcial. Ekvac. 41 (1998), 107-132. [12] Y. Sugiyama, Blow-up for the 1D compressible Euler equa tions with separated time and space dependent damping coefficient, preprint. (arXiv:2212.1107 2v1) [13] N. Zhang, Optimal convergence rates to diffusion waves fo r solutions of p-system with damping on quadrant, J. Math. Anal. Appl. 512 (2022), #126118. [14] N. Zhang and C. Zhu, Long-time behavior of solutions to t heM1model with boundary effect, Disc. Cont. Dyn. Sys. 43 (2023), 1824-1859. [15] H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damp- ing, J. Differential Equations 174 (2001), 200-236. 22

2004.08082v1.Collective_coordinate_study_of_spin_wave_emission_from_dynamic_domain_wall.pdf

arXiv:2004.08082v1 [cond-mat.mes-hall] 17 Apr 2020Collective coordinate study of spin wave emission from dyna mic domain wall Gen Tatara RIKEN Center for Emergent Matter Science (CEMS) 2-1 Hirosawa, Wako, Saitama, 351-0198 Japan Rubn M. Otxoa de Zuazola Hitachi Cambridge Laboratory, J. J. Thomson Avenue, CB3 OHE, Cambridge, United Kingdom and Donostia International Physics Center, 20018 San Sebasti´ an, Spain (Dated: April 20, 2020) Abstract We study theoretically the spin wave emission from a moving d omain wall in a ferromagnet. Introducing a deformation mode describing a modulation of t he wall thickness in the collective coordinate description, we show that thickness variation c ouples to the spin wave linearly and inducesspinwave emission. Thedominant emitted spinwave t urnsout tobepolarized in theout-of wall plane ( φ)-direction. The emission contributes to the Gilbert dampi ng parameter proportional to/planckover2pi1ωφ/K, the ratio of the angular frequency ωφofφand the easy-axis anisotropy energy K. 1I. INTRODUCTION Spin wave (magnon) is an excitation playing essential roles in the tran sport phenomena in magnets, and its control, magnonics, is a hot recent issue. Beside s application interest for devices, behaviours of spin waves have been drawing interests from fundamental science view points. Many theoretical studies have been carried out on gen eration of spin waves by dynamic magnetic objects such as a domain wall [1–6]. The subject is highly nontrivial because the wall is a soliton, which is stable in the absence of perturb ation, meaning that it couples to fluctuations, spin waves, only weakly in the ideal case, w hile in reality, various perturbations and dynamics leads to strong emission of spin waves. There are several pro- cesses that lead to the emission, and it is not obvious which is the domin ant process and how large is the dissipation caused by the emission. The low energy behavior of a domain wall in a ferromagnet is described in terms of collective coordinates, its center of mass position Xand angle of the wall plane, φ0[7]. In the absence of a pinning potential, a displacement of the wall costs n o energy owing to the translational invariance, and it is thus natural to regard Xas a dynamic variable X(t). This is in fact justified mathematically; X(t) is a collection of spin waves that corresponds to the translational motion of the wall [8, 9]. It turns out that the c anonical momentum of the ferromagnetic domain wall is the angle φ0. This is because the translational motion of collective spins requires a perpendicular spin polarization, i.e., a tilting o f the wall plane. Mathematically this is a direct consequence of the spin algebra, and is straightforwardly derived based on the equation of motion for spin (Landau-Lifshitz( -Gilbert) equation) [7] or on the Lagrangian formalism [10]. In the absence of hard-axis anisot ropy energy, φ0is also a zero mode. As zero modes, X(t) andφ0do not have linear coupling to the fluctuation, spin wave, and thus emission of spin wave does not occur to the lowes t order. In this case, the second-order interactions to the spin wave give rise to the dom inant effect. In Ref. [1], the coupled equations of motion for the wall and spin wave modes wer e solved classically and demonstrated that a damping indeed arises from the quadratic interaction. In the case of a strong hard-axis anisotropy, the plane of the wall is constrain ed near the easy-plane, φ0is frozen, resulting in a single variable system described solely by X(t) [9, 11]. The spin wave coupling and dissipation in this limit was discussed in Ref. [11]. In real materials, hard-axis anisotropy and pinning potential exist , andX(t) andφ0 2are not rigorously zero modes. In other words, wall dynamics induc es a deformation and emission of spin wave is possible due to linear couplings. It was argued in Ref. [2] that there emerges a linear coupling when the wall driven by a spin-transfer tor que has a velocity ˙X different from the steady velocity determined by the spin-transfe r torque, and the damping due to spin wave emission was discussed. Numerical analysis of Ref. [3 ] revealed that spin wave emission occurs by the modulation of the wall thickness during t he dynamics. The coupling to the wall velocity and second order in the spin wave was stu died analytically in detail and dissipation was estimated in Ref. [6]. The energy dissipation proportional to the second-order in the wall velocity was found. In this paper, we study the spin wave emission extending convention al collective coor- dinate representation of the wall [10]. As the domain wall is a soliton, t here is no linear coupling of its center of mass motion to the spin wave field if deformat ion is ignored. We thus introduce a deformation mode of the wall, a change of the thick nessλ. This is a natural variable in the presence of the hard-axis anisotropy energy, as th e thickness depends on the angleφ0as pointed out in Refs. [12, 13]. Following the prescription of spin wave expansion [9], we derive the Lagrangian for the three collective coordinates, t he center of mass position X(t), the angle of the wall plane φ0(t) and thickness λ(t), including the spin waves to the second order. It turns out that Xandφ0and their time-derivatives do not have linear coupling to the spin wave, while ˙λdoes. This result is natural as Xandφ0are (quasi) zero modes, and consistent with numerical observation [3]. It is shown th at the emitted spin wave is highly polarized; The dominant emission is the fluctuation of ang leφ, while that ofθis smaller by the order of the Gilbert damping parameter α. The forward emission of wavelength λ∗∝v−1 w, wherevwis the domain wall velocity, is dominant. The modulation of λis induced by the dynamics of φ0, and the contribution to the Gilbert damping parameter due to the spin wave emission from this process is estimated from the energy dissipation rate. It was found to be of the order of αφ sw≃λ a/planckover2pi1ωφ K, whereωφis the angular frequency of the modulation of φ0,Kis the easy-axis anisotropy energy and ais the lattice constant. This damping parameter contribution becomes very strong of the o rder of unity if /planckover2pi1ωφis comparable to the spin wave gap, K, as deformation of the wall becomes significant in this regime. 3II. COLLECTIVE COORDINATES FOR A DOMAIN WALL We consider a one-dimensional ferromagnet along the x-axis with easy and hard axis anisotropy energy along the zandyaxis, respectively. The Lagrangian in terms of polar coordinates ( θ,φ) of spin is L=LB−HS (1) where LB=/planckover2pi1S a/integraldisplay dx˙φ(cosθ−1) HS=S2 2a/integraldisplay dx/bracketleftbig J[(∇θ)2+sin2θ(∇φ)2]+Ksin2θ(1+κsin2φ)/bracketrightbig (2) are the kinetic term of the spin (spin Berry phase term) and the Ham iltonian, respectively, J >0,K >0andκK≥0being theexchange, easy-axis anisotropyandhardaxisanisotro py energies, respectively, abeing the lattice constant. A static domain wall solution of this system is cosθ= tanhx−X λ0,φ= 0 (3) whereλ0≡/radicalbig J/Kis the wall thickness at rest. The dynamics of the wall is described by allowing the wall position Xandφas dynamic variables. This corresponds to treat the energy zero mode of spin waves (zero mode) describing a trans lational motion and its conjugate variable φas collective coordinates [9]. This treatment is rigorous in the absenc e of pinning and hard-axis anisotropy but is an approximation otherwis e. Most previous studies considered a rigid wall, where the wall thickness is a constant λ0. Here we treat the wall thickness as a dynamic variable λ(t) to include a deformation and study the spin- wave emission. This treatment was applied in Ref. [13], but only static s olution of λwas discussed. As demonstrated in Ref. [9], the spin wave around a domain wall in ferr omagnet is conveniently represented using ξ=e−u(x,t)+iφ0(t)+η(x−X(t),t)(4) whereφ0(t) is the angle of the wall, u(x,t) =x−X(t) λ(t)(5) 4θ φ FIG. 1. Fluctuation corresponding to the real and imaginary part of the spin wave variable ˜η= ˜ηR+i˜ηI. (a): The profile of ˜ ηantisymmetric with respect to the wall center, which turns out to be dominant excitation. (b): The real part ˜ ηRdescribes the deformation within the wall plane, i.e., modulation of θ, while the imaginary part ˜ ηIdescribes the out-of plane ( φ) fluctuation as shown in (c). Transparent arrows denotes the equilibrium spin configuration. andη(x−X,t) describes thespin-wave viewed inthemoving frame. Asitis obvious f romthe definition, the real and imaginary part of ηdescribe the fluctuation of θandφ, respectively. The fluctuations antisymmetric with respect to the wall center, sh own in Fig. 1, turns out to be dominant. The ξ-representation of the polar angles are cosθ=1−|ξ|2 1+|ξ|2, sinθsinφ=−iξ−ξ 1+|ξ|2. (6) A. Domain wall dynamic variables We first study what spin-wave mode the new variable λ(t) couples to, by investigating the ’kinetic’ term of the spin Lagrangian, LB, which is written as LB=2i/planckover2pi1Sλ a/integraldisplay duIm[ξ˙ξ] 1+|ξ|2. (7) Using Eq. (6) and ∂tu=−1 λ/parenleftBig ˙X+u˙λ/parenrightBig , ∂ tξ=/parenleftbigg1 λ/parenleftBig ˙X+u˙λ/parenrightBig +i˙φ0+(∂t−˙X∇x)η/parenrightbigg ξ,(8) we have 2iIm[ξ˙ξ] = 2i( ˙ηI+˙φ0−˙X∇xηI)|ξ|2(9) 5sδ FIG. 2. Schematic figure showing the effect of asymmetric perpe ndicular spin polarization δs due to the spin wave mode ϕ. The asymmetric torque (curved arrows) induced by asymmetr icδs rotates the spins within the wall plane, resulting in a compr ession of the wall, i.e., to ˙λ. whereηi≡Im[η]. The kinetic term is expanded to the second order in the spin wave as (using integral by parts) LB=2/planckover2pi1S a[φ0˙X+ϕ˙λ]+L(2) B (10) where ϕ≡/integraldisplay duu coshu˜ηI, (11) represents an asymmetric configuration of ˜ ηIand L(2) B≡2/planckover2pi1Sλ a/integraldisplay du/bracketleftbigg ˜ηR↔ ∂t˜ηI−˙X˜ηR↔ ∇x˜ηI−2 λtanhu/parenleftBig 2˙X+u˙λ/parenrightBig ˜ηR˜ηI/bracketrightbigg ,(12) where ˜η≡η/(2coshu). When deriving Eq. (10), the orthogonality of fluctuation and the ze ro-mode, /integraldisplay du˜η coshu= 0, (13) was used. Equation (10) indicates that ϕis the canonical momentum of λ. In fact, it represents the asymmetric deformation of angle φ, as the imaginary part of the spin wave, ˜ηI, corresponds to fluctuation of φas seen in the definition, Eq. (4). Such an asymmetric configuration of φexerts a torque that induces a compression or expansion of the do main wall (Fig. 2), andthisiswhy ϕandλareconjugatetoeach other. Thecoupling ϕ˙λdescribes the spin wave emission when thickness changes, as we shall argue lat er. The second term proportional to ˙Xin the bracket in Eq. (12) represents a magnon current induced in t he moving frame (Doppler shift). 6The Hamiltonian of the system is similarly written in terms of spin wave va riables to the second order as HS=KS2λ a/bracketleftBigg/parenleftbiggλ0 λ/parenrightbigg2 +1+κsin2φ0/bracketrightBigg +2KS2λ a/integraldisplay dutanhu coshu˜ηR/bracketleftBigg −/parenleftbiggλ0 λ/parenrightbigg2 +1+κsin2φ0/bracketrightBigg +H(2) S, (14) where H(2) S≡2KS2λ a/integraldisplay du/bracketleftbigg λ2 0[(∇˜ηR)2+(∇˜ηI)2] + ˜ηR2/bracketleftbigg −λ2 0 λ2/parenleftbigg 1−1 cosh2u/parenrightbigg +/parenleftbigg 2−3 cosh2u/parenrightbigg (1+κsin2φ0)/bracketrightbigg + ˜ηI2/bracketleftbiggλ2 0 λ2/parenleftbigg 1−2 cosh2u/parenrightbigg +κcos2φ0/bracketrightbigg +2κ˜ηR˜ηItanhusin2φ0/bracketrightbigg (15) In the case of small κandλ≃λ0, the spin waves are described by a simple Hamiltonian as Hsw≡2KS2λ a/integraldisplay du/bracketleftbigg λ2[(∇˜ηR)2+(∇˜ηI)2]+( ˜ηR2+ ˜ηI2)/parenleftbigg 1−2 cosh2u/parenrightbigg/bracketrightbigg +HD,(16) where HD≡2/planckover2pi1Sλ a˙X/integraldisplay du˜ηR↔ ∇x˜ηI, (17) is the Doppler shift term. For a constant wall velocity ˙X, it simply shifts the wave vector of the spin wave. Without the Doppler shift, the eigenfunction of th is Hamiltonian (16) is labeled by a wave vector kas φk(u) =1√2π˜ωk(−ikλ+tanhu)eikλu, (18) where ˜ωk≡1+(kλ)2(19) is the dimensionless energy of spin wave. Dissipation function is W=α/planckover2pi1S 2a/integraldisplay dx(˙θ2+sin2θ˙φ2) =/planckover2pi1Sλ 2a α/parenleftBigg˙X λ/parenrightBigg2 +α˙φ02+αλ/parenleftBigg˙λ λ/parenrightBigg2 , (20) 7whereαis the Gilbert damping parameter and αλ≡α/integraltext duu2 cosh2u=π2 12α. Asdrivingmechanisms ofadomainwall, weconsider amagneticfieldandc urrent-induced torque (spin-transfer torque) [9, 14, 15]. A magnetic field applied a long the negative easy axis is represented by the Hamiltonian ( γ=e/mis the gyromagnetic ratio) HB=/planckover2pi1Sγ aBz/integraldisplay dxcosθ. (21) Using Eqs. (6)(13), we obtain HB=−2/planckover2pi1Sγ aBz/parenleftbigg X+λ/integraldisplay dutanhu˜η2 R/parenrightbigg . (22) (The first term is derived evaluating a diverging integral/integraltext dx1 1+e2u(x)carefully introducing the system size Las/integraltextL/2 −L/2dx1 1+e2u(x)and dropping a constant.) The magnetic field therefore exerts a force2/planckover2pi1Sγ aBzon the domain wall. The spin-transfer effect induced by injecting spin-polarized electr ic current is represented by a Hamiltonian having the same structure as the spin Berry’s phase termLB[9, 15] HSTT=−/planckover2pi1S avst/integraldisplay dxcosθ(∇xφ), (23) wherevst≡aP 2eSjis a steady velocity of magnetization structure under spin polarized current Pj(Pis the spin polarization and jis the applied current density (one-dimensional)). The spin wave expression is HSTT=2/planckover2pi1S avst/bracketleftbigg φ0+2/integraldisplay dx/parenleftbigg ˜ηR∇x˜ηI+1 λtanhu˜ηR˜ηI/parenrightbigg/bracketrightbigg . (24) As has been known, a spin-transfer torque contributing to the wa ll velocity and does not work as a force, as the applied current or vstcouples to φ0and not to X. The equation of motion for the tree domain wall variables is therefor e obtained from Eqs. (10) (14) (20) and driving terms (22)(24) as ˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0ζ+vst ˙φ0+α˙X λ=˜Bz αλ˙λ λ=KS /planckover2pi1/bracketleftBigg/parenleftbiggλ0 λ/parenrightbigg2 −(1+κsin2φ0)/bracketrightBigg −˙ϕ−2KS /planckover2pi1/bracketleftBigg/parenleftbiggλ0 λ/parenrightbigg2 +(1+κsin2φ0)/bracketrightBigg ζ, (25) 8wherevc≡KSκ 2/planckover2pi1λ,˜Bz≡γBzandϕ(Eq. (11)) and ζ≡/integraldisplay dutanhu coshu˜ηR, (26) are contributions linear in spin wave. III. SPIN WAVE EMISSION In this section we study the spin wave emission due to domain wall dyna mics. The emission is described by the linear coupling between the spin wave field a nd the domain wall in Eqs.(10) (14). Moreover, dynamic second-order couplings in Eqs . (12)(15) leads to spin wave excitation. In the first linear process, the momentum and ene rgy of the spin wave is supplied by the dynamic domain wall, while the second process present s a scattering of spin waves where the domain wall transfer momentum and energy to the incident spin wave. A. Linear emission We here discuss the emission due to the linear interactions in Eqs.(10) (14) in the labo- ratory (rest) frame. The laboratory frame is described by replac ingη(x−X(t),t) byη(x,t) in the derivation in Sec. IIA. It turns out that the Lagrangian Eq.(1 2) in the laboratory frame has no Doppler shift term and the term ˙X˜ηR˜ηIis half. The emitted wave has an angular frequency shifted by the Doppler shift from the moving wall. Using the equation of motion, Eq. (25), the spin wave emission arises from the thickness c hange. The interaction Hamiltonian reads in the complex notation ˜ η= ˜ηR+i˜ηI H(1) η(t) =˙λ(t)/integraldisplay dx(g˜η+g˜η), (27) where g(x)≡2/planckover2pi1S a1 coshx−X(t) λ/parenleftbigg −αλtanhx−X(t) λ+ix−X(t) λ/parenrightbigg (28) Let us study here the emission treating λas a constant as its dynamics is taken account in the first factor in the interaction Hamiltonian (27). The Fourier tra nsform of the interaction 9is calculated using /integraldisplay∞ −∞duei˜kuu coshu=iπ2 2sinhπ 2˜k cosh2π 2˜k /integraldisplay∞ −∞duei˜kutanhu coshu=π˜k coshπ 2˜k(29) as H(1) η(t) =−π2 2λ˙λ(t)/summationdisplay k1 coshπ 2kλeikX(t)/parenleftbigg ˜ηIk(t)tanhπ 2kλ+2 παλkλ˜ηRk(t)/parenrightbigg ,(30) We consider the case where the wall is approximated by a constant v elocityvw, i.e.,X(t) = vwt. The frequency representation of time-integral of Eq. (30) is /integraldisplay dtH(1) η(t) =−π2 2/integraldisplaydΩ 2π/integraldisplaydω 2πλ˙λ(Ω)/summationdisplay k1 coshπ 2kλ/parenleftbigg ˜ηIk(t)tanhπ 2kλ+2αλ πkλ˜ηRk(t)/parenrightbigg δ(ω−(kvw+Ω)), (31) It is seen that the angular frequency of the emitted spin wave ( ω) iskvw+ Ω, i.e., that of the thickness variation ˙λwith a Doppler shift due to the wall motion. The Doppler shift of angular frequency, δν≡kvw, is expected tobesignificant; For k= 1/λwithλ= 10−100nm andvw= 100 m/s, we have δν= 10−1 GHz. The function g(x) represents the distribution of the wave vector k, which has a broad peak at k= 0 with a width of the order of λ−1. To have a finite expectation value ∝angb∇acketleft˜η∝angb∇acket∇ight, the angular frequency ωand wave vector kneeds to match the dispersion relation of spin wave, ω=ωk, i.e., kvw+/planckover2pi1Ω =KS(1+(kλ)2). (32) The angular frequency Ω is determined by the equation for λin Eq. (25), and is of the order of the angular frequency of φ0,ωφ. (See Sec. VA for more details.) Equation (32) has solution for a velocity larger than the threshold velocity vth≡2KS /planckover2pi1λ/radicalBig 1−/planckover2pi1ωφ KS. The emitted wave lengths k∗are (plotted in Fig. 4) k∗λ=/planckover2pi1vw 2KSλ 1±/radicalBigg 1−/parenleftbiggvth vw/parenrightbigg2 . (33) The sign of k∗(direction of emission) is along the wall velocity, meaning that the emis sion is dominantly in the forward direction. The group velocity of the emitte d wave is of the same 10vwλ∗ FIG. 3. Schematic figure showing the spin wave emission from a domain wall with thickness oscillation ( ˙λ) moving with velocity vw. The linear coupling leads to a forward emission of spin wave with wave length λ∗≡2π/k∗, wherek∗is defined by Eq. (33). vwk* • • •ω~=0.2 ω~=0.5 ω~=0.8 FIG. 4. Plot of the wave length k∗of the emitted spin wave as function of wall velocity vwfor ˜ω≡/planckover2pi1ωφ/KS= 0.2,0.5 and 0.8. Dotted line is k∗=/planckover2pi1 KSλ2vw. Threshold velocity for the emission vthis denoted by circles. order as the wall velocity; dωk dk/vextendsingle/vextendsingle/vextendsingle/vextendsingle k=k∗=2KS /planckover2pi1λ2k∗=vw 1±/radicalBigg 1−/parenleftbiggvth vw/parenrightbigg2 . (34) The dominant spin wave emission considered here is the antisymmetric excitation of the imaginary part ˜ ηIrepresenting the fluctuation of angle φ. The antisymmetric excitation of φis a natural excitation arising from the intrinsic property, the aniso tropy energy. The easy-axis anisotropy energy acts as a local potential VKfor each spin in the wall as in Fig. 5. When the wall moves to the right, the spins ahead of the wall are d riven towards the high energy state, while the spins behind (left in Fig. 5) are towards lo w energy states. This asymmetry leads to an asymmetric local “velocity” of angle θ, and its canonical momentum 11FIG. 5. The local potential VKfor spins in a domain wall arising from the easy axis anisotro py energy,K. When the wall moves to the right, the spins right (left) of th e wall rotates towards high (low) energy states, resulting in an asymmetric local v elocity of rotation, exciting the angle φasymmetrically with respect to the wall center. φ. This role of Kto induce asymmetric φis seen in the equations of motion for polar angles [9]: Focusing on the contribution of the easy axis anisotropy, the ve locity of the in-plane spin rotation, sin θ˙φ=−KSsinθcosθis asymmetric with the wall center θ=π/2. Faster rotation in the left part of the wall (π 2< θ < π) than the right part (0 < θ <π 2) indicates that the wall becomes thinner. In the equation of motion for λ(Eq. (25)), this effect is represented by the term −˙ϕon the right-hand side, meaning that asymmetric deformation modeϕtends to compress the wall. B. Green’s function calculation We present microscopic analysis of the spin wave emission using the Gr een’s function. We consider here the slow domain wall dynamics limit compared to the sp in-wave energy scale and neglect the time-dependence of the variable uarising from variation of ˙X. The calculation here thus corresponds to the spin wave effects in the mo ving frame with the domain wall. The amplitude of the spin wave, ∝angb∇acketleft˜η∝angb∇acket∇ight, is calculated using the path-ordered Green’s function method as a linear response to the source field ˙λ. The amplitude is ∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−i/integraldisplay Cdt′˙λ(t′)/integraldisplay du′g(u′)/angbracketleftbig TC˜η(u,t)˜η(u′,t′)/angbracketrightbig (35) whereCdenotes the contour for the path-ordered (non-equilibrium) Gre en’s function in the complex time and TCdetnoes the path-ordering. Evaluating the path-order, we obta in the 12real-time expression of ∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=/integraldisplay∞ −∞dt′˙λ(t′)/integraldisplay du′g(u′)Gr η(u,t,u′,t′) (36) where Gr η(u,t,u′,t′)≡ −iθ(t−t′)/angbracketleftbig [˜η(u,t),˜η(u′,t′)]/angbracketrightbig (37) the retarded Green’s function of ˜ η. The Green’s function is calculated expressing ˜ ηin terms of the orthogonal base for spin wave wave function [9] as ˜η(u,t) =/summationdisplay kηk(t)φk(u), (38) whereφkis the eigenfunction of Eq. (18) and ηkis the annihilation operator satisfying [ηk,ηk′] =δk,k′. The time-development of the operator is ηk(t) =e−iωktηk(0), where ωk≡ KS˜ωkis the energy of spin wave. The retarded Green’s function thus is Gr η(u,t,u′,t′) =−iθ(t−t′)/summationdisplay ke−iωk(t−t′)φk(u)φk(u′)≡/integraldisplaydω 2πe−iω(t−t′)Gr η(u,u′,ω) (39) where Gr η(u,u′,ω) =/summationdisplay k1 ω−ωk+i0φk(u)φk(u′) (40) is the Fourier transform, + i0 denoting the small positive imaginary part. The Green’s function has a nonlocal nature in space, as seen from the overlap o f the spin wave function /summationdisplay kφk(u)φk(u′) =a 2πλ/bracketleftbigg δ(u−u′)−1 2/parenleftBig e−|u−u′|(1−tanhutanhu′)+sinh(u−u′)(tanhu−tanhu′)/parenrightBig/bracketrightbigg . (41) Here we use low-frequency approximation, namely, consider the eff ect of high-frequency magnon compared to the wall dynamics and use Gr η(u,u′,ω)≃ −/summationtext k1 ωkφk(u)φk(u′). The retarded Green’s function then becomes local in time as Gr η(u,t,u′,t′) =δ(t−t′)Gr η(u,u′,ω). We thus obtain ∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)/summationdisplay k1 ωkφk(u)/integraldisplay du′g(u′)φk(u′) (42) 13withuandu′havingX(t) of the equal time t. The integral/integraltext du′g(u′)φk(u′) describing the overlap of spin-wave wave function and the domain wall is calculated u sing /integraldisplay dutanhu coshuφk(u) =1√2π˜ωkπ coshπ 2kλ˜ωk 2/integraldisplay duu coshuφk(u) =1√2π˜ωkπ coshπ 2kλ(43) as /integraldisplay dug(u)φk(u) =2/planckover2pi1S a1√2π˜ωkπ coshπ 2kλ/parenleftbigg i−αλ˜ωk 2/parenrightbigg (44) The spin wave amplitude emitted by the wall dynamics is therefore ∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)2/planckover2pi1 Ka/summationdisplay k1√2π˜ωkπ coshπ 2kλ1 ˜ωkφk(u)/parenleftbigg i−αλ˜ωk 2/parenrightbigg (45) The integral/summationtext k(˜ωk)−β1 coshπ 2kλφk(u) (β=1 2,3 2) is real, and thus Re[ ∝angb∇acketleft˜η∝angb∇acket∇ight]/Im[∝angb∇acketleft˜η∝angb∇acket∇ight]≃α. As ∝angb∇acketleft˜η(u,t)∝angb∇acket∇ightis odd in u, the emitted spin wave is an antisymmetric fluctuation of the angle φwith respect to the wall center (Fig. 1). (Because of low frequenc y approximation in deriving Eq. (42), the nonlocal nature (Eq. (41)) is smeared out in the result Eq. (45). ) The quantities representing the effects of spin wave emission on the wall dynamics in Eq. (25) are ζ=/integraldisplay dutanhu coshuRe[˜η] =˙λπ/planckover2pi1 4Kaαλ/summationdisplay k1 cosh2π 2kλ≡αµζ˙λ λ ϕ=/integraldisplay duu coshuIm[˜η] =−˙λπ/planckover2pi1 Ka/summationdisplay k1 ˜ω2 k1 cosh2π 2kλ≡µϕ˙λ λ(46) whereµζ≡π3/planckover2pi1λ 48Ka/summationtext k1 cosh2π 2kλandµϕ≡π/planckover2pi1λ Ka/summationtext k1 ˜ω2 k1 cosh2π 2kλ. The first integral is evaluated as /summationtext k1 cosh2π 2kλ=a/integraltextdk 2π1 cosh2π 2kλ=2a π2λand the second one is/summationtext k1 ˜ω2 k1 cosh2π 2kλ≡2a π2λγϕ, whereγϕ is a constant of the order of unity. The constants are therefore µζ=π/planckover2pi1 24K µϕ=−2/planckover2pi1γϕ πK. (47) From Eq. (46), the averaged amplitude of the imaginary part of the emitted spin wave is of the order of/planckover2pi1˙λ Kλ(the real part is a factor of αsmaller). As seen from Eq. (25), the time scale ofλdynamics is K//planckover2pi1, and thus the emitted spin wave amplitude can be of the order of unity if the modulation of λis strong, resulting in a significant damping. (See Eq. (61) below.) 14C. Spin wave excitation due to second order interaction Besides emission due to the linear order interaction discussed above , spin waves are excited also due to the second order interaction in Eqs.(10) (14) wh en the wall is dynamic. Here we focus on the effect of a dynamic potential in the Hamiltonian ( Eq. (16)) V(x,t)≡4KS2 a1 cosh2x−X(t) λ(48) and calculate the excited spin wave density in the laboratory frame b y use of linear response theory. For a constant wall velocity, X(t) =vwt, the Fourier representation of the potential is Vq(Ω) = 8π2KS2λ aqλ sinhπ 2qλδ(Ω−qvw), (49) The potential thereforeinduces Dopplershift of qvwintheangular frequency ofthescattered spin wave. This dynamic potential induces an excited spin wave densit y asδn(x,t) = iG< η(x,t,x,t), where G< ηis the lesser Green’s function of spin wave. The linear response contribution in the Fourier representation is δn(q,Ω) =i/summationdisplay k/integraldisplaydω 2πVq(Ω)(n(ω+Ω)−n(ω))gr kωga k+q,ω+Ω (50) Inthisprocess, theexcitedspinwave density hasthesamewavelen gth andangularfrequency of the driving potential Vq(Ω). This means that the excitation moves together with the domain wall, and thus this is not an emission process. For slow limit, q≪kand Ω≪ω, usingn(ω+Ω)−n(ω) =n(ω+qvw)−n(ω)≃qvwn′(ω), we obtain a compact expression of δn(q,Ω) =i4πKS2 avw(qλ)2 sinhπ 2qλδ(Ω−qvw)/integraldisplaydω 2π/summationdisplay kn′(ω)|gr kω|2(51) and the real space profile is δn(x,t) =δn0vw vatanhx−vwt λ cosh2x−vwt λ(52) whereδn0=−4 π(KS)2/integraltextdω 2π/summationtext kn′(ω)|gr kω|2andva≡Kλ//planckover2pi1is a velocity scale determined by magnetic anisotropy energy. The induced spin wave density has t hus an antisymmetric spatial profile with respect to the wall center and propagate with a domain wall velocity in the present slowly varying limit. It is not therefore a spin wave emissio n, but represents the deformation of the wall asymmetric with respect to the center. 15IV. EQUATION OF MOTION OF THREE COLLECTIVE COORDINATES Theequationofmotion(25)including thespinwave emission effectsex plicitly istherefore ˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0αµζ˙λ λ+vst (53) ˙φ0+α˙X λ=˜Bz αλ˙λ λ=KS /planckover2pi1/bracketleftBigg/parenleftbiggλ0 λ/parenrightbigg2 −(1+κsin2φ0)/bracketrightBigg −µϕ¨λ λ−2KS /planckover2pi1/bracketleftBigg/parenleftbiggλ0 λ/parenrightbigg2 +(1+κsin2φ0)/bracketrightBigg αµζ˙λ λ. (54) The spin-wave contribution of the first equation, the second term of the right-hand side, is of the order αsmaller than the first term and is neglected. From the equations, we see that the dynamics of Xandφare not strongly coupled to the variation of the width. In particular , whenκis small, the dynamics of the wall center ( Xandφ) governed by the energy scale of K⊥=κKis much slower than that of a deformation mode λ, which is of the energy scale ofK, and thus it is natural that the two dynamics are decoupled. Then κis not small, λ affects much the wall center dynamics. For static case of λ, we have λ=λ0/radicalbig 1+κsin2φ0, (55) as was argued in Refs. [12, 13]. Using this relation assuming slow dynam ics to estimate the spin-wave contribution in the equation for λ, we obtain µϕ¨λ+ ˜αλ˙λ=KS /planckover2pi1λ/bracketleftBigg/parenleftbiggλ0 λ/parenrightbigg2 −(1+κsin2φ0)/bracketrightBigg , (56) where ˜αλ≡αλ/parenleftbig 1+2S π/parenrightbig =π2 12α/parenleftbig 1+2S π/parenrightbig is the effective damping for the width. The mass forλ,µϕ, was induced by the imaginary part of the spin-wave. V. DISSIPATION DUE TO SPIN WAVE EMISSION Considering the action, which is a time-integral of the Lagrangian, a nd by use of integral by parts with respect to time, the linear interaction Hamiltonian, Eq. (27), is equivalent to H(1) η=−λFλ, where Fλ≡2/integraldisplay duRe[g˙˜η], (57) 16is a generalized force for variable λ. Using Eqs. (45)(43), it reads Fλ=−¨λfλ, (58) where (neglecting the order of α2) fλ≡π/planckover2pi12S Ka2/summationdisplay k1 ˜ω2 k1 cosh2π 2kλ=2/planckover2pi12S πKλaγϕ. (59) The energy dissipation rate due to the spin wave emission is therefor e dEsw dt≡ −˙λFλ=fλ 2d dt˙λ2, (60) and thus Esw=fλ 2˙λ2. As is seen from Eq. (56), the intrinsic energy scale governing the dynamics of λisK, and thus the intrinsic scale of ˙λ/λis of the order of K//planckover2pi1. The energy dissipation by an intrinsic spin-wave emission is estimated roughly as Ei sw≃Kλ a, which is the typical spin wave energy multiplied by the number of spin waves ex cited in the wall. The quantitydEi sw dtcorresponds to a dissipation function Wi swinduced by the intrinsic spin wave emission. Considering the intrinsic frequency of λof the order of K//planckover2pi1, the Gilbert damping parameter induced by the intrinsic emission is αi sw≃2aλ /planckover2pi1SfλK /planckover2pi1=4γϕ π. (61) This value is of the order of unity ( γϕis a constant), meaning that spin wave emission from the thickness change is very efficient in dissipating energy from the w all. This result may not be surprising if one notices that the intrinsic energy scale of thic kness change is that of easy-axis anisotropy energy K, which is the energy scale where significant deformation of the wall is induced. A. Modulation of λdue toφ0dynamics In most cases, the dynamics of λis driven by the time-dependence of φ0as seen in Eq. (56). Let us consider this case of a forced oscillation. We consider b y simplyfying φ0grows linear with time, φ0=ωφt,ωφbeing a constant. Linearizing Eq. (56) using λ=λ+δλ, whereλ≡λ0//radicalbig 1+κ/2 is the average thickness, we have an equation of motion of a force d oscillation, µϕ¨δλ+ ˜αλ˙δλ+µϕ(Ωλ)2δλ=KS 2/planckover2pi1λκcos2ωφt, (62) 17where Ω λ=K /planckover2pi1/radicalBig πS γϕ/parenleftbig 1+κ 2/parenrightbig is an intrinsic angular frequency of δλ. The solution having an external angular frequency of 2 ωφis δλ=δλcos(2ωφt−εφ), (63) where δλ≡κλπS 4γϕ(K//planckover2pi1)2 /radicalBig (Ω2 λ−4ω2 φ)2+4(˜αλωφ µϕ)2(64) is the amplitude of the forced oscillation and εφ≡tan−12˜αλωφ µϕ Ω2 λ−4ω2 φis a phase shift. A resonance occurs for ωφ= Ωλ/2. The energy dissipation rate for the emission due to forced oscillat ion induced by dynamics of φis dEφ sw dt≃λ a/parenleftbiggδλ λ/parenrightbigg2ω3 φ K. (65) The contribution to the Gilbert damping parameter is obtained from t he relationdEφ sw dt= αφ sw(˙λ/λ)2as αφ sw≃λ a/planckover2pi1ωφ K. (66) Let us focus on the periodic oscillation of φ0, realized for large driving forces, namely, for Bz> αKSκ 2/planckover2pi1γ≡BW(γBz> αvc) for the field-driven case or j >eS2 /planckover2pi1Pλ aKκ≡ji(vst> vc) for the current-driven case ( BWis the Walker’s breakdown field and jiis the intrinsic threshold current [10]). The solution of the equation of motion (54) then read s φ0≃ωφt, (67) where (jis defined in one-dimension to have the unit of A=C/s) ωφ≃˜Bz+αvst λ=γBz+aP 2eSλαj. (68) TheGilbertdampingconstant duetospinwaveemission, Eq. (66), th usgrowslinearlyinthe driving fields in this oscillation regime. Using current-induced torque f or a pinned domain wall would be straightforward for experimental observation of th is behaviour, although the contribution to the Gilbert damping is proportional to αand not large, αφ sw≃α/planckover2pi1P ej K(for S∼1,P∼1). 18VI. SUMMARY Westudiedspinwaveemissionfromamovingdomainwallinaferromagne tbyintroducing a deformation mode of thickness modulation as a collective coordinat e. It was shown that the time-derivative of the thickness ˙λhas a coupling linear in the spin wave field, resulting in an emission, consistent with previous numerical result [3]. The domin ant emitted spin wave is in the forward direction to the moving domain wall and is strong ly polarized in the out-of plane direction, i.e., it is a fluctuation of φ. The dynamics of λis induced by the variation of the angle of the wall plane, φ0, as has been noted [12, 13]. For a φ0with an angular frequency of ωφ, the Gilbert damping parameter as a result of spin wave emission isαφ sw≃λ a/planckover2pi1ωφ K, whereKis the easy-axis anisotropy energy ( ais the lattice constant). The present study is in the low energy and weak spin wave regime, and treating the higher energy dynamics with strong spin wave emission is an important future subject. ACKNOWLEDGMENTS GT thanks Y. Nakatani for discussions. This work was supported b y a Grant-in-Aid for Scientific Research (B) (No. 17H02929) from the Japan Societ y for the Promotion of Science and a Grant-in-Aid for Scientific Research on Innovative Ar eas (No.26103006) from The Ministry of Education, Culture, Sports, Science and Technolog y (MEXT), Japan. [1] D. Bouzidi and H. Suhl, Phys. Rev. Lett. 65, 2587 (1990). [2] Y. L. Maho, J.-V. Kim, and G. Tatara, Phys. Rev. B 79, 174404 (2009). [3] X. S. Wang, P. Yan, Y. H. Shen, G. E. W. Bauer, and X. R. Wang, Phys. Rev. Lett. 109, 167209 (2012). [4] X. S. Wang and X. R. Wang, Phys. Rev. B 90, 014414 (2014). [5] N. J. Whitehead, S. A. R. Horsley, T. G. Philbin, A. N. Kuch ko, and V. V. Kruglyak, Phys. Rev. B 96, 064415 (2017). [6] S. K. Kim, O. Tchernyshyov, V. Galitski, and Y. Tserkovny ak, Phys. Rev. B 97, 174433 (2018). [7] J. C. Slonczewski, Int. J. Magn. 2, 85 (1972). 19[8] R. Rajaraman, Solitons and Instantons (North-Holland, 1982) p. Chap. 8. [9] G. Tatara, H. Kohno, and J. Shibata, Physics Reports 468, 213 (2008). [10] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). [11] H.-B. Braun and D. Loss, Phys. Rev. B 53, 3237 (1996). [12] N. L. Schryer and L. R. Walker, Journal of Applied Physic s45, 5406 (1974). [13] A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier, Journal of Applied Physics 95, 7049 (2004), https://doi.org/10.1063/1.1667804. [14] L. Berger, Phys. Rev. B 33, 1572 (1986). [15] G. Tatara, Physica E: Low-dimensional Systems and Nano structures 106, 208 (2019). 20

1705.03416v1.Low_spin_wave_damping_in_the_insulating_chiral_magnet_Cu___2__OSeO___3__.pdf

Low spin wave damping in the insulating chiral magnet Cu 2OSeO 3 I. Stasinopoulos,1S. Weichselbaumer,1A. Bauer,2J. Waizner,3 H. Berger,4S. Maendl,1M. Garst,3, 5C. P eiderer,2and D. Grundler6, 1Physik Department E10, Technische Universit at M unchen, D-85748 Garching, Germany 2Physik Department E51, Technische Universit at M unchen, D-85748 Garching, Germany 3Institute for Theoretical Physics, Universit at zu K oln, D-50937 K oln, Germany 4Institut de Physique de la Mati ere Complexe, Ecole Polytechnique F ed erale de Lausanne, 1015 Lausanne, Switzerland 5Institut f ur Theoretische Physik, Technische Universit at Dresden, D-01062 Dresden, Germany 6Institute of Materials and Laboratory of Nanoscale Magnetic Materials and Magnonics (LMGN), Ecole Polytechnique F ed erale de Lausanne (EPFL), Station 12, 1015 Lausanne, Switzerland (Dated: October 2, 2018) Chiral magnets with topologically nontrivial spin order such as Skyrmions have generated enor- mous interest in both fundamental and applied sciences. We report broadband microwave spec- troscopy performed on the insulating chiral ferrimagnet Cu 2OSeO 3. For the damping of magnetiza- tion dynamics we nd a remarkably small Gilbert damping parameter of about 1 104at 5 K. This value is only a factor of 4 larger than the one reported for the best insulating ferrimagnet yttrium iron garnet. We detect a series of sharp resonances and attribute them to conned spin waves in the mm-sized samples. Considering the small damping, insulating chiral magnets turn out to be promising candidates when exploring non-collinear spin structures for high frequency applications. PACS numbers: 76.50.+g, 74.25.Ha, 4.40.Az, 41.20.Jb The development of future devices for microwave ap- plications, spintronics and magnonics [1{3] requires ma- terials with a low spin wave (magnon) damping. In- sulating compounds are advantageous over metals for high-frequency applications as they avoid damping via spin wave scattering at free charge carriers and eddy currents [4, 5]. Indeed, the ferrimagnetic insulator yt- trium iron garnet (YIG) holds the benchmark with a Gilbert damping parameter intr= 3105at room temperature [6, 7]. During the last years chiral mag- nets have attracted a lot of attention in fundamental research and stimulated new concepts for information technology [8, 9]. This material class hosts non-collinear spin structures such as spin helices and Skyrmions be- low the critical temperature Tcand critical eld Hc2 [10{12]. Additionally, Dzyaloshinskii-Moriya interaction (DMI) is present that induces both the Skyrmion lattice phase and nonreciprocal microwave characteristics [13]. Low damping magnets oering DMI would generate new prospects by particularly combining complex spin order with long-distance magnon transport in high-frequency applications and magnonics [14, 15]. At low tempera- tures, they would further enrich the physics in magnon- photon cavities that call for materials with small intrto achieve high-cooperative magnon-to-photon coupling in the quantum limit [16{19]. In this work, we investigate the Gilbert damping in Cu2OSeO 3, a prototypical insulator hosting Skyrmions [20{23]. This material is a local-moment ferrimagnet withTc= 58 K and magnetoelectric coupling [24] that gives rise to dichroism for microwaves [25{27]. The magnetization dynamics in Cu 2OSeO 3has already been explored [13, 28, 29]. A detailed investigation on thedamping which is a key quality for magnonics and spin- tronics has not yet been presented however. To eval- uateintrwe explore the eld polarized state (FP) where the two spin sublattices attain the ferrimagnetic arrangement[21]. Using spectra obtained by two dier- ent coplanar waveguides (CPWs), we extract a minimum intr=(9.94.1)105at 5 K, i.e. only about four times higher than in YIG. We resolve numerous sharp reso- nances in our spectra and attribute them to modes that are conned modes across the macroscopic sample and allowed for by the low damping. Our ndings substanti- ate the relevance of insulating chiral magnets for future applications in magnonics and spintronics. From single crystals of Cu 2OSeO 3we prepared two bar-shaped samples exhibiting dierent crystallographic orientations. The samples had lateral dimensions of 2:30:40:3 mm3. They were positioned on CPWs that provided us with a dynamic magnetic eld hinduced by a sinusoidal current applied to the signal surrounded by two ground lines. We used two dierent CPWs with ei- ther a broad [30] or narrow signal line width of ws= 1 mm or 20m, respectively [31]. The central long axis of the rectangular Cu 2OSeO 3rods was positioned on the central axis of the CPWs. The static magnetic eld Hwas ap- plied perpendicular to the substrate with Hkh100iand Hkh111ifor sample S1 and S2, respectively. The direc- tion ofHdened the z-direction. The dynamic eld com- ponent h?Hprovided the relevant torque for excita- tion. Components hkHdid not induce precessional mo- tion in the FP state of Cu 2OSeO 3. We recorded spectra by a vector network analyzer using the magnitude of the scattering parameter S12. We subtracted a background spectrum recorded at 1 T to enhance the signal-to-noisearXiv:1705.03416v1 [cond-mat.str-el] 9 May 20172 ratio (SNR) yielding the displayed
jS12j. In Ref. [7], Klingler et al. have investigated the damping of the in- sulating ferrimagnet YIG and found that Gilbert param- etersintrevaluated from both the uniform precessional mode and standing spin waves conned in the macro- scopic sample provided the same values. For Cu 2OSeO 3 we evaluated in two ways[32]. When extracting the linewidth
Hfor dierent resonance frequencies fr, the Gilbert damping parameter intrwas assumed to vary according to [33, 34] 0

H= 4intr
fr+0

H0; (1) where is the gyromagnetic factor and
H0the contri- bution due to inhomogeneous broadening. Equation (1) is valid when viscous Gilbert damping dominates over scattering within the magnetic subsystem [35]. When performing frequency-swept measurements at dierent eldsH, the obtained linewidth
fwas considered to scale linearly with the resonance frequency as [36]
f= 2intr
fr+
f0; (2) with the inhomogeneous broadening
f0. The conver- sion from Eq. (1) to Eq. (2) is valid when frscales linearly withHandHis applied along a magnetic easy or hard axis of the material [37, 38]. In Fig. 1 (a) to (d) we show spectra recorded in the FP state of the material using the two dierent CPWs. For the same applied eld Hwe ob- serve peaks residing at higher frequency fforHkh100i compared to Hkh111i. From the resonance frequencies, we extract the cubic magnetocrystalline anisotropy con- stantK= (0:60:1)
103J/m3for Cu 2OSeO 3[31]. The magnetic anisotropy energy is found to be extremal forh100iandh111ire ecting easy and hard axes, respec- tively [31]. The saturation magnetization of Cu 2OSeO 3 amounted to 0Ms= 0:13 T at 5 K[22]. Figure 1 summarizes spectra taken with two dier- ent CPWs on two dierent Cu 2OSeO 3crystals exhibit- ing dierent crystallographic orientation in the eld H. For the narrow CPW [Fig. 1 (a) and (c)], we observed a broad peak superimposed by a series of resonances that all shifted to higher frequencies with increasing H. The eld dependence excluded them from being noise or arti- facts of the setup. Their number and relative intensities varied from sample to sample and also upon remounting the same sample in the cryostat (not shown). They disap- peared with increasing temperature Tbut the broad peak remained. For the broad CPW [Fig. 1 (b) and (d)], we measured pronounced peaks whose linewidths were sig- nicantly smaller compared to the broad peak detected with the narrow CPW. We resolved resonances below the large peaks [arrows in Fig. 1 (b)] that shifted with Hand exhibited an almost eld-independent frequency oset from the main peaks that we will discuss later. It is instructive to rst follow the orthodox approach and analyze damping parameters from modes re ecting the 69121518-0.4-0.20.0(d)Δ |S12|f (GHz)H || 〈111〉 69121518-6-30(c)Δ |S12| (10-2)f (GHz) -0.6-0.4-0.20.0H || 〈100〉 (b)broad CPWΔ |S12| -0.3-0.2-0.10.00 .35 T(a)narrow CPWΔ |S12|0 .25 T0 .45 T0.55 TFIG. 1. (Color online) Spectra
jS12jobtained at T = 5 K for dierent Husing (a) a narrow and (b) broad CPW when Hjjh100ion sample S1. Corresponding spectra taken on sam- ple S2 for Hjjh111iare shown in (c) and (d), respectively. Note the strong and sharp resonances in (b) and (d) when us- ing the broad CPW that provides a much more homogeneous excitation eld h. Arrows mark resonances that have a eld- independent oset with the corresponding main peaks and are attributed to standing spin waves. An exemplary Lorentz t curve is shown in blue color in (b). excitation characteristics of the CPW [29]. Second, we follow Ref. [7] and analyze conned modes. Lorentz curves (blue) were tted to the spectra recorded with the broad CPW to determine resonance frequencies and linewidths. Note that the corresponding linewidths were larger by a factor ofp 3 compared to the linewidth
fthat is conventionally extracted from the imaginary part of the scattering parameters [39]. The extracted linewidths
fwere found to follow linear ts based on Eq. (2) at dierent temperatures (details are shown in Ref. [31]). In Fig. 2 (a) we show a resonance curve that was obtained as a function of Htaken with the narrow CPW at 15 GHz. The curve does not show sharp features as Hwas varied in nite steps (symbols). The linewidth
H(symbols) is plotted in Fig. 2 (b) for dierent resonance frequencies and temperatures. The data are well described by linear ts (lines) based on Eq. (1). Note that the resonance peaks measured with the broad CPW were extremely sharp. The sharpness did not allow us to analyze the resonances as a function ofH. We refrained from tting the broad peaks of Fig. 1 (a) and (c) (narrow CPW) as they showed a clear asym- metry attributed to the overlap of subresonances at nite wavevector k, as will be discussed below. In Fig. 3 (a) and (b) we compare the parameter intr obtained from both dierent CPWs (circles vs. stars) and the two evaluation routes [40]. For Hkh100i[Fig. 3 (a)], between 5 and 20 K the lowest value for intramounts to (3.70.4)103. This value is three times lower com- pared to preliminary data presented in Ref. [29]. Beyond3 FIG. 2. (Color online) (a) Lorentz curve (magenta line) tted to a resonance (symbols) measured at f= 15 GHz as a func- tion ofHat 5 K. (b) Frequency dependencies of linewidths
H(symbols) for four dierent T. We performed thep 3- correction. The slopes of linear ts (straight lines) following Eq. 1 are considered to re ect the intrinsic damping parame- tersintr. 04812H || 〈100〉 αintr (10-3)Δ H narrow CPWΔ f broad CPW H || 〈111〉 1020304050T (K) 10203040500.00.20.40.60.8Δf0 (GHz)T (K)(b)( a)( d)( c) FIG. 3. (Color online) (a) and (b) Intrinsic damping param- etersintrand inhomogeneous broadening
f0for two dier- ent eld directions (see labels) obtained from the slopes and intercepts at fr= 0 of linear ts to the linewidth data (see Fig. 2 (b) and Ref. [31]). Dashed lines are guides to the eyes. 20 K the damping is found to increase. For Hkh111i [Fig. 3 (b)] we extract (0.6 0.6)103as the smallest value. Note that these values for intrstill contain an ex- trinsic contribution and thus represent upper bounds for Cu2OSeO 3, as we will show later. For the inhomogeneous broadening
f0in Fig. 3 (c) and (d) the datasets are consistent (we have used the relation
f0=
H0=2 to convert
H0into
f0). We see that
f0increases withTand is small for the broad CPW, independent of the crystallographic direction of H. For the narrow CPW the inhomogeneous broadening is largest at small Tand then decreases by about 40 % up to about 50 K. Note that a CPW broader than the sample is as- sumed to excite homogeneously at fFMR [41] transfer- ring a wave vector k= 0 to the sample. Accordinglywe ascribe the intense resonances of Fig. 1 (b) and (d) to fFMR. UsingfFMR= 6 GHz and intr= 3:7103at 5 K [Fig. 3 (a)], we estimate a minimum relaxation time of = [2intrfr]1= 6:6 ns. In the following, we examine in detail the additional sharp resonances that we observed in spectra of Fig. 1. In Fig. 1 (b) taken with the broad CPW for Hkh100i, we identify sharp resonances that exhibit a characteris- tic frequency oset fwith the main resonance at all elds (black arrows). We illustrate this in Fig. 4(a) in that we shift spectra of Fig. 1 (b) so that the positions of their main resonances overlap. The additional small res- onances (arrows) in Fig. 1 (b) are well below the uniform mode. This is characteristic for backward volume magne- tostatic spin waves (BVMSWs). Standing waves of such kind can develop if they are re ected at least once at the bottom and top surfaces of the sample. The resulting standing waves exhibit a wave vector k=n=d , with order number nand sample thickness d= 0:3 mm. The BVMSW dispersion relation f(k) of Ref. [13] provides a group velocity vg=300 km/s at k==d[triangles in Fig. 4 (b)]. Hence, the decay length ld=vgamounts to 2 mm considering = 6:6 ns. This is larger than twice the relevant lateral sizes, thereby allowing stand- ing spin wave modes to form in the sample. Based on the dispersion relation of Ref. [13], we calculated the fre- quency splitting f=fFMRf(n=d ) [open diamonds in Fig. 4 (b)] assuming n= 1 andt= 0:4 mm for the sample width tdened in Ref. [13]. Experimental val- ues (lled symbols) agree with the calculated ones (open symbols) within about 60 MHz. In case of the narrow CPW, we observe even more sharp resonances [Fig. 1 (a) and (c)]. A set of resonances was reported previously in the eld-polarized phase of Cu 2OSeO 3[26, 28, 42, 43]. Maisuradze et al. assigned secondary peaks in thin plates of Cu 2OSeO 3to dierent standing spin-wave modes [43] in agreement with our analysis outlined above. 0.30.40.51.101.151.201.25- 500-300-100100δf (GHz)(b)/s61549 0H (T)v g (km/s) -10 -0.8-0.6-0.4-0.20.0f - f (0) (GHz)H || 〈100〉 (a)b road CPWΔ |S12|δ f FIG. 4. (Color online) (a) Spectra of Fig. 1 (b) replotted as ffFMR(H) for dierent Hsuch that all main peaks are at zero frequency and the eld-independent frequency splitting fbecomes visible. The numerous oscillations seen particu- larly on the bottom most curve are artefacts from the cali- bration routine. (b) Experimentally evaluated (lled circles) and theoretically predicted (diamonds) splitting fusing dis- persion relations for a platelet. Calculated group velocity vg atk==(0:3 mm). Dashed lines are guides to the eyes.4 The inhomogeneous dynamic eld hof the narrow CPW provides a much broader distribution of kcom- pared to the broad CPW. This is consistent with the fact that the inhomogeneous broadening
f0is found to be larger for the narrow CPW compared to the broad one [Fig. 3 (c) and (c)]. Under these circumstances, the excitation of more standing waves is expected. We at- tribute the series of sharp resonances in Fig. 1 (a) and (c) to such spin waves. In Fig. 5 (a) and (b) we highlight prominent and particularly narrow resonances with #1, #2 and #3 recorded with the narrow CPW. We trace their frequencies fras a function of HforHkh100iand Hkh111i, respectively. They depend linearly on Hsug- gesting a Land e factor g= 2:14 at 5 K. We now concentrate on mode #1 for Hk h100iat 5 K that is best resolved. We t a Lorentzian line- shape as shown in Fig. 5(c) for 0.85 T, and summarize the corresponding linewidths
fin Fig. 5(d). The inset of Fig. 5(d) shows the eective damping e=
f=(2fr) evaluated directly from the linewidth as suggested in Ref. [29]. We nd that eapproaches a value of about 3.5 104with increasing frequency. This value includes both the intrinsic damping and inhomogeneous broad- ening but is already a factor of 10 smaller compared to intrextracted from Fig. 3 (a). Note that Cu 2OSeO 3 exhibiting 3.5104outperforms the best metallic thin- lm magnet [44]. To correct for inhomogeneous broad- ening and determine the intrinsic Gilbert-type damping, we apply a linear t to the linewidths
fin Fig. 5(d) at fr>10:6 GHz and obtain (9.9 4.1)105. Forfr 10.6 GHz the resonance amplitudes of mode #1 were small reducing the condence of the tting procedure. Furthermore, at low frequencies, we expect anisotropy to modify the extracted damping, similar to the results in Ref. [45]. For these reasons, the two points at low frwere left out for the linear t providing (9.9 4.1)105. We nd
fand the damping parameters of Fig. 3 to increase with T. It does not scale linearly for Hkh100i [31]. A deviation from linear scaling was reported for YIG single crystals as well and accounted for by the con- uence of a low- kmagnon with a phonon or thermally excited magnon [5]. In the case of Hkh111i(cf. Fig. 3 (b)) we obtain a clear discrepancy between results from the two evaluation routes and CPWs used. We relate this observation to a misalignment of Hwith the hard axish111i. The misalignment motivates a eld-dragging contribution [38] that can explain the discrepancy. For this reason, we concentrated our standing wave analysis on the case Hkh100i. We now comment on our spectra taken with the broad CPW that do not show the very small linewidth attributed to the conned spin waves. The sharp mode #1 yields
f= 15:3 MHz near 16 GHz [Fig. 5 (d)]. At 5 K the dominant peak measured at 0.55 T with the broad CPW provides however
f= 129 MHz.
fobtained by the broad CPW is thus increased by a factor of eight and explains the relatively large Gilbert FIG. 5. (Color online) (a)-(b) Resonance frequency as a func- tion of eld Hof selected sharp modes labelled #1 to #3 (see insets) for Hkh100iandHkh111iat T = 5 K. (c) Exemplary Lorentz t of sharp mode #1 for Hkh100iat 0.85 T. (d) Ex- tracted linewidth
f as a function of resonance frequency fr along with the linear t performed to determine the intrinsic dampingintrin Cu 2OSeO 3. Inset: Comparison among the extrinsic and intrinsic damping contribution. The red dotted lines mark the error margins of intr= (9:94:1)105. damping parameter in Fig. 3 (a) and (b). We conrmed this larger value on a third sample with Hkh100iand ob- tained (3.10.3)103[31] using the broad CPW. The discrepancy with the damping parameter extracted from the sharp modes of Fig. 5 might be due to the remaining inhomogeneity of hover the thickness of the sample lead- ing to an uncertainty in the wave vector in z-direction. For a standing spin wave such an inhomogeneity does not play a role as the boundary conditions discretize k. Accordingly, Klingler et al. extract the smallest damp- ing parameter of 2 :7(5)105reported so far for the ferrimagnet YIG when analyzing conned magnetostatic modes [7]. To summarize, we investigated the spin dynamics in the eld-polarized phase of the insulating chiral mag- net Cu 2OSeO 3. We detected numerous sharp reso- nances that we attribute to standing spin waves. Their eective damping parameter is small and amounts to 3:5104. A quantitative estimate of the intrinsic Gilbert damping parameter extracted from the conned modes provides even intr=(9.94.1)105at 5 K. The small damping makes an insulating ferrimagnet exhibit- ing Dzyaloshinskii-Moriya interaction a promising can- didate for exploitation of complex spin structures and related nonreciprocity in magnonics and spintronics. We thank S. Mayr for assistance with sample prepa- ration. Financial support through DFG TRR80, DFG 1143, DFG FOR960, and ERC Advanced Grant 291079 (TOPFIT) is gratefully acknowledged.5 Electronic mail: dirk.grundler@ep .ch [1] I. Zutic and H. Dery, Nat. Mater. 10, 647 (2011). [2] M. Krawczyk and D. Grundler, J. Phys.: Condens. Mat- ter26, 123202 (2014). [3] A. V. 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1507.06748v1.Boosting_Domain_Wall_Propagation_by_Notches.pdf

arXiv:1507.06748v1 [cond-mat.mes-hall] 24 Jul 2015Boosting Domain Wall Propagation by Notches H. Y. Yuan and X. R. Wang1,2,∗ 1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2HKUST Shenzhen Research Institute, Shenzhen 518057, China Wereportacounter-intuitivefindingthatnotchesinanothe rwise homogeneousmagnetic nanowire can boost current-induced domain wall (DW) propagation. DW motion in notch-modulated wires can be classified into three phases: 1) A DW is pinned around a n otch when the current density is below the depinning current density. 2) DW propagation ve locity is boosted by notches above the depinning current density and when non-adiabatic spin- transfer torque strength βis smaller than the Gilbert damping constant α. The boost can be manyfold. 3) DW propagation velocity is hindered when β > α. The results are explained by using the Thiele equation. PACS numbers: 75.60.Ch, 75.78.-n, 85.70.Ay, 85.70.Kh I. INTRODUCTION Magnetic domain wall (DW) motion along a nanowire underpins many proposals of spintronic devices1,2. High DW propagation velocity is obviously important because it determines the device speed. In current-driven DW propagation,many efforts havebeen devoted to high DW velocity and low current density in order to optimize de- vice performance. The issue of whether notches can en- hance current-induced DW propagation is investigated here. Traditionally, notches are used to locate DW positions1–4. Common wisdom expects notches to strengthen DW pinning and to hinder DW motion. In- deed, in the field-driven DW propagation, intentionally created roughness slows down DW propagation although they can increase the Walker breakdown field5. Unlike the energy-dissipation mechanism of field-induced DW motion6, spin-transfer torque (STT)7–10is the driven force behind the current-driven DW motion. The torque consists of an adiabatic STT and a much smaller non- adiabatic STT9,10. In the absence of the non-adiabatic STT, there exists an intrinsic pinning even in a homoge- neous wire, below which a sustainable DW motion is not possible11,12. Interestingly, there are indications13that the depinningcurrentdensityofaDWtrappedinanotch is smaller than the intrinsic threshold current density in the absence of the non-adiabatic STT. Although there is no intrinsic pinning1,10in the presence ofa non-adiabatic STT, It is interesting to ask whether notches can boost DW propagation in the presence of both adiabatic STT and non-adiabatic STT. In this paper, we numerically study how DW propa- gates along notch-modulated nanowires. Three phases are identified: pinning phase when current density is be- low depinning current density ud; boosting phase and hindering phase when the current density is above ud andthe non-adiabaticSTT strength βissmallerorlarger than the Gilbert damping constant α, respectively. The average DW velocity in boosting and hindering phases is respectively higher and lower than that in the wire without notches. It is found that DW depinning is facili-tated by antivortex nucleation. In the case of β < α, the antivortexgenerationis responsiblefor velocityboost be- cause vortices move faster than transverse walls. In the other case of β > α, the longitudinal velocity of a vor- tex/antivortex is slower than that of a transverse wall in ahomogeneouswallandnotcheshinderDWpropagation. II. MODEL AND METHOD We consider sufficient long wires (with at least 8 notches) of various thickness and width. It is well known that14narrowwiresfavoronlytransversewallswhilewide wires prefer vortex walls. Transverse walls are the main subjects of this study. A series of identical triangular notches of depth dand width ware placed evenly and alternately on the two sides of the nanowires as shown in Fig. 1a with a typical clockwise transverse wall pinned at the center of the first notch. The x−,y−andz−axis are along length, width, and thickness directions, respec- tively. The magnetization dynamics of the wire is gov- erned by the Landau-Lifshitz-Gilbert (LLG) equation ∂m ∂t=−γm×Heff+αm×∂m ∂t−(u·∇)m+βm×(u·∇)m, wherem,γ,Heff, andαare respectively the unit vec- tor of local magnetization, the gyromagnetic ratio, the effective field including exchange and anisotropy fields, and the Gilbert damping constant. The third and fourth terms on the right hand side are the adiabatic STT and non-adiabatic STT10. The vector uis along the electron flow direction and its magnitude is u=jPµB/(eMs), wherej,P,µB,e, andMsare current density, current polarization,the Bohrmagneton, the electronchargeand the saturation magnetization, respectively. For permal- loy ofMs= 8×105A/m,u= 100 m/s corresponds toj= 1.4×1012A/m2. In this study, uis lim- ited to be smaller than both 850 m/s (corresponding to j≃1.2×1013A/m2!) and the Walkerbreakdowncurrent density because current density above the values gener- atesintensivespinwavesaroundDWsandnotches,which makes DW motion too complicated to be even described.2 xy z (b) (a) L w dj FIG. 1. (color online) (a) A notch-modulated nanowire. L is the separation between two adjacent notches. The color codes the y−component of mwith red for my= 1, blue for my=−1 and green for my= 0. The white arrows denote magnetization direction. (b) The phase diagram in β−u plane. A is the pinning phase; B is the boosting phase; and C is the hindering phase. Vortices are (are not) generated nea r notches by a propagating DW in C1 (C2). Inset: The notch depth dependence of depinning current udwhen notch width is fixed at w= 48 nm. Dimensionless quantity βmeasures the strength of non- adiabatic STT and whether βis larger or smaller than α is still in debate10,15,16. The LLGequation isnumerically solved by both OOMMF17andMUMAX18packages19. The electric current density is modulated according to wire cross section area while the possible change of current direction around notch is neglected. The material pa- rameters are chosen to mimic permalloy with exchange stiffness A= 1.3×10−11J/m,α= 0.02 andβvarying from 0.002 to 0.04. The mesh size is 4 ×4×4 nm3. III. RESULTS A. Transverse walls in wide wires: boosting and hindering This is the focus of this work. Our simulations on wires of 4 nm thick and width ranging from 32 nm to 128 nm and notches of d= 16 nm and wvarying from 16 nm to 128 nm show similar behaviors. Domain walls in these wires are transverse. Results presented below are on a wire of 64 nm wide and notches of w= 48 nm. Three phases can be identified. A DW is pinned at a notch when uis below a depinning current density ud. This pinning phase is denoted as A (green region) in Fig. 1b. Surprisingly, udincreases slightly with β, indicatingthat the β-term actually hinders DW depinning out of a notch although it is responsible for the absence of the intrinsic pinning in a uniform wire (see discussion below for possible cause). When uis above ud, a DW starts to propagate and it can either be faster or slower than the DW velocity in the corresponding uniform wire, depend- ing on relative values of βandα. Whenβ < α, DW velocity is boosted through antivor- texgenerationat notches. Thisphaseisdenoted asphase B. When β > α, the boosting of DW propagation is sup- pressed no matter vortices are generated (phase C1) or not (phase C2). The upper bound of the phase plane is determined by the Walker breakdown current density andu= 850 m/s. If the current density is larger than the upper bound, spin waves emission from DW20and notches are so strong that new DWs may be created. Also, the Walker breakdown is smaller than the depin- ning value udforβ >0.04. Thus the phase plane in Fig. 1b is bounded by β= 0.04. Although the general phase diagram does not change, the phase boundaries depend on the wire and notch specificities. The inset is notch depth dependence of the depinning current when w= 48 nm andβ= 0.0121. Boosting phase: The boost of DW propagation for β < α can be clearly seen in Fig. 2. Figure 2a is the average DW velocity ¯ vas a function of notch separation Lfor u= 600 m/s > ud. ¯vis maximal around an optimal notch separation Lp, which is close to the longitudinal distance that an antivortex travels in its lifetime. Lp increases with βand it is respectively about 1.5 µm, 2 µm, and 4 µm forβ= 0.005 (squares), 0.01 (circles) and 0.015 (up-triangles). This result suggests that the an- tivortex generation and vortex dynamics are responsible for the DW propagation boost. Filled symbols in Fig. 2b are ¯vfor various current density when Lpis used. For a comparison, DW velocities in the corresponding homoge- neous wires are also plotted as open symbols which agree perfectlywith ¯ v=βu/αdiscussedbelow. Take β= 0.005 as an example, ¯ vis zero below ud= 550 m/s and jumps to an average velocity ¯ v≃550 m/s at ud, which is about four times of the DW velocity in the homogeneous wire. As the current density further increases, the average ve- locityalsoincreasesandisapproximatelyequalto u. The inset of Fig. 2b shows the instantaneous DW velocity for β= 0.005 and u= 600 m/s. Blue dots denote the mo- ments at which the DW is at notches. Right after the current is turned on at t= 0 ns, the instantaneous DW velocity is very low until an antivortex of winding num- berq=−123,24is generatednear the notch edge at 0.5ns (see discussion and Fig. 9 below). The motion of the an- tivortex core drags the whole DW to propagate forward at a velocity around 600 m/s. The antivortex core anni- hilatesitselfatthebottomedgeofthewireaftertraveling about 1.5 µm and the initial transverse wall reverses its chirality at the same time24. Surprisingly, the reversal of DW chiralityleads to a significantincreasesofDW veloc- ity as shown by the peaks of the instantaneous velocity at about 2.0ns in the inset. Another antivortex of wind-3 (a) (b) FIG. 2. (color online) (a) L−dependence of average DW ve- locity ¯vforu= 600 m/s, α= 0.02, andβ= 0.005 (squares), 0.01 (circles), 0.015 (up-triangles). The dash lines are βu/α. (b)u−dependence of ¯ vforβ= 0.005 (squares), 0.01 (circles), 0.015 (up-triangles). Open symbols are DW velocity in the corresponding homogeneous wires. Straight lines are βu/α. ¯v is above βu/αwhenu > u d. Inset: instantaneous DW speed foru= 600 m/s, β= 0.005, and L= 1.5µm. The blue dots indicate the moments when the DW is at notches. ing number q=−1 is generated at the second notch and DW propagation speeds up again. Once the antivortex core forms, it pulls the DW out of notch. This process then repeats itself and the DW propagates at an average longitudinal velocity of about 600 m/s. A supplemental movie corresponding to the inset is attached25. Hindering phase: Things are quite different for β > α. Figure 3a shows that ¯ vincreases monotonically with L forβ= 0.025, 0.03 and 0.035, which are all larger than α. In order to make a fair comparison with the results of β < α, Fig. 3b is the current density dependence of ¯ vfor L= 2µm andβ= 0.025 (filled squares), 0.03 (filled cir- cles)and0.035(filledup-triangles). Again,DWvelocities in the corresponding homogeneouswires are presented as open symbols. Take β= 0.025 as an example, although the average velocity jumps at the depinning current den- sity 565 m/s, it’s still well below the DW velocity in the corresponding uniform wire. The inset of Fig. 3b shows(b) (a) FIG. 3. (color online) (a) L−dependence of ¯ vforu= 600 m/s and β= 0.025 (squares), 0.03 (circles), and 0.035 (up- triangles), all larger than α= 0.02. The dash lines are βu/α. (b)u−dependence of ¯ vforL= 2µm. Fill symbols (squares forβ= 0.025, circles for β= 0.03, and up-triangles for β= 0.035) are numerical data in notched wire of w= 48 nm andd= 16 nm. Open circles are DW velocity of the cor- respondinghomogeneous wire. Straight lines are βu/α. Inset: instantaneous DW velocity for u= 600 m/s and β= 0.025. ThebluedotsdenotethemomentswhentheDWisatnotches. the instantaneous DW velocity for u= 600 m/s. An antivortex is generated at the first notch. In contrast to the case of β < α, the antivortex slows down DW propagation velocity below the value in the correspond- ing uniform wire. Moreover, the transverse wall keeps its original chirality unchanged when the antivortex is anni- hilated at wire edge, and no vortex/antivortex is gener- ated at the second notch. However, another antivortex is generated at the third notch. This is the typical cycle of phase C1. As uincreases above 640 m/s, phase C1 disappears and the DW passes all the notches without generating any vortices. This motion is termed as phase C2. For β >0.025, only phase C2 is observed. In C2, DW profile is not altered, and the average DW velocity is slightly below that in a uniform wire.4 (b) (a) FIG. 4. (color online) (a) u−dependence of ¯ vforβ= 0.01 (filled circles) and 0.015 (filled up-triangles). (b) u−dependence of ¯ vforβ= 0.03 (filled squares) and 0.035 (filled up-triangles). Open symbols are DW velocity in the corresponding homogeneous wires. Straight lines are βu/α. ¯v is below βu/αwhenu > u d. The nanowire is 8 nm wide and 1 nm thick while the notch size is 10 nm wide and 2 nm deep for (a) and 50 nm wide and 2 nm deep for (b). The separation of adjacent notches is 100 nm. B. Transverse walls in very narrow wires One interesting question is whether notches can boost DW propagation in very narrow wires such that the nu- cleation of a vortex/antivortex is highly unfavorable. To address this issue, Fig. 4a are u−dependence of the av- erage DW velocity on a 8 nm wide wire for β < α(circles forβ= 0.01 and up-triangles for β= 0.015) with (filled symbols) and without (open symbols) notches. When notches are placed, notch depth is 2 nm, L= 100 nm, w= 10 nm. DW velocity in the corresponding ho- mogeneous wire (open symbols) follows perfectly with ¯v=βu/α(straight lines). It is clear that averaged DW velocity in the notched wire (filled symbols) is below the values of the DW velocity in the corresponding homo- geneous wire. Take β= 0.015 as an example, ¯ vis zero belowud= 310 m/s and jumps to an average velocity ¯v≃168 m/s at ud, which is below the DW velocity in the corresponding uniform wire. Things are similar for β > α. Figure 4b is the cur- rent density dependence of ¯ vforβ= 0.03 (filled squares) and 0.035 (filled up-triangles). Again, DW velocities in homogeneous wire are presented as open symbols for a comparison. The averaged DW velocity in the notched wire (filled symbols) is below the values of the DW ve- locity in the corresponding homogeneous wire. C. Vortex walls in very wide wires Although our main focus is on transverse walls, it should be interesting to ask whether DW propagation boost can occur for vortex walls. It is well-known that a vortex/antivortex wall is more stable for a much wider wire in the absence of a field and a current14. One may expect that DW propagation boost would not oc- cur in such a wire because the boost comes from vor- tex/antivortex generation near notches and a such vor- tex/antivortex exists already in a wider wire even in the my +1 -1 0200 nm 1.5 ns 13.5 ns 18.0 ns 26.0 ns 0 ns 47.5 ns 14.5 ns 0 ns (c) (d) (a) (b) FIG. 5. (color online) (a) u−dependence of ¯ vforβ= 0.01 (filled circles) and 0.015 (filled up-triangles). (b) u−dependence of ¯ vforβ= 0.025 (filled squares) and 0.03 (filled circles). Open symbols are DW velocity in the corre- sponding homogeneous wires. Straight lines are βu/α. ¯vis above (below) βu/αwhenu > u dandβ < α(β > α). The nanowire is 520 nm wide and 10 nm thick while the rectan- gular notch is 160 nm wide and 60 nm deep. The separation of adjacent notches is 8 µm. (c) and (d) The spin configu- rations in a uniform wire (a) and in a notched wire (b) at various moments for β= 0.01 andu= 650 m/s. The time is indicated on the bottom-right corner of each configuratio n. The color codes the value of myand color bar is shown in the bottom-right corner. absence of a current. However, DW propagation boost was still observed as shown in Fig. 5 for a wire of 520 nm wide and 10 nm thick. Rectangular notches of 60 nm deep and 160 nm wide are separated by L= 8µm. Whenβ < α(Fig. 5a: circles for β= 0.01 and up- triangles for β= 0.015), the average DW propagation velocities in the notched wire (filled symbols) is higher than the DW velocity in the corresponding homogeneous wire (open symbols) when u > ud. Figure 5b shows that the average DW propagation velocities in a notched wire (filled symbols) is lower than that in the corresponding homogeneous wire (open symbols) for β > α(squares for β= 0.025 and circles for β= 0.03). Figure 5c shows the spin configurations of the DW in the homogeneous wire ofβ= 0.01before a current is applied (the left configura- tion) and during the current-driven propagation (middle and right configurations). When a current u= 650 m/s is applied at 0 ns, a vortex wall moves downward. The vortex was annihilated at wire edge, and the vortex wall transformintoatransversewall. TheDWkeepsitstrans- verse wall profile and propagates with velocity of βu/α (solid lines in Fig. 5a and 5b). The middle and right configurations are two snapshots at 14.5 ns and 47.5 ns. Time is indicated in the bottom-right corner. Figure 5d are snapshots of DW spin configurations in the notched wire ofβ= 0.01 when a current u= 650 m/s is applied5 u m xxy zu m x(a) (b) FIG. 6. (color online) Directions of vortex core magnetizat ion (red symbols) and non-adiabatic torque (blue symbols) for a clockwise transverse wall (a) and a counterclockwise trans - verse wall (b). The dots (crosses) represent ±z-direction. att= 0 ns. At t= 0 ns, a vortex wall is pinned near the first notch. Right after the current is turned on, the vortex wall starts to depin and complicated structures may appear during the depinning process as shown by the snapshot at t= 1.5 ns. At t= 13.5 ns, the DW transforms to a transverse wall and propagates forward. When the transverse wall reaches the second notch at aboutt= 18.0 ns, new vortex core nucleates near the notch and drags the whole DW to propagate forward. In contrast to the case of homogeneous wire where a prop- agating DW prefers a transverse wall profile, DW with more than one vortices can appear as shown by the snap- shot att= 26.0 ns. The vortex core in this structure boosts DW velocity above the average DW velocity of a uniform wire. This finding may also explain a surprising observation in an early experiment4that depinning cur- rent does not depend on DW types. A vortex wall under a current transforms into a transverse wall before depin- ning from a notch. Thus both vortex wall and transverse wall have the same depinning current. IV. DISCUSSION A. Depinning process analysis Empirically, we found that vortex/antivortex polarity is uniquely determined by the types of transverse wall and current direction. This result is based on more than twenty simulations that we have done by varying var- ious parameters like notch geometry, wire width, mag- netic anisotropy, damping etc. Within the picture that DW depinning starts from vortex/antivortex nucleation, theβ−dependence of depinning current density udcan be understood as follows. For a clockwise (counter- clockwise) transverse wall and current in −xdirection, p= +1 (p=−1), as shown in Fig. 6. If one as- sumes that vortex/antivortex formation starts from the vortex/antivortex core, it means that the core spin ro- tates into + z-direction for a vortex of p= 1. For a clock- wise wall, β-torque ( βm×∂m ∂x) tends to rotate core spin in−z-direction, as shown in Fig. 6a, so the presence of a smallβ-torque tries to prevent the nucleation of vor- tices. Thus, the larger βis, the higher udwill be. This may be the reason why the depinning current density ud increases as βincreases.(a) (b) FIG. 7. (a) Depinning current density as a function of an external field. A 0.4 ns field pulse in the x-direction is turned on simultaneously with the current. The shape of a pulse of H= 100 Oe is shown in the inset. Since the depinning field of the wire (64 nm wide and 4 nm thick) is 150 Oe, the field amplitude is limited to slightly below 150 Oe in the curve. (b ) Depinningcurrentdensityas afunctionof nanowire thickne ss. Our simulations suggest that DW depinning starts from vortex/antivortexnucleation. Adiabatic spin trans- fertorquetendstorotatethespinsattheedgedefectnear a notch out of plane and to form a vortex/antivortex core. Thus, any mechanisms that help (hinder) the creation of a vortex/antivortex core shall decrease (in- crease) the depinning current density ud. To test this hypothesis, we use a magnetic field pulse of 0.4 ns along ±x−direction (shown in the inset of Fig. 7a) such that the field torque rotates spins out of plane. Figure 7a is the numerical results of the magnetic field depen- dence of the depinning current density for a 64 nm wide wire with triangular notches of 48 nm wide and 16 nm deep. The non-adiabatic coefficient is β= 0.01. As expected, uddecreases (increases) with field when it is along -x−direction (+ x−direction) so that spins rotate into +z−direction (- z−direction). All other parameters are the same as those for Fig. 2. If the picture is correct, one should also expect the depinning current density depends on the wire thick- ness. The shape anisotropy impedes vortex core for- mation because it does not favor a spin aligning in the z−direction. The shape anisotropy decreases as the thickness increases. Thus, one should expect the depin- ning current density decreases with the increase of wire thickness. Indeed, numerical results shown in Fig. 7b verifiestheconjecture. All otherparametersarethe same as that in Fig. 7a ( H= 0). B. Width effects on the depinning current density The DW propagating boost shown above is from the wire in which the notch depth (16 nm) is relatively big in comparisonwith wire width (64 nm). Naturally, one may ask whether the DW propagation boost exists also in a wire when the notch depth is much smaller than the wire width. To address the issue, we fix the notch geometry and vary the wire width. Figure 8 is the nanowire width dependence of depinning current density when the notch6 (a) (b) (c) (d) 50 nm 50 nm FIG. 8. (color online) (a) and (b) are nanowire width de- pendence of depinning current density for β= 0.005 (a) and β= 0.01 (b), respectively. The wire thickness is 4 nm and notch size is fixed at 48 ×16 nm2. (c) and (d) are the real configurations of initial domain walls pinned at the notch fo r 64 nm and 160 nm wide wires, respectively. The color coding is the same as that of Fig. 5. The blue jagged lines indicate the profiles of triangular notches. size is fixed at 48 ×16 nm2. Figures 8a and 8b show the phase boundary between vortex-assisted boosting phase and the pinning phase. DW propagation boost exists when nanowire width is one order of magnitude larger than the notch depth. The top view of the wire and spin configurations for 64 nm wide and 160 nm wide wires are shown in Fig. 8c and Fig. 8d, respectively. C. DW Propagation and vortex dynamics DW propagation boost and slow-down by vortices can be understood from the Thiele equation10,26,27, F+G×(v−u)+D·(αv−βu) = 0,(1) whereFis the external force related to magnetic field that is zero in our case, Gis gyrovector that is zero for a transverse wall and G=−2πqplM s/γˆ zfor a 2D vortex wall, where qis the winding number (+1 for a vortex and -1foranantivortex), pisvortexpolarity( ±1forcorespin in±zdirection) and lis the thickness of the nanowire. D is dissipation dyadic, whose none zero elements for a vor- tex/antivortex wall are Dxx=Dyy=−2MsWl/(γ∆)27, whereWis nanowire width and ∆ is the Thiele DW width26.vis the DW velocity. For a transverse wall, v=βu/α(solid lines) agrees perfectly with numerical results (open symbols) in ho- mogeneous wires as shown in Figs. 2b and 3b without any fitting parameters. For a vortex wall, the DW veloc- ity is vy=1 1+α2W2/(π2∆2)W πqp∆(α−β)u,(2) vx=u 1+α2W2/(π2∆2)/parenleftbigg 1−β α/parenrightbigg +βu α.(3)vydepends on DW width, αas well as β/α. For a given vortexwall, vyhas opposite sign for β < αandβ > α. In terms of topological classification of defects23, the edge defect of the transverse DW at the first notch (Fig. 1a) has winding number q=−1/2, and this edge defect can onlygivebirthtoanantivortexof q=−1andp= 1while itself changes to an edge defect of q= 1/2 as shown in Fig. 9a. Empirically, we found that antivortexpolarityis uniquely determined by the types of transverse wall and current direction. A movie visualizing the DW propa- gation in boosting phase is shown in the Supplemental Movie25. All the parameters are the same as the inset of Fig. 2b. The three segments of identical length 1200 nm are connected in series to form a long wire. When β < α, the antivortex moves downward ( vy<0) to the lower edge defect of winding number of q= 1/2. The lower edge defect changes its winding number to q=−1/2 and the transverse DW reverses its chirality24when the vortex merges with the edge defect. Then another an- tivortex of winding number q=−1 andp=−1 is gen- erated at the second notch on the lower wire edge and it moves upward ( vy>0). The DW reverses its chiral- ity again at upper wire edge when the antivortex dies. Then this cycle repeats itself. The spin configurations corresponding to various stages are shown in the lower panels of Fig. 9a. When β > α, as shown in Fig. 9b, the antivortex of q=−1 andp= +1 moves upward sincevy>0. The chirality of the original transverse wall shall not change when the antivortex is annihilated at the upper edge defect because of winding number con- servation. No antivortex is generated at the even number notches and same type of the antivortex is generated at odd number notches, hence the transverse wall preserves its chirality throughout propagation. The corresponding spin configurations are shown in the lower panels of Fig. 9b. The second term in Eq. (3) (for vx) isβu/α, the same as the transverse DW velocity in a homogeneous wire (straight lines in Figs. 2b and 3b). The first term de- pends on DW properties as well as βandα. It changes sign atβ=α.vxis larger than βu/αin the presence of vortices if β < α. Therefore, in this case vortex genera- tions and vortex dynamics boost DW propagation. For smallαand to the leading order correction in αandβ, Eq. (3) becomes vx=u−(α2−αβ)uW2/(π2∆2). Thus, the longitudinal velocity equals approximately uand de- pends very weakly on β. This is what was observed in Fig. 2b. vx=ucorresponds to the complete conversion of itinerant electron spins into local magnetic moments. Although the Thiele equation cannot explain why a DW generates vortices around notches in phase B, it explains well DW propagation boost for β < α. This result is in contrast to the field-driven DW propagation where vor- tex/antivortexgenerationreduces the Walker breakdown field and inevitably slows down DW motion5,24. Before conclusion, we would also like to point out that it is possible to realize both β < α(boosting phase) and β > α(hindering phase) experimentally in magnetic ma-7 (a) (b) +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 -1/2 -1/2 -1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 -1/2 -1/2 +1/2 -1/2 -1/2 -1/2 -1 -1 -1 -1 -1 -1/2 (s1) (s2) (s3) (s4) (s5) s1 s1 (s1) (s2) (s3) (s4) (s5) s2 s2 s3 s3 s4 s4 s5 s5 (s6) (s6) 50 nm s6 s6 FIG. 9. (color online) (a) Illustrations of changes of topol og- ical defects (transverse DW edge defects and vortices) duri ng thebirthanddeathofvortices inPhase Bas aDWpropagates from the left to the right and the corresponding spin config- urations at various moments. Lines represent DWs. Big blue dots for vortices and open circles for edge defects of wind- ing number −1/2 and filled black circles for edge defects of winding number 1 /2. The color coding is the same as that of Fig. 5. The blue jagged lines indicate the profiles of trian- gular notches. The nanowire is 64 nm wide and 4 nm thick. The notch dimensions are 48 ×16 nm3. The interval between adjacent notches is L= 1500 nm. u= 600 m/s, β= 0.005. (b) Illustrations of changes of topological defects in Phas e C1 and the the corresponding spin configurations at various mo- ments. The nanowire is 64 nm wide and 4 nm thick. The notch dimensions are 48 ×16 nm2. The interval between ad- jacent notches is L= 2000 nm. u= 600 m/s, β= 0.025.terials like permalloy with damping coefficient engineer- ing. A recent study28demonstrated that αof permalloy can increaseby four times througha dilute impurity dop- ing of lanthanides (Sm, Dy, and Ho). V. CONCLUSIONS In conclusion, notches can boost DW propagation whenβ < α. The boost is facilitated by antivortex generation and motion, and boosting effect is optimal when two neighboring notches is separated by the dis- tance that an antivortex travels in its lifetime. In the boosting phase, DW can propagate at velocity uthat corresponds to a complete conversion of itinerant elec- tron spins into local magnetic moments. When β > α, the notches always hinder DW propagation. According to Thiele’s theory, the generation of vortices increases DW velocity for β < αand decreases DW velocity when β > α. This explains the origin of boosting phase and hindering phase. Furthermore, it is found that a vortex wall favored in a very wide wire tends to transform to a transverse wall under a current. This may explain exper- imental observation that the depinning current density is not sensitive to DW types. VI. ACKNOWLEDGMENTS We thank Gerrit Bauer for useful comments. HYY ac- knowledges the support of Hong Kong PhD Fellowship. This work was supported by NSFC of China (11374249) as well as Hong Kong RGC Grants (163011151 and 605413). ∗Corresponding author: phxwan@ust.hk 1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005). 3M. Kl¨ aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. Heyderman, F. Nolting, and U. R¨ udiger, Phys. Rev. Lett. 94, 106601 (2005). 4M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett. 97, 207205 (2006). 5Y. Nakatani, A. Thiaville, and J. Miltat, Nat. Mater. 2, 521 (2003). 6X. R. Wang, P. Yan, J. Lu and C. He, Ann. Phys. (N.Y.) 324, 1815 (2009); X. R. Wang, P. Yan, and J. Lu, Euro- phys. Lett. 86, 67001 (2009). 7L. Berger, J. Appl. Phys. 55, 1954 (1984). 8J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 9S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).10A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro- phys. Lett. 69, 990 (2005). 11Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004). 12G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). 13H. Y. Yuan and X. R. Wang, European Physical Journal B (in press); arXiv:1407.4559 [cond-mat.mes-hall] 14R. D. McMichael and M. J. Donahue, IEEE Trans. Magn. 33, 4167 (1997). 15G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 97, 057203 (2006). 16L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, Science330, 1810 (2010). 17http://math.nist.gov/oommf. 18A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, F. B. V. Waeyenberge, AIP Adbances 4, 107133 (2014).8 19OOMMF package was used in the early stage of this re- search. In order to simulate a long and wide wire, we switched to MUMAX package. Two packages give almost identical results on shorter wires, and the results present ed here were generated from MUMAX. 20B. Hu and X. R. Wang, Phys. Rev. Lett. 111, 027205 (2013); X. S. Wang, P. Yan, Y. H. Shen, G. E.W. Bauer, and X. R. Wang, Phys. Rev. Lett. 109, 167209 (2012). 21Notch geometry affects depinning current because of the change of current density and perpendicular shape anisotropy (see Ref. 22) in notch area. Both effects help to generate vortices and thus reduce the depinning current.This may explain the result. 22A. Aharoni, J. Appl. Phys. 83, 3432 (1998). 23O. Tchernyshyov and G. -W. Chern, Phys. Rev. Lett. 95, 197204 (2005). 24H. Y. Yuan and X. R. Wang, J. Magn. Magn. Mater. 368, 70 (2014). 25See Supplemental Material at [URL] for DW propagation in boosting phase. 26A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 27D. L. Huber, Phys. Rev. B 26, 3758 (1982). 28S. G. Reidy, L. Cheng, and W. E. Bailey, Appl. Phys. Lett. 82, 1254 (2003).

1603.07977v1.Large_spin_pumping_effect_in_antisymmetric_precession_of_Ni___79__Fe___21___Ru_Ni___79__Fe___21__.pdf

Large spin pumping eect in antisymmetric precession of Ni79Fe21/Ru/Ni 79Fe21 H. Yang,1Y. Li,1and W.E. Bailey1,a) Materials Science and Engineering, Dept. of Applied Physics and Applied Mathematics, Columbia University, New York NY 10027 (Dated: 16 September 2021) In magnetic trilayer structures, a contribution to the Gilbert damping of ferromagnetic resonance arises from spin currents pumped from one layer to another. This contribution has been demonstrated for layers with weakly coupled, separated resonances, where magnetization dynamics are excited predominantly in one layer and the other layer acts as a spin sink. Here we show that trilayer structures in which magnetizations are excited simultaneously, antisymmetrically, show a spin-pumping eect roughly twice as large. The antisym- metric (optical) mode of antiferromagnetically coupled Ni 79Fe21(8nm)/Ru/Ni 79Fe21(8nm) trilayers shows a Gilbert damping constant greater than that of the symmetric (acoustic) mode by an amount as large as the intrinsic damping of Py (
'0.006). The eect is shown equally in eld-normal and eld-parallel to lm plane geometries over 3-25 GHz. The results conrm a prediction of the spin pumping model and have implications for the use of synthetic antiferromagnets (SAF)-structures in GHz devices. Pumped spin currents1,2are widely understood to in- uence the magnetization dynamics of ultrathin lms and heterostructures. These spin currents increase the Gilbert damping or decrease the relaxation time for thin ferromagnets at GHz frequencies. The size of the eect has been parametrized through the eective spin mixing conductance g"# r, which relates the spin current pumped out of the ferromagnet, transverse to its static (time- averaged) magnetization, to its precessional amplitude and frequency. The spin mixing conductance is inter- esting also because it determines the transport of pure spin current across interfaces in quasistatic spin trans- port, manifested in e.g. the spin Hall eect. In the spin pumping eect, spin current is pumped away from a ferromagnet / normal metal (F 1/N) in- terface, through precession of F1, and is absorbed else- where in the structure, causing angular momentum loss and damping of F1. The spin current can be absorbed through dierent processes in dierent materials. When injected into paramagnetic metals (Pt, Pd, Ru, and oth- ers), the spin current relaxes exponentially with para- magnetic layer thickness3{5. The relaxation process has been likened to spin- ip scattering as measured in CPP- GMR, where spin- ip events are localized to heavy-metal impurities6and the measurement reveals the spin diu- sion length SD. When injected into other ferromagnets (F2in F 1/N/F 2), the spin current is absorbed through its torque on magnetization5,7. A similar process appears to be relevant for antiferromagnets as well8. In F 1/N/F 2structures, only half of the total possible spin pumping eect has been detected up until now. For well-separated resonances of F1andF2, only one layer will precess with large amplitude at a given frequency !, and spin current is pumped from a precessing F1into a staticF2. If both layers precess symmetrically, with a)Electronic mail: Contact author. web54@columbia.eduthe same amplitude and phase, equal and opposite spin currents are pumped into and out of each layer, causing no net eect on damping. The dierence between the symmetric mode and the uncoupled mode, increased by a spin pumping damping spwas detected rst in epi- taxial Fe/Au/Fe structures9. However, if the magnetiza- tions can be excited with antisymmetric precession, the coupled mode should be damped by twice that amount, 2sp. Takahashi10has published an explicit prediction of this "giant spin pumping eect" very recently, including an estimate of a fourfold enhanced spin accumulation in the central layer. In this paper, we show that a very large spin pump- ing eect can be realized in antisymmetric precession of Py(8 nm)/Ru(0.70-1.25 nm)/Py(8 nm) synthetic antifer- romagnets (SAF, Py=Ni 79Fe21). The eect is roughly twice that measured in Py trilayers with uncoupled precession. Variable-frequency ferromagnetic resonance (FMR) measurements show, for structures with magne- tization saturated in the lm plane or normal to the lm plane, that symmetric (acoustic mode) precession of the trilayer has almost no additional damping, but the an- tisymmetric (optical mode) precession has an additional Gilbert damping of 0.006, compared with an uncou- pled Py(8nm) layer in a F 1/N/F 2structure of0.003. The interaction stabilizes the antiparallel magnetization state of SAF structures, used widely in dierent elements of high-speed magnetic information storage, at GHz fre- quencies. Method: Ta(5 nm)/Cu(3 nm)/Ni 79Fe21(8 nm)/Ru(tRu)/Ni 79Fe21(8 nm)/Cu(3 nm)/SiO 2(5 nm), tRu= 0.7 - 1.2 nm heterostructures were deposited by ultrahigh vacuum (UHV) sputtering at a base pressure of 5109Torr on thermally oxidized Si substrates. The Ru thckness range was centered about the second antiferromagnetic maximum of interlayer exchange coupling (IEC) for Py/Ru/Py superlattices, 8-12 A, established rst by Brillouin light scattering (BLS) measurement11. Oscillatory IEC in this system, as aarXiv:1603.07977v1 [cond-mat.mtrl-sci] 25 Mar 20162 function of tRu, is identical to that in the more widely studied Co/Ru( tRu)/Co superlattices12, 11.5 A, but is roughly antiphase to it. An in-plane magnetic eld bias of 200 G, rotating in phase with the sample, was applied during deposition as described in13. The lms were characterized using variable fre- quency, swept-eld, magnetic-eld modulated ferromag- netic resonance (FMR). Transmission measurements were recorded through a coplanar waveguide (CPW) with center conductor width of 300 m, with the lms placed directly over the center conductor, using a microwave diode signal locked in to magnetic eld bias modulation. FMR measurements were recorded for magnetic eld bias HBapplied both in the lm plane (parallel condition, pc) and perpendicular to the plane (normal condition, nc.) An azimuthal alignment step was important for the nc measurements. For these, the sample was rotated on twoaxes to maximize the eld for resonance at 3 GHz. For all FMR measurements, the sample magnetization was saturated along the applied eld direction, simplify- ing extraction of Gilbert damping . The measurements dier in this sense from low-frequency measurements of similar Py/Ru/Py trilayer structures by Belmenguenai et al14, or broadband measurements of (stier) [Co/Cu] 10 multilayers by Tanaka et al15. In these studies, eects oncould not be distinguished from those on inhomo- geneous broadening. Model: In the measurements, we compare the mag- nitude of the damping, estimated by variable-frequency FMR linewidth through
H1=2=
H0+ 2!=j j, and the interlayer exchange coupling (IEC) measured through the splitting of the resonances. Coupling terms between layersiandjare introduced into the Landau-Lifshitz- Gilbert equations for magnetization dynamics through _mi=mi( iHe+!ex;imj) +0mi_mi+sp;i(mi_ mimj_ mj) (1) incgsunits, where we dened magnetization unit vec- tors as m1=M1=Ms;1,m2=M2=Ms;2withMs;ithe saturation moments of layer i. The coupling constants are, for the IEC term, !ex;i iAex=(Ms;iti), where the energy per unit area of the system can be written uA=Aexmi
mj, andtiis the thickness of layer i. Pos- itive values of Aexcorrespond to ferromagnetic coupling, negative values to antiferromagnetic coupling. The spin pumping damping term is sp;i h~gFNF "#=(4Ms;iti), where ~gFNF "# is the spin mixing conductance of the tri- layer in units of nm2.0is the bulk damping for the layer. The collective modes of 1 ;2 are found from small- amplitude solutions of Equations 1 for i= 1;2. General solutions for resonance frequencies with arbitrary mag- netization alignment, not cognizant of any spin pump- ing damping or dynamic coupling, were developed by Zhang et al12. In our experiment, to the extent possi- ble, layers 1 ;2 are identical in deposited thickness, mag- netization, and interface anisotropy (each with Cu the opposite side from Ru). Therefore if !irepresents the FMR frequency (dependent on bias eld HB) of each layeri, the two layers have !1=!2=!0. In this limit, there are two collective modes: a perfectly sym- metric mode Sand a perfectly antisymmetric mode A with complex frequencies f!S= (1i0)!0andf!A= (1i02isp) (!0+ 2!ex). The Gilbert damping for the modes, k=Im(f!k)=Re(!k f), wherek= (S;A), and the resonance elds Hk Bsatisfy HA B=HS B+ 2HexHex=Aex=(MstF) (2) A=S+ 2spsp= h~gFNF "#=(4Ms;iti) (3) and!ex= Hex. Note that there is no relationshipin this limit between the strength of the exchange cou- plingAexand the spin-pumping damping 2 spexpressed in the antisymmetric mode. The spin pumping damping and the interlayer exchange coupling can be read sim- ply from the dierences in the the Gilbert damping  and resonance elds between the antisymmetric and sym- metric modes. The asymmetric mode will have a higher damping by 2 spfor anyAexand a higher resonance eld forAex<0, i.e. for antiferromagnetic IEC: because the ground state of the magnetization is antiparallel at zero applied eld, antisymmetric excitation rotates mag- netizations towards the ground state and is lower in fre- quency than symmetric excitation. Results: SamplepcandncFMR data are shown in Figure 1. Raw data traces (lock-in voltage) as a func- tion of applied bias eld HBat 10 GHz are shown in the inset. We observe an intense resonance at low eld and resonance weaker by a factor of 20-100 at higher eld. On the basis of the intensities, as well as supporting MOKE measurements, we assign the lower-eld resonance to the symmetric, or "acoustic" mode and the higher-eld res- onance to the antisymmetric, or "optical" mode. Similar behavior is seen in the nc- andpcFMR measurements. In Figure 1a) and c), which summarizes the elds- for-resonance !(HB), there is a rigid shift of the antisymmetric-mode resonances to higher bias elds HB, as predicted by the theory. The lines show ts to the Kittel resonance, !pc= r Heff Heff+ 4Meff s , !nc= Heff4Meff s
with an additional eective eld along the magnetization direction for the antisym- metric mode: Heff;S =HB, andHeff;A =HB 8Aex=(4MstF). In Figure 2, we show coupling parameters, as a func-3 HBHeffm(t) HBHeff m(t)(a) (c)(b) (d)10 GHz10 GHz 1/21/2 FIG. 1. FMR measurement of Ni 79Fe21(8 nm)/Ru(tRu)/Ni 79Fe21(8 nm) trilayers; example shown fortRu= 1.2 nm. Inset : lock-in signal, transmitted power at 10 GHz, as a function of bias eld HB, for a) pc-FMR and c) nc-FMR. A strong resonance is observed at lower HB and a weaker one at higher HB, attributed to the symmetric (S) and antisymmetric (A) modes, respectively. a), c): Field for resonance !(HB) for the two resonances. Lines are ts to the Kittel resonance expression, assuming an additional, constant, positive eld shift for !A,Hex=2Aex=(MstF) due to antiferromagnetic interlayer coupling Aex<0. b) pc-FMR and d) nc-FMR linewidth as a function of frequency
Hpp(!) for ts to Gilbert damping . tion of Ru thickness, extracted from the FMR measure- ments illustrated in Figure 1. Coupling elds are mea- sured directly from the dierence between the symmet- ric and antisymmetric mode positions and plotted in Figure 2a. We convert the eld shift to antiferromag- netic IEC constant Aex<0 through Equation 2, us- ing the thickness tF= 8 nm and bulk magnetization 4Ms= 10.7 kG4. The extracted exchange coupling strength in pc-FMR has a maximum antiferromagnetic value ofAex=0.2 erg/cm2, which agrees to 5% with that measured by Fassbender et al11for [Py/Ru] Nsu- perlattices. The central result of the paper is shown in Figure 2 b). We compare the damping of the symmetric ( S) and an- tisymmetric ( A) modes, measured both through pc-FMR andnc-FMR. The values measured in the two FMR ge- ometries agree closely for the symmetric modes, for which signals are larger and resolution is higher. They agree roughly within experimental error for the antisymmetric modes, with no systematic dierence. The antisymmetric modes clearly have a higher damping than the symmetric modes. Averaged over all thickness points, the enhanced damping is roughly AS= 0.006. Discussion: The damping enhancement of the anti- symmetric ( A) mode over the symmetric ( S) mode, shown in Figure 2b), is a large eect. The value is close to the intrinsic bulk damping 00.007 for Ni 79Fe21. 0.7 0.8 0.9 1.0 1.1 1.2 1.301002003004005006002Hex (Oe)a) nc,Hexpc,HexMOKE,Hex 0.7 0.8 0.9 1.0 1.1 1.2 1.3 tRu (nm)0.0060.0080.0100.0120.0140.0160.018α α0α0+αspα0+2αspb)pc, S nc, Spc, A nc, A0.000.050.100.150.20 −Aex (erg/cm2)FIG. 2. Coupling parameters for Py/Ru/Py trilayers. a): Interlayer (static) coupling from resonance eld shift of an- tisymmetric mode; see Fig 1 a),c). The antiferromagnetic exchange parameter Aexis extracted through Eq 2, in agree- ment with values found in Ref11. The line is a guide to the eye. b) Spin pumping (dynamic) coupling from damping of the symmetric (S) and antisymmetric (A) modes; see Fig 1 b), d). The spin pumping damping for uncoupled layer pre- cession in Py/Ru/Py, spis shown for comparsion. Dotted lines show the possible eect of 100 Oe detuning for the two Py layers. See text for details. We compare the value with the value 2 spexpected from theory for the antisymmetric mode and written in Eq 3. The interfacial spin mixing conductance for Ni 79Fe21/Ru, was found in Ref.16to be ~gFN "#= 24 nm2. For a F/N/F structure, in the limit of ballistic transport with no spin relaxation through N, the eective spin mixing conduc- tance is ~gFN "#=2: spin current must cross two interfaces to relax in the opposite Flayer, and the conductance re- ects two series resistances17. This yields sp= 0:0027. The observed enhancement matches well with, and per- haps exceeds slightly, the "giant" spin pumping eect of 2sp, as shown. Little dependence of the Gilbert damping enhancement ASon the resonance eld shift HAHScan be ob- served in Figure 2 a,b. We believe that this independence re ects close tuning of the resonance frequencies for Py layers 1 and 2, as designed in the depositions. For - nite detuning
!dened through !2=!0+
!and !1=!0
!, the modes change. Symmetric and anti- symmetric modes become hybridized as S0andA0, and4 the dierence in damping is reduced. Dening g
!2= (1i0 ) (1i0 2isp )
!2, it is straightfor- ward to show that for the nc-case, the mode frequen- cies are!S0;A0= (f!S+f!A)=2q (f!Sf!A)2=4 +g
!2. The relevant parameter is the frequency detuning nor- malized to the exchange (coupling) frequency, z
!=(2!ex); ifz1, the layers have well-separated modes, and each recovers the uncoupled damping en- hancement of sp,S0;A0=0+spidentied in Refs5,9. The possibility of nite detuning, assuming ( !2 !1)= = 100 Oe, is shown in Figure 2b), with the dot- ted lines. The small zlimit for detuning nds sym- metric eects on damping of the S0andA0modes, with S0=0+ 2spz2andA0=0+ 2sp(1z2), respec- tively, recovering perfect symmetric and antisymmetric mode values for z= 0. We assume that the eld split- ting shown in Figure 2 a) gives an accurate measure of 2!ex= , as supported by the MOKE results. This value goes into the denominator of z. We nd a reasonable t to the dependence of SandAdamping on Ru thickness, implicit in the coupling. For the highest coupling pionts, the damping values closely reach the low- zlimit, and we believe that the "giant" spin pumping result of 2 spis evident here. We would like to point out next that it was not a- priori obvious that the Py/Ru/Py SAF would exhibit the observed damping. Ru could behave in two limits in the context of spin pumping: either as a passive spin- sink layer, or as a ballistic transmission layer supporting transverse spin-current transmission from one Py layer to the other. Our results show that Ru behaves as the latter in this thickness range. The symmetric-mode damping of the SAF structure, extrapolated back to zero Ru thick- ness, is identical within experimental resolution ( 104) to that of a single Py lm 16 nm thick measured in nc- FMR ("0" line in Fig 2b.) If Ru, or the Py/Ru interface, depolarized pumped spin current very strongly over this thickness range as has been proposed for Pt18, we would expect an immediate increase in damping of the acous- tic mode by the amount of sp. Instead, the volume- dependent Ru depolarization in spin pumping has an (ex- ponential) characteristic length of SD10 nm5, and attenuation over the range explored of 1 nm is negli- gible. Perspectives: Finally, we would like to highlight some implications of the study. First, as the study conrms the prediction of a "giant" spin pumping eect as pro- posed by Takahashi10, it is plausible that the greatly en- hanced values of spin accumulation predicted there may be supported by Ru in Py/Ru/Py synthetic antiferro- magnets (SAFs). These spin accumulations would dier strongly in the excitation of symmetric and antisymmet- ric modes, and may then provide a clear signature intime-resolved x-ray magnetooptical techniques19, similar to the observation of static spin accumulation in Cu re- ported recently20. Second, in most device applications of synthetic an- tiferromagnets, it is not desirable to excite the antisym- metric (optical) mode. SAFs are used in the pinned layer of MTJ/spin valve structures to increase exchange bias and in the free layer to decrease (magnetostatic) stray elds. Both of these functions are degraded if the opti- cal, or asymmetric mode of the SAF is excited. Accord- ing to our results, at GHz frequencies near FMR, the susceptibility of the antisymmetric mode is reduced sub- stantially, here by a factor of two (from 1/ ) for nc-FMR , due to spin pumping. This reduction of on resonance will scale inversely with layer thickness. The damping, and susceptibility, of the desired symmetric (acoustic) mode is unchanged, on the other hand, implying that spin pumping favors the excitation the symmetric mode for thin Ru, the typical operating point. We acknowledge NSF-DMR-1411160 for support. 1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 2Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, Reviews in Modern Physics 77, 1375 (2005). 3S. Mizukami, Y. Ando, and T. Miyazaki, Journal of Magnetism and Magnetic Materials 239, 42 (2002). 4A. Ghosh, J. F. Sierra, S. Auret, U. Ebels, and W. E. Bailey, Applied Physics Letters 98, (2011). 5A. Ghosh, S. Auret, U. Ebels, and W. E. Bailey, Phys. Rev. Lett. 109, 127202 (2012). 6J. Bass and W. Pratt, Journal of Physics: Condensed Matter 19, 41 pp. (2007). 7G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H. Back, Phys- ical Review Letters 99, 246603 (2007). 8P. Merodio, A. Ghosh, C. Lemonias, E. Gautier, U. Ebels, V. Baltz, and W. Bailey, Applied Physics Letters 104(2014). 9B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Ur- ban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003). 10S. Takahashi, Applied Physics Letters 104(2014). 11J. Fassbender, F. Nortemann, R. Stamps, R. Camley, B. Hille- brands, G. Guntherodt, and S. Parkin, Journal of Magnetism and Magnetic Materials 121, 270 (1993). 12Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev. B50, 6094 (1994). 13C. Cheng, N. Sturcken, K. Shepard, and W. Bailey, Review of Scientic Instruments 83, 063903 (2012). 14M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and G. Bayreuther, Physical Review B 76(2007). 15K. Tanaka, T. Moriyama, M. Nagata, T. Seki, K. Takanashi, S. Takahashi, and T. Ono, Applied Physics Express 7(2014). 16N. Behera, M. S. Singh, S. Chaudhary, D. K. Pandya, and P. K. Muduli, Journal of Applied Physics 117(2015). 17See eqs. 31, 74, 81 in Ref2. 18J.-C. Rojas-S anchez, N. Reyren, P. Laczkowski, W. Savero, J.- P. Attan e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and H. Jar es, Phys. Rev. Lett. 112, 106602 (2014). 19W. Bailey, C. Cheng, R. Knut, O. Karis, S. Auret, S. Zohar, D. Keavney, P. Warnicke, J.-S. Lee, and D. Arena, Nature Com- munications 4, 2025 (2013). 20R. Kukreja, S. Bonetti, Z. Chen, D. Backes, Y. Acremann, J. A. Katine, A. D. Kent, H. A. D urr, H. Ohldag, and J. St ohr, Phys. Rev. Lett. 115, 096601 (2015).

2206.09969v1.First_principles_calculation_of_the_parameters_used_by_atomistic_magnetic_simulations.pdf

arXiv:2206.09969v1 [cond-mat.mtrl-sci] 20 Jun 2022APS/123-QED First-principles calculation of the parameters used by ato mistic magnetic simulations Sergiy Mankovsky and Hubert Ebert Department of Chemistry/Phys. Chemistry, LMU Munich, Butenandtstrasse 11, D-81377 Munich, Germany (Dated: June 22, 2022) While the ground state of magnetic materials is in general we ll described on the basis of spin den- sity functional theory (SDFT), the theoretical descriptio n of finite-temperature and non-equilibrium properties require an extension beyond the standard SDFT. T ime-dependent SDFT (TD-SDFT), which give for example access to dynamical properties are co mputationally very demanding and can currently be hardly applied to complex solids. Here we focus on the alternative approach based on the combination of a parameterized phenomenological spin H amiltonian and SDFT-based electronic structure calculations, giving access to the dynamical and finite-temperature properties for example via spin-dynamics simulations using the Landau-Lifshitz- Gilbert (LLG) equation or Monte Carlo simulations. We present an overview on the various methods t o calculate the parameters of the various phenomenological Hamiltonians with an emphasis on the KKR Green function method as one of the most flexible band structure methods giving access to practically all relevant parameters. Concerning these, it is crucial to account for the spin-orbi t coupling (SOC) by performing rela- tivistic SDFT-based calculations as it plays a key role for m agnetic anisotropy and chiral exchange interactions represented by the DMI parameters in the spin H amiltonian. This concerns also the Gilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral multispin interaction parameters of the extended Heisenbe rg Hamiltonian, as well as spin-lattice interaction parameters describing the interplay of spin an d lattice dynamics processes, for which an efficient computational scheme has been developed recently b y the present authors. PACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds I. INTRODUCTION Density functional theory (DFT) is a ’formally exact approachto the static electronic many-body problem’ for theelectrongasintheequilibrium, whichwasadoptedfor a huge number of investigations during the last decades to describe the ground state of solids, both magnetic and non-magnetic,aswellasvariousgroundstateproperties1. However, dealing with real systems, the properties in an out-of-equilibrium situation are of great interest. An example for this is the presence of external pertur- bation varying in time, which could be accounted for by performing time-dependent first-principles electronic structure calculations. The time-dependent extension of density functional theory (TD-DFT)2is used suc- cessfully to study various dynamical processes in atoms and molecules, in particular, giving access to the time evolution of the electronic structure in a system af- fected by a femtosecond laser pulse. However, TD-DFT can be hardly applied to complex solids because of the lack of universal parameter-free approximations for the exchange-correlation kernel. Because of this, an ap- proach based on the combination of simulation methods for spin- and lattice dynamics, using model spin and lat- tice Hamiltonians is more popular for the moment. A great progress with this approach has been achieved dur- ing last decade due to the availability of parameters for the model Hamiltonians calculated on a first principles level, that is a central issue of the present contribution. As it was pointed out in Ref. 1, this approach has the ad- vantage, that the spin-related many-body effects in thiscase are much simpler to be taken into account when compared to the ab-initio approach. Thus, the isotropic exchangecouplingparameters JijfortheclassicalHeisen- berg Hamiltonian worked out Liechtenstein et al.3,4have been successfully used by many authors to predict the ground state magnetic structure of material and to in- vestigateitsfinite-temperatureproperties. Dependingon the materials, the isotropic Jijcan exhibit only spatial anisotropy. Extension of the Heisenberg Hamiltonian ac- counting for anisotropy in spin subspace is often done by adding the so-called Dzyaloshinskii-Moriya interactions (DMI) and the magnetic anisotropy term, HH,rel=−/summationdisplay i,jJij(ˆei·ˆej)−/summationdisplay i,j/vectorDij(ˆei׈ej)+/summationdisplay iˆeiKiiˆei. (1) with ˆei(j)the orientation of the spin magnetic moment at sitei(j). Alternatively, one may describe exchange inter- actions in the more general tensorial form, Jij, leading to: HH,rel=−/summationdisplay i,jˆeiJijˆej+/summationdisplay iˆeiKiiˆei,(2) In the second case the DMI is represented as the an- tisymmetric part of the exchange tensor, i.e. Dα ij= 1 2(Jβγ ij−Jγβ ij)ǫαβγ. It should be stressed, that calcula- tions of the spin-anisotropic exchange interaction param- eters as well as of the magnetic anisotropy parameters require a relativistic treatment of the electronic struc- ture in contrast to the case of the isotropic exchange pa- rameters which can be calculated on a non-relativistic2 level. Various schemes to map the dependence of the electronicenergyonthemagneticconfigurationweresug- gested in the literature to calculate the parameters of the spin Hamiltonians5–8, depending of its form given in Eqs. (1) or (2). Despite of its simplicity, the spin Hamiltonian gives access to a reasonable description of the temperature dependence of magnetic properties of materials when combined with Monte Carlo (MC) simulations9, or non- equilibrium spin dynamics simulations based on the phe- nomenological Landau-Lifshitz-Gilbert equations10,11 1 γd/vectorM dτ=−/vectorM×/vectorHeff+/vectorM×/bracketleftBigg˜G(/vectorM) γ2M2sd/vectorM dτ/bracketrightBigg .(3) Here/vectorHeffis the effective magnetic field defined as /vectorHeff= −1 M∂F ∂ˆm, whereFis the free energy of the system and ˆm=/vectorM /vectorMswithMsthesaturationmagnetizationtreatedat first-principles level, and γis the gyromagnetic ratio and ˜Gis the Gilbert damping parameter. Alternatively, the effective magnetic field can be representedin terms ofthe spin Hamiltonian in Eq. (2), i.e. /vectorHeff=−1 M∂/angbracketleftHH,rel/angbracketrightT ∂ˆm, with/an}b∇acketle{t.../an}b∇acket∇i}htTdenoting the thermal averagefor the extended Heisenberg Hamiltonian HH,rel. The first-principles calculation of the parameters for the Heisenberg Hamiltonian as well as for the LLG equa- tion for spin dynamics have been reported in the litera- ture by various groups who applied different approaches based on ab-initio methods. Here we will focus on calcu- lations based on the Green function multiple-scattering formalism being a rather powerful tool to supply all pa- rameters for the extended Heisenberg Hamiltonian as well as for the LLG equation. A. Magnetic anisotropy Let’s first consider the magnetic anisotropy term in spin Hamiltonian, characterized by parameters (written intensorialforminEqs.(1)and(2))deducedfromtheto- talenergydependentontheorientationofthemagnetiza- tion ˆm. The latter is traditionallysplit into the magneto- crystalline anisotropy(MCA) energy, EMCA(ˆm), induced by spin-orbit coupling (SOC) and the shape anisotropy energy,Eshape(ˆm), caused by magnetic dipole interac- tions, EA(ˆm) =EMCA(ˆm)+Eshape(ˆm). (4) Although a quantum-mechanical description of the mag- neticshapeanisotropydeservesseparatediscussion12this contribution can be reasonably well estimated based on classical magnetic dipole-dipole interactions. Therefore, we will focus on the MCA contribution which is fully determined by the electronic structure of the considered system. In the literature the focus is in general on the MCA energy of the ground state, which can be estimated straightforwardlyfromthe totalenergycalculatedfordif- ferent orientations of the magnetization followed by amapping onto a model spin Hamiltonian, given e.g. by an expansion in terms of spherical harmonics Ylm(ˆm)13 EMCA(ˆm) =/summationdisplay levenm=l/summationdisplay m=−lκm lYlm(ˆm).(5) Alternative approach to calculate the MCA parameters is based on magnetic torque calculations, using the defi- nition Tˆm(θˆu) =−∂E(ˆm) ∂θˆu, (6) avoiding the time-consuming total energy calculations. This scheme is based on the so-called magnetic force the- oremthatallowstorepresenttheMCAenergyintermsof a correspondingelectronic single-particleenergies change under rotation of magnetization, as follows14: ∆ESOC(ˆm,ˆm′) =−/integraldisplayEˆm F dE/bracketleftBig Nˆm(E)−Nˆm′(E)/bracketrightBig −1 2nˆm′(Eˆm′ F)(Eˆm F−Eˆm′ F)2 +O(Eˆm F−Eˆm′ F)3(7) withNˆm(E) =/integraltextEdE′nˆm(E′) the integrated DOS for the magnetization along the direction ˆ m, andnˆm(E) the densityofstates(DOS) representedin termsofthe Green function as follows nˆm(E) =−1 πIm TrGˆm(E). (8) This expressioncan be used in a very efficient way within the framework of the multiple-scattering formalism. In this case the Green function is given in terms of the scat- tering path operator τ(E)nn′connecting the sites nand n′as follows G0(/vector r,/vector r′,E) =/summationdisplay ΛΛ′Zn Λ(/vector r,E)τnn′ ΛΛ′(E)Zn′× Λ′(/vector r′,E) −/summationdisplay Λ/bracketleftBig Zn Λ(/vector r,E)Jn× Λ(/vector r′,E)Θ(r′−r) +Jn Λ(/vector r,E)Zn× Λ(/vector r′,E)Θ(r−r′)/bracketrightBig δnn′,(9) where the combined index Λ = ( κ,µ) represents the rela- tivistic spin-orbit and magnetic quantum numbers κand µ, respectively15;Zn Λ(/vector r,E) andJn Λ(/vector r,E) are the regular and irregular solutions of the single-site Dirac equation (27)16–18. The scattering path operator is given by the expression τ(E) = [m(E)−G0(E)]−1(10) withm(E) =t−1(E) andG0(E) the inverse single-site scattering and structure constant matrices, respectively. The double underline used here indicates matrices with respect to site and angular momentum indices17.3 Using the Lloyd’s formula that gives the integrated DOS in terms of the scattering path operator, Eq. (7) can be transformed to the form ∆ESOC(ˆm,ˆm′) =−1 πIm Tr/integraldisplayEF dE ×/parenleftbig lnτ(ˆm,E)−lnτ(ˆm′,E)/parenrightbig (11) with the scattering path operator evaluated for the mag- netization along ˆ mand ˆm′, respectively. With this, the magnetic torque T(θ) can be expressed by means of multiple scattering theory leading for the torque component with respect to a rotation of the mag- netization around an axis ˆ u, to the expression19 Tˆm(θˆu) =−1 πℑ/integraldisplayEF dE∂ ∂θˆu/bracketleftbig lndet/parenleftbig t(ˆm)−1−G0/parenrightbig/bracketrightbig . (12) Mapping the resulting torque onto a corresponding pa- rameterized expression as for example Eq. (5), one ob- tains the corresponding parameters of the spin Hamilto- nian. However,oneshouldnotethatthemagneticanisotropy of materials changes when the temperature increases. This occurs first of all due to the increasing amplitude of thermally induced spin fluctuations responsible for a modification of the electronic structure. A correspond- ing expression for magnetic torque st finite temperature was worked out by Staunton et al.19, on the basis of the relativistic generalization of the disordered local moment (RDLM) theory20. To perform the necessary thermal av- eraging over different orientational configurations of the local magnetic moments it uses a technique similar to the one used to calculate the configurational average in the case of random metallic alloys, so-called Coherent Potential Approximation (CPA) alloy theory21,22. Ac- cordingly, the free energy difference for two different ori- entations of the magnetization is given by ∆F(ˆm,ˆm′) =−/integraldisplay dEfFD(E,ˆm) (13) /bracketleftbigg /an}b∇acketle{tNˆm/an}b∇acket∇i}ht(E)−/an}b∇acketle{tNˆm′/an}b∇acket∇i}ht(E)/bracketrightbigg .(14) By using in this expression the configurational aver- aged integrated density of states20,23given by Lloyd’s formula, the corresponding expression for the magnetic torque at temperature T Tˆm,T(θˆu) =−∂ ∂θˆu/parenleftbigg/summationdisplay i/integraldisplay Pˆm i(ˆei)/an}b∇acketle{tΩˆm/an}b∇acket∇i}htˆeidˆei/parenrightbigg .(15) can be written explicitly as: Tˆm,T(θˆu) =−1 πIm/integraldisplayEF dEfFD(E,ˆm) /parenleftbigg/summationdisplay i/integraldisplay∂Pˆm i(ˆei) ∂θˆuln detMˆm i(ˆei,E)dˆei/parenrightbigg .(16)where Mˆm i(ˆei,E) = 1+([ti(ˆei)]−1−tˆm i,c(ˆei)]−1)τˆm ii,c,(17) and τˆm ii,c= ([tˆm i,c(ˆei)]−1−G0)−1. (18) where the index cindicates quantities related to the CPA medium. Fig. 1 (top) shows as an example the results for the temperature-dependent magnetization ( M(T)) cal- culated within the RDLM calculations for L10-ordered FePt24. Fig. 1 (bottom) gives the corresponding param- eterK(T) for a uni-axial magneto-crystalline anisotropy, which is obviously in good agreement with experiment. 200 400 600800 Temperature T (K)00.20.40.60.8M(T) 0.2 0.4 0.60.8 (M(T))2-2-1.5-1-0.5∆ESOC (meV) FIG. 1. RDLM calculations on FePt. Top: the magneti- zationM(T) versus Tfor the magnetization along the easy [001] axis (filled squares). The full line shows the mean field approximation to a classical Heisenberg model for compar- ison. Bottom: the magnetic anisotropy energy ∆ ESOCas a function of the square of the magnetization M(T). The filled circles show the RDLM-based results, the full line giv e K(T)∼[M(T)/M(0)]2, and the dashed line is based on the single-ion model function. All data taken from24. B. Inter-atomic bilinear exchange interaction parameters Most first-principles calculations on the bilinear ex- change coupling parameters reported in the literature,4 FIG. 2. Adiabatic spin-wave dispersion relations along hig h- symmetry lines of the Brillouin zone for Ni. Broken line: frozen-magnon-torque method, full line: transverse susce pti- bility method31. All data are taken from Ref. 31. are based on the magnetic force theorem (MFT) by evaluating the energy change due to a perturbation on the spin subsystem with respect to a suitable reference configuration25. Many results are based on calculations of the spin-spiral energy ǫ(/vector q), giving access to the ex- change parameters in the momentum space, J/vector q7,26–28, followed by a Fourier transformation to the real space representation Jij. Alternatively, therealspaceexchange parameters are calculated directly by evaluating the en- ergy change due to the tilting of spin moments of inter- acting atoms. The corresponding non-relativistic expres- sion (so-called Liechtenstein or LKAG formula) has been implemented based on the KKR as well as LMTO Green function (GF)3,4,25,29band structure methods. It should be noted that the magnetic force theorem provides a rea- sonable accuracy for the exchange coupling parameters in the case of infinitesimal rotations of the spins close to some equilibrium state, that can be justified only in the long wavelength and strong-coupling limits30. Accord- ingly, calculations of the exchange coupling parameters beyond the magnetic force theorem, represented in terms of the inverse transverse susceptibility, were discussed in the literature by various authors25,30–33. Grotheer et al., for example, have demonstrated31a deviation of the spin-wave dispersion curves away from Γ point in the BZ, calculated for fcc Ni using the exchange parameters J/vector q∼χ−1 /vector q, from the MFT-based results for J/vector q. On the otherhand, the resultsareclosetoeachotherin the long- wavelength limit (see Fig. 2). The calculations beyond thestandardDFTaredonebymakinguseoftheso-called constrained-field DFT. The latter theory was also used by Bruno33who suggested the ’renormalization’ of the exchange coupling parameters expressed in terms of non- relativistic transverse magnetic susceptibility, according toJ=1 2Mχ−1M=1 2M(˜χ−1−Ixc)M, with the various quantities defined as follows ˜χ−1 ij=2 π/integraldisplayEF dE/integraldisplay Ωid3r/integraldisplay Ωjd3r′(19) ×Im[G↑(/vector r,/vector r′,E)G↓(/vector r′,/vector r,E)],(20)Mi=/integraldisplay Ωid3rm(/vector r), (21) and ˜Ixc ij=δij∆i 2Mi, (22) with ∆ i=4 Mi/summationtext j˜Jij, where ˜Jij=1 πIm/integraldisplayEF dE/integraldisplay Ωid3r/integraldisplay Ωjd3r′(23) ×[Bxc(/vector r)G↑(/vector r,/vector r′,E)Bxc(/vector r′)G↓(/vector r′,/vector r,E)].(24) This approach results in a Curie temperature of 634 K for fcc Ni (vs. 350 K based on the MFT) which is in good agreement with the experimental value of (621 −631 K). As was pointed out by Solovyev30, such a corrections can be significant only for a certain class of materials, while, for instance, the calculations of spin-wave energies31and TC33for bcc Fe demonstrate that these corrections are quite small. As most results in the literature were ob- tained using the exchange parameters based on the mag- netic force theorem, we restrict below to this approxima- tion. Similar to the case of the MCA discussed above, ap- plication of the magnetic force theorem gives the energy change due to tilting of two spin moments represented in terms of the integrated DOS4. Within the multiple scat- tering formalism, this energy can be transformed using the Lloyd’s formula leading to the expression ∆E=−1 πIm Tr/integraldisplayEF dE/parenleftbig lnτ(E)−lnτ0(E)/parenrightbig (25) withτ(0)(E) andτ(E) the scattering path operators for non-distorted and distorted systems, respectively. As reported in Ref. 4, the expression for Jijrepresent- ing the exchange interaction between the spin moments on sitesiandj, is given by the expression Jij=−1 4πImTrL/integraldisplayEF dE∆iτ↑ ij∆jτ↓ ji,(26) with ∆i(j)= ([t↑]−1 i(j)−[t↓]−1 i(j)), wheret↑ i(j)andt↓ i(j)are thespin-upandspin-downsingle-sitescatteringmatrices, respectively, while τ↑ ijandτ↓ jiare the spin-up and spin- down, respectively, scattering path operators. As rela- tivistic effects are not taken into account, the exchange interactions are isotropic with respect to the orientation of the magnetization as well as with respect to the di- rection of the spin tilting. On the other hand, spin-orbit coupling gives rise to an anisotropy for exchange inter- actions requiring a representation in the form of the ex- change tensor Jijwith its antisymmetric part giving ac- cess to the Dzyaloshinskii-Moriya (DM) interaction /vectorDij. Udvardi et al.5and later Ebert and Mankovsky6sug- gested an extension of the classical Heisenberg Hamilto- nianbyaccountingforrelativisticeffects forthe exchange coupling (see also Ref. 25). These calculations are based5 onafullyrelativistictreatmentoftheelectronicstructure obtained by use of of the Dirac Hamiltonian HD=−ic/vector α·/vector∇+1 2c2(β−1) +¯V(/vector r)+β/vector σ·/vectorB(/vector r)+e/vector α·/vectorA(/vector r).(27) Here,αiandβare the standard Dirac matrices15while ¯V(/vector r) and/vectorB(/vector r) are the spin independent and spin depen- dent parts of the electronic potential. Considering a ferromagnetic (FM) state as a reference state with the magnetization along the zdirection, a tilt- ing of the magnetic moments on sites iandjleads to a modification ofthe scattering path operatorimplying the relation lnτ−lnτ0=−ln/parenleftbig 1+τ[∆mi+∆mj+...]/parenrightbig ,(28) withmi=t−1 i. This allows to write down the expression for the energy change due to a spin tilting on sites iand jas follows Eij=−1 πImTr/integraldisplayEF dE∆miτij∆mjτji(29) Within the approach of Udvardi et al.5, the depen- dence of the single-site inverse scattering matrix mion the orientation of magnetic moment ˆ eiis accounted for by performing a corresponding rotation operation us- ing the rotation matrix R(θ,φ), i.e., one has mi(θ,φ) = R(θ,φ)m0 iR+(θ,φ). The change of the scattering matrix miunder spin rotation, ∆ mi, linearized with respect to the rotation angles, is given by the expression ∆mi=R(θi,φi)m0 iR+(θi,φi)−m0 i =mθ iδθi+mφ iδφi (30) with mθ i=∂ ∂θmi=∂R ∂θmiR++Rmi∂R+ ∂θ, mφ i=∂ ∂φmi=∂R ∂φmiR++Rmi∂R+ ∂φ.(31) To calculate the derivatives of the rotation matrix, the definition ˆR(αˆn,ˆn) =eiαˆn(ˆn·ˆ/vectorJ)(32) for the corresponding operator is used, withˆ/vectorJthe total angular momentum operator. ˆR(αˆn,ˆn) describes a rota- tion of the magnetic moment ˆ mby the angle αˆnabout the direction ˆ n⊥ˆm, that gives in particular R(θ,ˆn) for ˆn= ˆyandR(φ,ˆn) for ˆn= ˆz. This leads to the second derivatives of the total energy with respect to the titling angles αi={θi,φi}andβj= {θj,φj} ∂2E ∂αi∂βj=−1 πImTr/integraldisplayEF dEmα iτijmβ jτji(33)As is discussed by Udvardi et al.5, these derivatives give access to all elements Jµν ijof the exchange tensor, where µ(ν) ={x,y,z}. Note, however, that only the tensor el- ements with µ(ν) ={x,y}can be calculated using the magnetization direction along the ˆ zaxis, giving access to thezcomponent Dz ijof the DMI. In order to obtain all other tensor elements, an auxiliary rotation of the mag- netization towards the ˆ xand ˆydirections of the global frame of reference is required. For example, the com- ponentDx ijif the DMI vector can be evaluated via the tensor elements Jzy ij=∂2E ∂θi∂φjandJyz ij=∂2F ∂φi∂θj(34) forθ=π 2andφ= 0. An alternative expression within the KKR multiple scattering formalism has been worked out by Ebert and Mankovsky6, by using the alternative convention for the electronicGreenfunction(GF) assuggestedbyDederichs and coworkers34. According to this convention, the off- site part of the GF is given by the expression: G(/vector ri,/vector rj,E) =/summationdisplay ΛΛ′Ri Λ(/vector ri,E)Gij ΛΛ′(E)Rj× Λ′(/vector rj,E),(35) whereGij ΛΛ′(E) is the so-called structural Green’s func- tion,Ri Λisaregularsolutiontothesingle-siteDiracequa- tionlabeledbythecombinedquantumnumbersΛ15. The energy change ∆ Eijdue to a spin tilting on sites iandj , given by Eq. (29), transformed to the above mentioned convention is expressed as follows ∆Eij=−1 πImTr/integraldisplay dE∆tiGij∆tjGji,(36) where the change of the single-site t-matrix ∆ tican be represented in terms of the perturbation ∆ Vi(/vector r) at site iusing the expression ∆ti Λ′Λ=/integraldisplay d3rRi× Λ′(r)∆V(r)Ri Λ(r) = ∆V(R)i Λ′Λ,(37) wherethe perturbation causedby the rotationof the spin magnetic moment ˆ eiis represented by a change of the spin-dependent potential in Eq. (27) (in contrast to the approach used in Ref. 5) ∆V(r) =Vˆn(r)−Vˆn0(r) =β/vector σ(ˆn−ˆn0)B(r).(38) Using again the frozen potential approximation implies that the spatial part of the potential Vˆn(r) does not change upon rotation of spin orientation. Coming back to the convention for the GF used by Gy¨ orffy and coworkers35according to Eq. (9) the expres- sion for the elements of the exchange tensor represented in terms of the scattering path operator τij Λ′Λ(E) has the form Jαiαj ij=−1 πImTr/integraldisplay dETαiτijTαjτji,(39)6 where Tαi ΛΛ′=/integraldisplay d3rZ× Λ(/vector r)βσαB(r)ZΛ′(/vector r).(40) When compared to the approach of Udvardi et al.5, the expression in Eq. (39) is given explicitly in Cartesian coordinates. However, auxiliary rotations of the magne- tization are still required to calculate all tensor elements, and as a consequence, all components of the DMI vec- tor. This can be avoided using the approach reported recently36for DMI calculations. In this case, using the grand-canonical potential in the operator form K=H−µN, (41) withµthe chemical potential, the variation of single- particle energy density ∆ E(/vector r) caused by a perturbation is written in terms of the electronic Green function for T= 0 K as follows ∆E(/vector r) =−1 πImTr/integraldisplayµ dE(E−µ)∆G(/vector r,/vector r,E).(42) Assuming the perturbation ∆ Vresponsible for the change of the Green function ∆ G=G−G0(the in- dex 0 indicates here the collinear ferromagnetic reference state) to be small, ∆ Gcan be expanded up to any order w.r.t. the perturbation ∆G(E) =G0∆VG0 +G0∆VG0∆VG0 +G0∆VG0∆VG0∆VG0 +G0∆VG0∆VG0∆VG0∆VG0+...,(43) leading to a corresponding expansion for the energy change with respect to the perturbation as follows ∆E= ∆E(1)+∆E(2)+∆E(3)+∆E(4)+...,(44) Here and below we drop the energy argument for the Green function G(E) for the sake of convenience. This expression is completely general as it gives the energy change as a response to any type of perturbation. When ∆Vis associated with tiltings of the spin magnetic mo- ments, it can be expressed within the frozen potential approximation and in line with Eq. (38) as follows ∆V(/vector r) =/summationdisplay iβ/parenleftbig /vector σ·ˆsi−σz/parenrightbig Bxc(/vector r).(45) With this, the energy expansion in Eq (44) gives access to the bilinear DMI as well as to higher order multispin interactions37. To demonstrate the use of this approach, we start with the xandycomponents of the DMI vector, which can be obtained by setting the perturbation ∆ Vin the form of a spin-spiral described by the configuration of the magnetic moments ˆmi=/parenleftBig sin(/vector q·/vectorRi),0,cos(/vector q·/vectorRi)/parenrightBig ,(46)with the wave vector /vector q= (0,q,0). As it follows from the spin Hamiltonian, the slope of the spin wave energy dispersion at the Γ point is determined by the DMI as follows lim q→0∂E(1) DM ∂qy= lim q→0∂ ∂qy/summationdisplay ijDy ijsin(/vector q·(/vectorRj−/vectorRi)) =/summationdisplay ijDy ij(/vectorRj−/vectorRi)y. (47) Identifying this with the corresponding derivative of the energy ∆ E(1)in Eq. 44 ∂∆E(1) ∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle q→0=∂E(1) DM ∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle q→0, (48) and equating the corresponding terms for each atomic pair (i,j), one obtains the following expression for the y component of the DMI vector: Dy ij=/parenleftbigg −1 2π/parenrightbigg ImTr/integraldisplayµ dE(E−µ) ×/bracketleftbigg Oj(E)τji(E)Ti,x(E)τij(E) −Oi(E)τij(E)Tj,x(E)τji(E)/bracketrightbigg ,(49) In a completely analogous way one can derive the x- component of the DMI vector, Dx ij. The overlap inte- gralsOj ΛΛ′and matrix elements Ti,α ΛΛ′of the operator Ti,α=βσαBi xc(/vector r) (which are connected with the compo- nents of the torque operator β[/vector σ׈m]Bi xc(/vector r)) are defined as follows:6 Oj ΛΛ′=/integraldisplay Ωjd3rZj× Λ(/vector r,E)Zj Λ′(/vector r,E) (50) Ti,α ΛΛ′=/integraldisplay Ωid3rZi× Λ(/vector r,E)/bracketleftBig βσαBi xc(/vector r)/bracketrightBig Zi Λ′(/vector r,E).(51) As is shown in Ref. 37, the Dz ijcomponent of the DMI, as well isotropic exchange parameter Jijcan also be ob- tained on the basis of Eqs. (43) and (44) using the second order term w.r.t. the perturbation, for a spin spiral with the form ˆsi= (sinθcos(/vector q·/vectorR),sinθsin(/vector q·/vectorR),cosθ).(52) In this case case, the DMI component Dz ijand the isotropic exchange interaction are obtained by taking the first- and second-orderderivatives of the energy ∆ E(2)(/vector q) (see Eq. (44)), respectively, with respect to /vector q: ∂ ∂/vector q∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle q→0=−sin2θN/summationdisplay i/negationslash=jDz ijˆq·(/vectorRi−/vectorRj) (53) and ∂2 ∂/vector q2∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle q→0= sin2θ/summationdisplay i,jJij(ˆq·(/vectorRi−/vectorRj))2(54)7 with ˆq=/vector q/|/vector q|the unit vector giving the direction of the wave vector /vector q. Identifying these expressions again with the corresponding derivatives of ∆ E(2)(/vector q), one obtains the following relations for Dz ij Dz ij=1 2(Jxy ij−Jyx ij) (55) and forJij Jij=1 2(Jxx ij+Jyy ij), (56) where the tensor elements Jαβare given by Eqs. (39) and (40). Similar to the magnetic anisotropy, the exchange cou- pling parameters depend on temperature, that should be taken into account within the finite temperature spin dy- namic simulations. An approach that gives access to calculations of exchange coupling parameters for finite temperature has been reported in Ref. 37. It accounts for the electronic structure modification due to temper- ature induced lattice vibrations by using the alloy anal- ogy model in the adiabatic approximation. This implies calculations of the thermal average /an}b∇acketle{t.../an}b∇acket∇i}htTas the configu- rational average over a set of appropriately chosen set of atomic displacements, using the CPA alloy theory38–40. To make use of this scheme to account for lattice vi- brations, a discrete set of Nvvectors ∆/vectorRq v(T) is intro- duced for each atom, with the temperature dependent amplitude, which characterize a rigid displacement of the atomic potential in the spirit of the rigid muffin- tin approximation41,42. The corresponding single-site t- matrix in the common global frame of the solid is given by the transformation: tq v=U(∆/vectorRv)tq,locU(∆/vectorRv)−1, (57) with the so-called U-transformation matrix U(/vector s) given in its non-relativistic form by:41,42 ULL′(/vector s) = 4π/summationdisplay L′′il+l′′−l′CLL′L′′jl′′(|/vector s|k)YL′′(ˆs).(58) HereL= (l,m) represents the non-relativistic angu- lar momentum quantum numbers, jl(x) is a spheri- cal Bessel function, YL(ˆr) a real spherical harmonics, CLL′L′′a corresponding Gaunt number and k=√ Eis the electronic wave vector. The relativistic version of the U-matrix is obtained by a standard Clebsch-Gordan transformation.15 Every displacement characterized by a displacement vectors ∆/vectorRv(T) can be treated as a pseudo-component of a pseudo alloy. Thus, the thermal averaging can be performed as the site diagonal configurational average forasubstitutional alloy,bysolvingthe multi-component CPA equations within the global frame of reference40. The same idea can be used also to take into account thermalspinfluctuations. Asetofrepresentativeorienta- tion vectors ˆ ef(withf= 1,...,Nf) for the local magneticmoment is introduced. Using the rigid spin approxima- tion, the single-site t-matrix in the global frame, corre- sponding to a given orientation vector, is determined by: tq f=R(ˆef)tq,locR(ˆef)−1, (59) wheretq,locis the single-site t-matrix in the local frame. Here the transformation from the local to the global frame of reference is expressed by the rotation matrices R(ˆef) that are determined by the vectors ˆ efor corre- sponding Euler angles.15Again, every orientation can be treated as a pseudo-component of a pseudo alloy, that allows to use the alloy analogy model to calculate the thermal average over all types of spin fluctuations40. The alloy analogy for thermal vibrations applied to the temperature dependent exchange coupling parame- ters leads to ¯Jαiαj ij=−1 2πℑ/integraldisplay dETrace/an}b∇acketle{t∆Vαiτij∆Vαjτji/an}b∇acket∇i}htc,(60) where/an}b∇acketle{t.../an}b∇acket∇i}htcrepresents the configurational average with respect to the set of displacements. In case of the ex- change coupling parameters one has to distinguish be- tween the averaging over thermal lattice vibrations and spin fluctuations. In the first case the configurational av- erage is approximated as follows /an}b∇acketle{t∆Viτij∆Vjτji/an}b∇acket∇i}htvib≈ /an}b∇acketle{t∆Viτij/an}b∇acket∇i}htvib/an}b∇acketle{t∆Vjτji/an}b∇acket∇i}htvib, assuming a negligible impact of the so-called vertex corrections43. This averaging ac- counts for the impact of thermally induced phonons on the exchange coupling parameters for every temperature before their use in MC or spin dynamics simulations that dealsubsequentlywith thethermalaveraginginspinsub- space. The impact of spin fluctuations can be incorpo- rated as well within the electronic structure calculations. For a non-polarized paramagnetic reference state, this can be done, e.g., by using the so-called disorder local moment (DLM) scheme formulated in general within the non-relativistic (or scalar-relativistic) framework. Mag- netic disorder in this case can be modeled by creating a pseudo alloy with an occupation of the atomic sites by two types of atoms with opposite spin moments oriented upwards,M↑and downwards M↓, respectively, i.e. con- sidering the alloy M↑ 0.5M↓ 0.5. In the relativistic case the corresponding RDLM scheme has to describe the mag- netic disorder by a discrete set of Nforientation vectors, and as a consequence, the average /an}b∇acketle{tτij/an}b∇acket∇i}htspinhas to be calculated taking into account all these orientations. A comparison of the results obtained for the isotropic ex- change coupling constants Jijfor bcc Fe using the DLM and RDLM schemes is shown in Fig. 3, demonstrating close agreement, with the small differences to be ascribed to the different account of relativistic effects, i.e. in par- ticular the spin-orbit coupling.8 11.5 22.5 3 Rij/a051015202530Jij (meV)SR-DLM RDLMFe (bcc), T = 1500 K FIG. 3. Isotropic exchange coupling parameters calculated for the disordered magnetic state of bcc Fe within the scalar - relativistic approach, using the DLM scheme (circles, SR- DLM) and within the fully-relativistic approach, using the RDLM scheme19,24(squares, RDLM). C. Multi-spin expansion of spin Hamiltonian: General remarks Despite the obvious success of the classical Heisenberg model for many applications, higher-order multi-spin ex- pansionHmsof the spin Hamiltonian H, given by the expression Hms=−1 3!/summationdisplay i,j,kJijkˆsi·(ˆsj׈sk), −2 p!/summationdisplay i,j,k,lJs ijkl(ˆsi·ˆsj)(ˆsk·ˆsl) −2 p!/summationdisplay i,j,k,l/vectorDijkl·(ˆsi׈sj)(ˆsk·ˆsl)+..., =H3+H4,s+H4,a+... (61) can be of great importance to describe more subtle prop- erties of magnetic materials44–56. This concerns first of all systems with a non-collinear ground state characterized by finite spin tilting angles, that makes multispin contributions to the energy non- negligible. Inparticular,manyreportspublishedrecently discuss the impact of the multispin interactions on the stabilization of exotic topologically non-trivial magnetic textures, e.g. skyrmions, hopfions, etc.57–59 Corresponding calculations of the multi-spin exchange parameters have been reported by different groups. The approach based on the Connolly-Williams scheme has been used to calculate the four-spin non-chiral (two-site and three-site) and chiral interactions for Cr trimers52 and for a deposited Fe atomic chain60, respectively, for the biquadratic, three-site four spin and four-site four spin interaction parameters58,61. The authors discuss the role of these type of interactions for the stabilization of different types of non-collinear magnetic structures as skyrmions and antiskyrmions. A more flexible mapping scheme using perturbation theory within the KKR Green function formalism wasonly reported recently by Brinker et al.62,63, and by the present authors37. Here we discuss the latter approach, i.e. the energy expansion w.r.t. ∆ Vin Eq. (44). One has to point out that a spin tilting in a real system has a finite amplitude and therefore the higher order terms in this expansion might become non-negligible and in gen- eral should be taken into account. Their role obviously depends on the specific materialandshould increasewith temperature that leads to an increasing amplitude of the spin fluctuations. As these higher-order terms are di- rectly connected to the multispin terms in the extended Heisenberg Hamiltonian, one has to expect also a non- negligibleroleofthe multispin interactionsforsomemag- netic properties. Extending the spin Hamiltonian to go beyond the clas- sical Heisenberg model, we discuss first the four-spin ex- change interaction terms Jijkland/vectorDijkl. They can be calculated using the fourth-order term of the Green func- tion expansion ∆ E(4)given by: ∆E(4)=−1 πImTr/integraldisplayEF dE ×(E−EF)∆VG∆VG∆VG∆VG =−1 πImTr/integraldisplayEF dE∆VG∆VG∆VG∆VG. (62) where the sum rule for the Green functiondG dE=−GG followed by integration by parts was used to get a more compact expression. Using the multiple-scattering repre- sentation for the Green function, this leads to: ∆E(4)=/summationdisplay i,j,k,l−1 πImTr/integraldisplayEF dE ×∆Viiτij∆Vjjτjk∆Vkkτkl∆Vllτli.(63) with the matrix elements ∆ Vii=/an}b∇acketle{tZi|∆V|Zi/an}b∇acket∇i}ht. Using the ferromagnetic state with /vectorM||ˆzas a reference state, and creating the perturbation ∆ Vin the form of a spin-spiral according to Eq. (52), one obtains the corresponding /vector q- dependent energy change ∆ E(4)(/vector q), written here explic-9 itly as an example ∆E(4)=−1 π/summationdisplay i,j,k,lImTr/integraldisplayEF dEsin4θ ×/bracketleftbigg Ixxxx ijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl) +Ixxyy ijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl) +Iyyxx ijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl) +Iyyyy ijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl) +Ixyxx ijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl) +Iyxyy ijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl) +Iyxxx ijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl) +Ixyyy ijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl)+.../bracketrightbigg (64) where Iαβγδ ijkl=Ti,α(E)τij(E)Tj,β(E)τjk(E) ×Tk,γ(E)τkl(E)Tl,δ(E)τli(E).(65) As is shownin Ref. 37, the four-spinisotropicexchange interaction Jijklandz-component of the DMI-like in- teraction Dz ijklcan be obtained calculating the energy derivatives∂4 ∂q4∆E(4)and∂3 ∂q3∆E(4)in the limit of q= 0, and then identified with the corresponding derivatives of the termsH4,sandH4,ain Eq. (61). These interaction terms are given by the expressions Js ijkl=1 4/bracketleftbigg Jxxxx ijkl+Jxxyy ijkl+Jyyxx ijkl+Jyyyy ijkl/bracketrightbigg (66) and Dz ijkl=1 4/bracketleftbigg Jxyxx ijkl+Jxyyy ijkl−Jyxxx ijkl−Jyxyy ijkl)/bracketrightbigg ,(67) where the following definition is used: Jαβγδ ijkl=1 2πImTr/integraldisplayEF dETα iτijTβ jτjkTγ kτklTδ lτli(68) These expressionobviously give also access to a special cases, i.e. the four-spin three-site interactions with l=j, and the four spin two-site, socalled biquadratic exchange interactions with k=iandl=j. The scalar biquadratic exchange interaction parame- tersJs ijijcalculated on the basis of Eq. (66) for the three 3dbulk ferromagnetic systems bcc Fe, hcp Co and fcc Ni have been reported in Ref. 37. The results are plotted in Fig. 4 as a function of the distance Rij+Rjk+Rkl+Rli. For comparison, the insets give the corresponding bilin- ear isotropic exchange interactions for these materials. One can see rather strong first-neighbor interactions for bcc Fe, demonstrating the non-negligible characterof the ✵ ✁ ✶✶ ✁ ✷✷ ✁ ✸✸ ✁❘✐ ✂ ✴ ✄ ✵ ✵ ✁ ✶ ✶ ✁ ✷ ✷ ✁ ✸ ✸ ✁ ✹ ✹ ✁ ✁❏ ☎✆☎✆s ✥✝✞✟✠✡ ① ①✡ ② ② ☛☛☞ ✌ ✍✍☞ ✌ ✎✎☞ ✌✏✑ ✒ ✓✔ ✕ ✌ ☛✕ ☛✌✖ ✗✘ ✙✚✛✜✢❜ ✣✣ ✤✦ (a)✶✶ ✁ ✷✷ ✁ ✸✸ ✁❘✐ ✂ ✴ ✄ ✲ ☎ ☎✷ ☎ ☎ ☎ ✷☎ ☎✵ ☎ ☎ ✆ ☎ ☎ ✝ ☎ ✶❏ ✞✟✞✟s ✥✠✡☛☞ ✌ ① ①✌ ② ② ✍✍✎ ✏ ✑✑✎ ✏ ✒✒✎ ✏✓✔ ✕ ✖✗ ✘ ✏ ✍✘ ✍✏✙ ✚✛ ✜✢✣✤✦❤ ✧★ ✩ ✪ (b)✵ ✁ ✶✶ ✁ ✷✷ ✁ ✸❘✐ ✂ ✴ ✄ ✲ ✵ ✵✷ ✲ ✵ ✵✶ ✁ ✲ ✵ ✵✶ ✲ ✵ ✵✵ ✁ ✵❏ ☎✆☎✆s ✥✝✞✟✠✡ ① ①✡ ② ② ☛☛☞ ✌ ✍✍☞ ✌ ✎✏✑ ✒ ✓✔ ✕ ☛ ✍ ✎✖ ✗✘ ✙✚✛✜✢❢ ✣✣ ✤✦ (c) FIG. 4. Scalar biquadratic exchange interactions Js ijijin bcc Fe (a), hcp Co (b) and fcc Ni (Ni). For comparison, the insets show the bilinear exchange interaction parameters calcula ted for the FM state with the magnetization along the ˆ z-axis. All data are taken from Ref. 37. biquadratic interactions. This is of course a material- specific property, and one notes as decrease for the bi- quadratic exchange parameters when going to Co and Ni as shown in Fig. 4 (b) and (c), respectively. In order to calculate the xandycomponents of the four-spin and as a special case the three-site-DMI (TDMI) and biquadratic-DMI (BDMI) type interactions, the scheme suggested in Ref. 37 for the calculation of the DMI parameters36,64can be used, which exploited the DMI-governed behavior of the spin-wave dispersion hav- ing a finite slope at the Γ point of the Brillouin zone.10 Note, however, that a more general form of perturbation isrequiredin thiscasedescribedbya2Dspin modulation field according to the expression ˆsi=/parenleftbig sin(/vector q1·/vectorRi) cos(/vector q2·/vectorRi),sin(/vector q2·/vectorRi), cos(/vector q1·/vectorRi)cos(/vector q2·/vectorRi)/parenrightbig , (69) where the wave vectors /vector q1and/vector q2are orthogonal to each other, as for example /vector q1=q1ˆyand/vector q2=q2ˆx. Taking the second-order derivative with respect to the wave-vector /vector q2and the first-order derivative with respect to the wave-vectors /vector q1and/vector q2, and considering the limit q1(2)→0, one obtains ∂3 ∂q3 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle q2=0H4,a=/summationdisplay i,j,k,lDx ijkl(ˆq2·/vectorRij)(ˆq2·/vectorRlk)2, and ∂ ∂q1/vextendsingle/vextendsingle/vextendsingle/vextendsingle q1=0∂2 ∂q2 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle q2=0H4,a=/summationdisplay i,j,k,lDy ijkl(ˆq1·/vectorRij)(ˆq2·/vectorRlk)2, where/vectorRij=/vectorRj−/vectorRiand/vectorRlk=/vectorRk−/vectorRl. The microscopic expressions for the xandycompo- nents of/vectorDijkldescribing the four-spin interactions is de- rived on the basis of the third-order term in Eq. (43) ∆E(3)=−1 πImTr/integraldisplayEF dE(E−EF) ×G0∆VG0∆VG0∆VG0. (70) The final expression for Dα ijklis achieved by taking the second-order derivative with respect to the wave-vector /vector q2and the first-orderderivative with respect to the wave- vectors/vector q1(2), considering the limit q1(2)→0, i.e. equat- ing within the ab-initio and model expressions the cor- responding terms proportional to ( /vectorRi−/vectorRj)y(/vectorRk−/vectorRl)2 x and (/vectorRi−/vectorRj)x(/vectorRk−/vectorRl)2 x(we keep a similar form in both casesforthe sakeofconvenience)givestheelements Dy,x ijkl andDy,y ijkl, as well as Dx,x ijklandDx,y ijkl, respectively, of the four-spin chiral interaction as follows Dα,β ijkj=ǫαγ1 8πImTr/integraldisplayEF dE(E−EF) /bracketleftBig OiτijTj,γτjkTk,βτklTl,βτli −Ti,γτijOjτjkTk,βτklTl,βτli/bracketrightBig +/bracketleftBig OiτijTj,βτjkTk,βτklTl,γτli −Ti,γτijTj,βτjkTk,βτklOlτli/bracketrightBig (71) withα,β=x,y, andǫαγthe elements of the transverse Levi-Civita tensor ǫ=/bracketleftbigg 0 1 −1 0/bracketrightbigg . The TDMI and BDMI parameterscan be obtained as the special cases l=jand l=j,k=i, respectively, from Eq. (71).The expression in Eq. (71) gives access to the xandy components of the DMI-like three-spin interactions Dα ijkj=Dα,x ijkj+Dα,y ijkj. (72) Finally, three-spin chiral exchange interaction (TCI) represented by first term in the extended spin Hamilto- nianhas been discussedin Ref. 37. As it followsfrom this expression, the contribution due to this type of interac- tion is non-zero only in case of a non-co-planar and non- collinear magnetic structure characterized by the scalar spatial type product ˆ si·(ˆsj׈sk) involving the spin mo- ments on three different lattice sites. In order to work out the expression for the Jijkinter- action, one has to use a multi-Q spin modulation65–67 which ensure a non-zero scalar spin chirality for every three atoms. The energy contribution due to the TCI, is non-zero only if Jijk/ne}ationslash=Jikj, etc. Otherwise, the termsijkandikjcancel each other due to the relation ˆsi·(ˆsj׈sk) =−ˆsi·(ˆsk׈sj). Accordingly, the expression for the TCI is derived us- ing the 2Q non-collinear spin texture described by Eq. (69), which is characterized by two wave vectors oriented along two mutually perpendicular directions, as for ex- ample/vector q1= (0,qy,0) and/vector q2= (qx,0,0). Applying such a spin modulation in Eq. (69) for the term H3associated with the three-spin interaction in the spin Hamiltonian in Eq. (61), the second-order derivative of the energy E(3)(/vector q1,/vector q2) with respect to the wave vectors q1andq2is given in the limit q1→0,q2→0 by the expression ∂2 ∂/vector q1∂/vector q2H(3) =−/summationdisplay i/negationslash=j/negationslash=kJijk/parenleftbig ˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig .(73) The microscopic energy term of the electron system, giving access to the chiral three-spin interaction in the spin Hamiltonian is described by the second-order term ∆E(2)=−1 πImTr/integraldisplayEF dE(E−EF) G0∆VG0∆VG0 (74) of the free energy expansion. Taking the first-order derivative with respect to q1andq2in the limit q1→ 0,q2→0, and equating the terms proportional to/parenleftbig ˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig with the correspondingterms inthespinHamiltonian,oneobtainsthefollowingexpres- sion for the three-spin interaction parameter Jijk=1 8πImTr/integraldisplayEF dE(E−EF) /bracketleftBig Ti,xτijTj,yτjkOkτki−Ti,yτijTj,xτjkOkτki −Ti,xτijOjτjkTk,yτki+Ti,yτijOjτjkTk,xτki +OiτijTi,xτjkTk,yτki−OjτijTi,yτjkTk,xτki/bracketrightBig ,(75)11 giving access to the three-spin chiral interaction deter- mined asJ∆=Jijk−Jikj. Its interpretation was dis- cussed in Ref. 68, where its dependence on the SOC as well as on the topological orbital susceptibility χTO ∆= χTO ijk−χTO ikjwas demonstrated. In fact that the expres- sion forχTO ijkworked out in Ref. 68 has a rather similar form asJijk, as that can be seen from the expression χTO ijk=−1 4πImTr/integraldisplayEF dE ×/bracketleftBig Ti,xτijTj,yτjklk zτki−Ti,yτijTj,xτjklk zτki −Ti,xτijlj zτjkTk,yτki+Ti,yτijlj zτjkTk,xτki +li zτijTj,xτjkTk,yτki−li zτijTj,yτjkTk,xτki/bracketrightBig . (76) For everytrimerofatoms, both quantities, χTO ijkandJijk, are non-zero only in the case of non-zero scalar spin chi- rality ˆsi·(ˆsj׈sk) and depend on the orientation of the trimermagneticmomentwith respecttothetrimerplain. This is shown in Fig. 668representing ∆ Jand ∆χTOas a function of the angle between the magnetization and normal ˆnto the surface, which are calculated for the two smallest trimers, ∆ 1and ∆ 2, centered at the Ir atom and the hole site in the Ir surface layer for 1ML Fe/Ir(111), respectively (Fig. 5). FIG. 5. Geometry of the smallest three-atom clusters in the monolayer of 3 d-atoms on M(111) surface ( M= Au, Ir): M- centered triangle ∆ 1and hole-centered triangle ∆ 2. The role of the SOC for the three-site 4-spin DMI-like interaction, Dz ijik, and the three-spin chiral interaction, J∆is shown in Fig. 7. These quantities are calculated for 1ML Fe on Au (111), for the two smallest triangles ∆1and ∆ 2centered at an Au atom or a hole site, re- spectively (see Fig. 5). Here, setting the SOC scaling factorξSOC= 0 implies a suppression of the SOC, while ξSOC= 1 corresponds to the fully relativistic case. Fig. 7 (a) shows the three-site 4-spin DMI-like interaction pa- rameter, Dz ijik(ξSOC) when the SOC scaling parameter ξSOCapplied to all components in the system, shown by full symbols, and with the SOC scaling applied only to the Au substrate. One can see a dominating role of the SOC of substrate atoms for Dz ijik. Also in Fig. 7 (b), a nearly linear variation can be seen for J∆(ξSOC) when the SOC scaling parameter ξSOCis applied to all com- ponents in the system (full symbols). Similar to Dz ijik, ✵ ✷✵ ✹✵✻✵ ✽✵❣ ✥ ✁✂ ✄ ✵ ✵ ☎ ✆ ✵ ☎ ✷ ✵ ☎ ✝ ✵ ☎ ✹ ✵ ☎ ✞ ✵ ☎ ✻✲✟ ❉ ❚ ✠ ✡☛☞✌✍ ❏✎✶ ✥ ❣ ✄ ✥ ✮ ✏ ✑✒ ✄❏✎ ✓ ✥ ❣ ✄ ✥ ✔ ✑✕ ✖✄❏✎✶ ✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄❏✎ ✓ ✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄ (a)✵ ✷✵ ✹✵✻✵ ✽✵❣ ✥ ✁✂ ✄ ✵ ✵ ☎✵✵ ✆ ✵ ☎✵ ✝ ✵ ☎✵ ✝✆ ✵ ☎✵✷ ✵ ☎✵✷ ✆ ✵ ☎✵ ✞❝ ❉ ❚ ✟ ✠♠ ❇ ✴✡✡☛☞ ✌✍ ✶ ✥ ❣ ✄ ✥ ✮ ✎ ✏✑✄✌✍ ✒ ✥ ❣ ✄ ✥ ✓ ✏✔ ✕ ✄✌✍ ✶ ✥ ✵ ✄ ✭ ✏✖ ❣✌✍ ✒ ✥ ✵ ✄ ✭ ✏✖ ❣ (b) FIG. 6. (a) Three-spinchiral exchange interaction paramet ers −J∆(γ) (’minus’ is used to stress the relation between J∆and χTO ∆), and (bc) topological orbital susceptibility (TOS, for SOC = 0), calculated for Fe on Ir (111), as a function of the angle between the magnetization and normal ˆ nto the surface, for the smallest triangles ∆ 1and ∆ 2. The dashed lines rep- resentJ∆(0) cos(γ) (a) and χTO ∆(0) cos(γ) (b), respectively. All data are taken from Ref. 68. this shows that the SOC is an ultimate prerequisite for a non-vanishing TCI J∆. When scaling the SOC only for Au (open symbols), Fig. 7 (b) show only weak changes fortheTCIparameters J∆(ξSOC), demonstratingaminor impact of the SOC of the substrate on these interactions, in contrast to the DMI-like interaction shown in Fig. 7 (a). One can see also that Dz ijikis about two orders of magnitude smaller than J∆for this particular system. The origin of the TCI parameters have been discussed in the literature suggesting a different interpretation of the correspondingterms derivedalsowithin the multiple- scattering theory Green function formalism62,69,70. How- ever, the expression worked out in Ref. 69 has obviously not been applied for calculations so far. As pointed out in Ref. 68, the different interpretation of this type of in- teractions can be explained by their different origin. In particular, one has to stress that the parameters in Refs. 68 and 69 were derived in a different order of pertur- bation theory. On the other hand, the approach used for calculations of the multispin exchange parameters re- ported in Ref. 62, 69, and 71 is very similar to the one used in Refs. 37 and 68. The corresponding expressions have been worked out within the framework of multiple- scattering Green function formalism using the magnetic force theorem. In particular, the Lloyd formula has been used to express the energy change due to the perturba- tion ∆Vleading to the expression ∆E=−1 πIm Tr/integraldisplayEF dE/summationdisplay p1 pTr/bracketleftbig G(E)∆V/bracketrightbigp.(77) Using the off-site part of the GF in Eq. (35), as defined12 0 0.2 0.4 0.6 0.8 1 ξSOC-0.004-0.00200.0020.0040.0060.008Dijik (meV) Dx(∆1), SOC Dx(∆1), SOC(Au)Dx(∆2), SOC(Au) Dx(∆2), SOC (a) 0 0.2 0.4 0.6 0.8 1 ξSOC-0.4-0.3-0.2-0.10J∆ (meV)∆1, SOC ∆2, SOC ∆1, SOC(Au) ∆2, SOC(Au) (b) FIG. 7. (a) Three-site 4-spin DMI-like interaction, Dz ijkjand (c) three-spin chiral exchange interaction (TCI) paramete rs J∆calculated for Fe on Au (111) on the basis of Eq. (75) as a function of SOC scaling parameter ξSOCfor the smallest triangles ∆ 1and ∆ 2. In figure (b), full symbols represent the results obtained when scaling the SOC for all elements in the system, while open symbols show the results when scaling only the SOC for Au. All data are taken from Ref. 68. by Dederichs et al.34, Eq. (77) is transformed to the form ∆E=−1 πIm Tr/integraldisplayEF dE/summationdisplay p1 pTr/bracketleftBig Gstr(E)∆t(E)/bracketrightBigp .(78) By splitting the structural Green function Gstr ijinto a spin-dependent ( /vectorBstr ij) and a spin-independent ( Astr ij) parts according to Gstr ij=Astr ijσ0+/vectorBstr ij·/vector σ (79) and expressing the change of the single-site scattering matrix ∆ti(E) = (t↑ i(E)−t↓)δˆsi×/vector σ, (80) by means of the rigid spin approximation, the different terms in Eq. (78) corresponding to different numbers p give access to corresponding multispin terms, chiral and non-chiral, in the extended spin Hamiltonian. In particu- lar, the isotropic six-spin interactions, that are responsi- bleforthenon-collinearmagneticstructureofB20-MnGe according to Grytsiuk et al69, is given by the expression κ6−spin ijklmn=1 3πIm Tr/integraldisplayEF dE ×Aijtσ jAjktσ kAkltσ lAlmtσ mAmntσ nAnitσ i.(81)A rather different point of view concerning the multi- spin extension of the spin Hamiltonian was adopted by Streib et al.72,73, who suggested to distinguish so-called local and global Hamiltonians. According to that classi- fication, a global Hamiltonian implies to include in prin- ciple all possible spin configurations for the energy map- ping in orderto calculate exchangeparametersthat char- acterize in turn the energy of any spin configuration. On the other hand, a local Hamiltonian is ’designed to de- scribe the energetics of spin configurations in the vicinity of the ground state or, more generally, in the vicinity of a predefined spin configuration’72. This implies that taking the ground state as a reference state, it has to be deter- mined first before the calculating the exchange parame- ters which are in principle applicable only for small spin tiltings around the reference state and can be used e.g. to investigate spin fluctuations around the ground state spin configuration. In Ref. 72, the authors used a con- strainingfieldtostabilizethenon-collinearmagneticcon- figuration. This leads to the effective two-spin exchange interactions corresponding to a non-collinear magnetic spin configuration72,73. According to the authors, ’lo- cal spin Hamiltonians do not require any spin interac- tions beyond the bilinear order (for Heisenberg exchange as well as Dzyaloshinskii-Moriya interactions)’ . On the other hand, they point out the limitations for these ex- change interactions in the case of non-collinear system in the regime when the standard Heisenberg model is not valid73, and multi-spin interactions get more important. II. GILBERT DAMPING Another parameter entering the Landau-Lifshitz- Gilbert (LLG) equation in Eq. (3) is the Gilbert damping parameter ˜Gcharacterizingenergydissipation associated with the magnetization dynamics. Theoretical investigations on the Gilbert damping pa- rameter have been performed by various groups and ac- cordinglythepropertiesofGDisdiscussedindetailinthe literature. Many of these investigations are performed assuming a certain dissipation mechanism, like Kamber- sky’sbreathingFermisurface(BFS)74,75, ormoregeneral torque-correlationmodels (TCM)76,77to be evaluated on the basis of electronic structure calculations. The earlier works in the field relied on the relaxation time param- eter that represents scattering processes responsible for the energy dissipation. Only few computational schemes for Gilbert damping parameter account explicitly for dis- orderin the systems, which is responsible forthe spin-flip scatteringprocess. This issuewasaddressedin particular by Brataas etal.78who described the Gilbert damping mechanism by means of scattering theory. This develop- mentsuppliedtheformalbasisforthefirstparameter-free investigations on disordered alloys38,39,79. A formalism for the calculation of the Gilbert damping parameter based on linear response theory has been re- ported in Ref. 39 and implemented using fully relativistic13 multiple scattering or Korringa-Kohn-Rostoker (KKR) formalism. Considering the FM state as a reference state of the system, the energy dissipation can be expressed in terms of the GD parameter by: ˙Emag=/vectorHeff·d/vectorM dτ=1 γ2˙/vector m[˜G(/vector m)˙/vector m].(82) On the other hand, the energy dissipation in the elec- tronic system is determined by the underlying Hamilto- nianˆH(τ) as follows ˙Edis=/angbracketleftBig dˆH dτ/angbracketrightBig . Assuming a small deviation of the magnetic moment from the equilibrium /vector u(τ), the normalized magnetization /vector m(τ) can be written in a linearized form /vector m(τ) =/vector m0+/vector u(τ), that in turn leads to the linearized time dependent electronic Hamiltonian ˆH(τ) ˆH=ˆH0(/vector m0)+/summationdisplay µ/vector uµ∂ ∂/vector uµˆH(/vector m0).(83) As shown in Ref. 38, the energy dissipation within the linear response formalism is given by: ˙Edis=−π/planckover2pi1/summationdisplay ij/summationdisplay µν˙uµ˙uν/angbracketleftBigg ψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH ∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg ψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH ∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg ×δ(EF−Ei)δ(EF−Ej).(84) Identifying it with the corresponding phenomenological quantity in Eq. (82), ˙Emag=˙Edisone obtains for the GD parameterαa Kubo-Greenwood-like expression: αµν=−/planckover2pi1γ πMsTrace/angbracketleftBigg ∂ˆH ∂uµImG+(EF)∂ˆH ∂uνImG+(EF)/angbracketrightBigg c, (85) whereα=˜G/(γMs), and/an}b∇acketle{t.../an}b∇acket∇i}htcindicates a configura- tional average required in the presence of chemical or thermally induced disorder responsible for the dissipa- tion processes. Within the multiple scattering formalism with the representation of the Green function given by Eq. (9), Eq. (85) leads to αµµ=g πµtot/summationdisplay nTrace/angbracketleftbig T0µ˜τ0nTnµ˜τn0/angbracketrightbig c(86) with the g-factor 2(1 + µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, the total magnetic moment µtot=µspin+µorb, and ˜τ0n ΛΛ′= 1 2i(τ0n ΛΛ′−τ0n Λ′Λ) and with the energy argument EFomit- ted. The matrix elements Tnµare identical to those oc- curring in the context of exchange coupling6and can be expressed in terms of the spin-dependent part Bof the electronic potential with matrix elements: Tnµ Λ′Λ=/integraldisplay d3rZn× Λ′(/vector r) [βσµBxc(/vector r)]Zn Λ(/vector r).(87) As is discussed in Ref. 39, fora system havingchemical disorder, the configurational average is performed using00.10.2 0.3 0.4 0.5 concentration xV02040α × 103without vertex corrections with vertex correctionsFe1-xVx (a) FIG. 8. The Gilbert damping parameter for (a) bcc Fe 1−xVx (T= 0 K) as a function of V concentration. Full (open) symbols give results with (without) the vertex corrections . All data are taken from Ref. 39. the scattering path operators evaluated on the basis of the coherent potential approximation (CPA) alloy the- ory. In the case of thermally induced disorder, the al- loy analogy model is used, which was discussed already above. When evaluating Eq. (86), the so-called vertex corrections have to be included43that accounts for the difference between the averages /an}b∇acketle{tTµImG+TνImG+/an}b∇acket∇i}htcand /an}b∇acketle{tTµImG+/an}b∇acket∇i}htc/an}b∇acketle{tTνImG+/an}b∇acket∇i}htc. Within the Boltzmann formal- ism these corrections account for scattering-in processes. The crucial role of these corrections is demonstrated39 in Fig. 8 representing the Gilbert damping parameter for an Fe 1−xVxdisordered alloy as a function of the con- centrationx, calculated with and without vertex correc- tions. As one can see, neglect of the vertex corrections may lead to the nonphysical result α <0. This wrong behavior does not occur when the vertex corrections are included, that obviously account for energy transfer pro- cesses connected with scattering-in processes. The impact of thermal vibrations onto the Gilbert damping can be taken into account within the alloy- analogy model (see above) by averaging over a discrete set of thermal atom displacements for a given temper- atureT. Fig. 9 represents the temperature dependent behavior of the Gilbert damping parameter αfor bcc Fe with 1% and 5% of impurities of Os and Pt38,39. One can seeastrongimpactofimpuritiesonGD.Inthecaseof1% of Pt in Fig. 9 (a), αdecreases in the low-temperature regime much steeper upon increasing the temperature, indicating that the breathing Fermi surface mechanism dominates. When the concentration of the impurities in- creases up to 5% (Fig. 9 (a)), the spin-flip scattering mechanism takes the leading role for the magnetization dissipation practically for the whole region of tempera- tures under consideration. The different behavior of GD forFe with OsandPt isaresult ofthe different densityof states (DOS) of the impurities at the Fermi energy (see Ref. 39 for a discussion). The role of the electron-phonon scattering for the ul- trafast laser-induced demagnetization was investigated14 0100200 300 400 500 temperature (K)12345α × 103Fe0.99Me0.01Pt Os (a) 0100200 300 400 500 temperature (K)22.533.54α × 103Fe0.95Me0.05 PtOs (b) FIG. 9. Gilbert damping parameter for bcc Fe 1−xMxwith M= Pt (circles) and M= Os (squares) impurities as a func- tion of temperature for 1% (a) and 5% (b) of the impurities. All data are taken from Ref. 39. by Carva et al.80based on the Elliott-Yafet theory of spin relaxation in metals, that puts the focus on spin- flip(SF) transitionsupon theelectron-phononscattering. As the evaluation of the spin-dependent electron-phonon matrix elements entering the expression for the rate of the spin-flip transition is a demanding problem, various approximations are used for this. In particular, Carva et al.80,81use the so-called Elliott approximation to evalu- ate a SF probability Pb S=τ τsfwith the spin lifetime τsf and a spin-diagonal lifetime τ: Pb S=τ τsf= 4/an}b∇acketle{tb2/an}b∇acket∇i}ht (88) with the Fermi-surface averaged spin mixing of Bloch wave eigenstates /an}b∇acketle{tb2/an}b∇acket∇i}ht=/summationdisplay σ,n/integraldisplay d3k|bσ /vectorkn|δ(Eσ /vectorkn−EF).(89) In the case of a non-collinear magnetic structure, the description of the Gilbert damping can be extended byadding higher-order non-local contributions. The role of non-local damping contributions has been investigated by calculating the precession damping α(/vector q) for magnons in FM metals, characterized by a wave vector /vector q. Follow- ing the same idea, Thonig et al.82used a torque-torque correlationmodel based on atight binding approach,and calculated the Gilbert damping for the itinerant-electron ferromagnets Fe, Co and Ni, both in the reciprocal, α(/vector q), and realαijspace representations. The important role of non-local contributions to the GD for spin dynam- ics has been demonstrated using atomistic magnetization dynamics simulations. Aformalismforcalculatingthe non-localcontributions to the GD has been recently worked out within the KKR Green function formalism83. Using linear response the- ory for weakly-noncollinear magnetic systems it gives ac- cess to the GD parameters represented as a function of a wave vector /vector q. Using the definition for the spin sus- ceptibility tensor χαβ(/vector q,ω), the Fourier transformation of the real-space Gilbert damping can be represented by the expression84,85 ααβ(/vector q) =γ M0Vlim ω→0∂ℑ[χ−1]αβ(/vector q,ω) ∂ω.(90) Hereγ=gµBis the gyromagneticratio, M0=µtotµB/V is the equilibrium magnetization and Vis the volume of the system. As is shown in Ref. 83, this expression can be transformed to the form which allows an expansion of GD in powers of wave vector /vector q: α(/vector q) =α+/summationdisplay µαµqµ+1 2/summationdisplay µναµνqµqν+....(91) with the following expansion coefficients: α0±± αα=g πµtot1 ΩBZTr/integraldisplay d3k/angbracketleftbigg Tβτ(/vectork,E± F)Tβτ(/vectork,E± F)/angbracketrightbigg c αµ±± αα=g πµtot1 ΩBZTr/integraldisplay d3k/angbracketleftbigg Tβ∂τ(/vectork,E± F) ∂kµTβτ(/vectork,E± F)/angbracketrightbigg c αµν±± αα=−g 2πµtot1 ΩBZ ×Tr/integraldisplay d3k/angbracketleftbigg Tβ∂τ(/vectork,E± F) ∂kµTβ∂τ(/vectork,E± F) ∂kν/angbracketrightbigg c.(92) For the prototype multilayer system (Cu/Fe 1−xCox/Pt)nthe calculated zero-order (uni- form) GD parameter αxxand the corresponding first-order (chiral) αx xxcorrection term for /vector q/ba∇dblˆxare plotted in Fig. 10 top and bottom, respectively, as a function of the Co concentration x. Both terms, αxx andαx xx, increase approaching the pure limits w.r.t. the Fe1−xCoxalloy subsystem. As is pointed out in Ref. 83, this increase is associated with the dominating so-called breathing Fermi-surface damping mechanism due to the modification of the Fermi surface (FS) induced by the SOC, which follows the magnetization direction that slowly varies with time. As αis caused for a ferromagnet15 0 0.2 0.4 0.6 0.8 100.20.4αxx 0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.) FIG. 10. The Gilbert damping parameters αxx(top) and αx xx(bottom) calculated for the model multilayer system (Cu/Fe 1−xCox/Pt)nusing first and second expressions in Eq. (92), respectively. All data are taken from Ref. 83. exclusively by the SOC one can expect that it vanishes for vanishing SOC. This was indeed demonstrated before39. The same holds also for αxthat is caused by SOC as well. Alternatively, a real-space extension for classical Gilbert dampingtensorwasproposedrecentlybyBrinker et al.86, by introducing two-site Gilbert damping tensor Gijentering the site-resolved LLG equation 1 γd/vectorMi dτ=−γ/vectorMi×/parenleftbigg /vectorHi,eff+/summationdisplay j/bracketleftBigg Gij(/vectorM)·d/vectorMi dτ/bracketrightBigg/parenrightbigg ,(93) which is related to the inverse dynamical susceptibility χijvia the expression d dωIm[χ]αβ ij=δij/parenleftbigg1 γMiǫαβγ/parenrightbigg +/parenleftbigg RiGijRT j/parenrightbigg αβ,(94) whereRiandRjarethe rotationmatricesto gofromthe global to the local frames of reference for atoms iandj, respectively, assuming a non-collinear magnetic ground state in the system. Thus, an expression for the GD can be directly obtained using the adiabatic approxima- tion for the slow spin-dynamics processes. This justifies the approximation ([ χ]−1(ω))′ ω≈([χ0]−1(ω))′ ω, with the un-enhanced dynamical susceptibility given in terms ofelectronic Green function Gij χαβ ij(ω+iη) =−1 πTr/integraldisplayEF dE /bracketleftbigg σαGij(E+ω+iη)σβImGij(E) +σαGij(E)σβImGij(E−ω−iη)/bracketrightbigg ,(95) with the Green function G(E±iη) = (E− H ±iη)−1 corresponding to the Hamiltonian H. Moreover, this approach allows a multisite expansion oftheGDaccountingforhigher-ordernon-localcontribu- tions for non-collinearstructures86. For this purpose, the Hamiltonian His split into the on-site contribution H′ and the intersite hopping term tij, which is spin depen- dent in the general case. The GF can then be expanded in a perturbative way using the Dyson equation Gij=G0 iδij+G0 itijG0 j+G0 itikG0 ktkjG0 j+....(96) As a result, the authors generalize the LLG equation by splitting the Gilbert damping tensor in terms pro- portional to scalar, anisotropic, vector-chiral and scalar- chiral products of the magnetic moments, i.e. terms like ˆei·ˆej, (ˆnij·ˆei)(ˆnij·ˆej), ˆnij·(ˆei׈ej), etc. It should be stressed that the Gilbert damping param- eter accounts for the energy transfer connected with the magnetization dynamics but gives no information on the angular momentum transfer that plays an important role e.g. for ultrafast demagnetization processes. The formal basis to account simultaneously for the spin and lattice degrees of freedom was considered recently by Aßmann and Nowak87and Hellsvik et al.88. Hellsvik et al.88,89re- portonanapproachsolvingsimultaneouslytheequations for spin and lattice dynamics, accounting for spin-lattice interactions in the Hamiltonian, calculated on a first- principles level. These interactions appear as a correc- tion to the exchange coupling parameters due to atomic displacements. As a result, this leads to the three-body spin-lattice coupling parameters Γαβµ ijk=∂Jαβ ij ∂uµ kand four- body parameters Λαβµν ijkl=∂Jαβ ij ∂uµ k∂uν lrepresented by rank 3 and rank 4 tensors, respectively, entering the spin-lattice Hamiltonian Hsl=−1 2/summationdisplay i,j,k,αβ,µΓαβµ ijkeα ieβ juµ k −1 4/summationdisplay i,j,k,l,αβ,µ,νΛαβµν ijkleα ieβ juµ kuν l.(97) The parameters Γαβµ ijkin Ref. 88 are calculated using a finite difference method, using the exchange coupling pa- rametersJijfor the system without displacements ( J0 ij) and with a displaced atom k(J∆ ij(/vector uk)), used to estimate the coefficient Γαβµ ijk≈(J∆ ij(/vector uk)−J0 ij) uµ.16 Alternatively, to describe the coupling of spin and spa- tial degrees of freedom the present authors (see Ref. 90) adopt an atomistic approach and start with the expan- sion of a phenomenological spin-lattice Hamiltonian Hsl=−/summationdisplay i,j,α,β/summationdisplay k,µJαβ,µ ij,keα ieβ juµ k −/summationdisplay i,j/summationdisplay k,lJαβ,µν ij,kleα ieβ juµ kuν l,(98) that can be seen as a lattice extension of a Heisenberg model. Accordingly, the spin and lattice degrees of free- dom are represented by the orientation vectors ˆ eiof the magnetic moments /vector miand displacement vectors /vector uifor each atomic site i. The spin-lattice Hamiltonian in Eq. (98) is restricted to three and four-site terms. As rel- ativistic effects are taken into account, the SLC is de- scribed in tensorial form with Jαβ,µ ij,kandJαβ,µν ij,klrepre- sented by rank 3 and rank 4 tensors, similar to those discussed by Hellsvik et al.88. Thesamestrategyasforthe exchangecouplingparam- etersJij4orJαβ ij5,6, is used to map the free energy land- scapeF({ˆei},{/vector ui}) accounting for its dependence on the spin configuration {ˆei}as well as atomic displacements {/vector ui}, making use of the magnetic force theorem and the Lloyd formulato evaluate integrated DOS ∆ N(E). With this, the free energy change due to any perturbation in the system is given by Eq. (25). Using as a reference the ferromagnetically ordered state of the system with a non-distorted lattice, and the perturbed state characterized by finite spin tiltings δˆei and finite atomic displacements /vector uiat sitei, one can write the corresponding changes of the inverse t-matrix as ∆s µmi=mi(δˆeµ i)−m0 iand ∆u νmi=mi(uν i)−m0 i. This allows to replace the integrand in Eq. (11) by lnτ−lnτ0=−ln/parenleftBig 1+τ[∆s µmi+∆u νmj+...]/parenrightBig ,(99) where all site-dependent changes in the spin configura- tion{ˆei}and atomic positions {/vector ui}are accounted for in a one-to-one manner by the various terms on the right hand side. Due to the use of the magnetic force theorem these blocks may be written in terms of the spin tiltings δˆeµ iand atomic displacements of the atoms uν itogether with the corresponding auxiliary matrices Tµ iandUν i, respectively, as ∆s µmi=δˆeµ iTµ i, (100) ∆u νmi=uν iUν i. (101) Inserting these expressionsinto Eq. (99) and the result in turnintoEq.(25)allowsustocalculatetheparametersof the spin-lattice Hamiltonian as the derivatives of the free energy with respect to tilting angles and displacements. This way one gets for example for the three-site term: Jαβ,µ ij,k=∂3F ∂eα i∂eβ j∂uµ k=1 2πIm Tr/integraldisplayEF dE ×/bracketleftBig Tα iτijTβ jτjkUµ kτki+Tα iτikUµ kτkjTβ jτji/bracketrightBig (102) FIG. 11. The absolute values of site-off-diagonal and site- diagonal SLC parameters: DMI |/vectorDx ij,j|and isotropic SLC Jiso,x ij,j(top), anti-symmetric diagonal components Jdia−a,x ij,j andJdia−a,x ii,k(middle), and symmetric off-diagonal compo- nentsJoff−s,x ij,jandJoff−s,x ii,k(bottom) for bcc Fe, as a function of the interatomic distance rij and for the four-site term: Jαβ,µν ij,kl=∂4F ∂eα i∂eβ j∂uµ k∂uν l=1 4πIm Tr/integraldisplayEF dE ×/bracketleftBigg Uµ kτkiTα iτijTβ jτjlUν lτlk +Tα iτikUµ kτkjTβ jτjlUν lτli +Uµ kτkiTα iτilUν lτljTβ jτjk +Tα iτikUµ kτklUν lτljTβ jτji/bracketrightBigg .(103) Fig. 11 shows corresponding results for the SLC pa- rameters of bcc Fe, plotted as a function of the distance rijfori=kwhich implies that a displacement along the xdirection is applied for one of the interacting atoms. The absolute values of the DMI-like SLC parameters (DSLC) |/vectorD|µ=x ij,k(note that Dz,µ ij,k=1 2(Jxy,µ ij,k− Jyx,µ ij,k) ) show a rather slow decay with the distance rij. The isotropic SLC parameters Jiso,µ=x ij,j, which have only a weak dependence on the SOC, are about one order of magnitude larger than the DSLC. All other SOC- driven parameters shown in Fig. 11, characterizing the displacement-induced contributions to MCA, are much smaller than the DSLC.17 III. SUMMARY To summarize, we have considered a multi-level atom- istic approach commonly used to simulate finite temper- atureand dynamical magneticpropertiesof solids, avoid- ing in particular time-consuming TD-SDFT calculations. Theapproachisbasedonaphenomenologicalparameter- ized spin Hamiltonian which allows to separate the spin and orbital degrees of freedom and that way to avoid the demanding treatment of complex spin-dependent many- body effects. As these parameters are fully determined by the electronic structure of a system, they can be de- duced from the information provided by relativistic band structure calculations based on SDFT. 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2001.11899v1.An_efficient_automated_data_analytics_approach_to_large_scale_computational_comparative_linguistics.pdf

An efficient automated data analytics approach to large scale computational comparative linguistics Gabija Mikulyte gabija.mikulyte@gmail.comandDavid Gilbert david.gilbert@brunel.ac.uk Department of Computer Science Brunel University London Uxbridge UB8 3PH U.K.arXiv:2001.11899v1 [cs.CL] 31 Jan 2020Contents List of Figures List of Tables 1 Introduction 1 2 Background 2 2.1 Human languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.1 Indo-European languages and Kurgan Hypothesis . . . . . . . 3 2.1.2 Brittonic languages . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.3 Sheep Counting System . . . . . . . . . . . . . . . . . . . . . 4 3 Aims and Objectives 5 3.1 Overall Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Specific Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Data 6 4.1 Language files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2.1 Sheep counting words . . . . . . . . . . . . . . . . . . . . . . 7 4.2.2 Geographical data . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Colours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.4 IPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Methodology 10 6 Methods 12 6.1 Edit Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2 Phonetic Substitution Table . . . . . . . . . . . . . . . . . . . . . . . 14 6.3 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.3.1 Using the OC program . . . . . . . . . . . . . . . . . . . . . . 14 6.3.2 Using R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.4 Further analysis with R . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.5 Process automation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7 Results 17 7.1 Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7.1.1 Analysis of average and subset linguistic distance . . . . . . . 18 7.2 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7.2.1 All to all comparison analysis . . . . . . . . . . . . . . . . . . 19 7.2.2 Linguistic and Geographical distance relationship . . . . . . . 22 7.3 Colours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3.1 Mean and Standard Deviation . . . . . . . . . . . . . . . . . . 23 7.3.2 Density Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7.3.3 Bhattacharya Coefficients . . . . . . . . . . . . . . . . . . . . 25 7.3.4 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . 26 7.4 IPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.5 Small Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.5.1 All to all comparison . . . . . . . . . . . . . . . . . . . . . . . 29 8 Conclusions 30 9 Further Work 32 10 Summary of contributions 35 Acknowledgements 36 Bibliography 37 Appendix A Phonetic Substitution tables 39 A.1 Editable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.2 editableGaby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Appendix B Dendrograms and Cluster plots 44 B.1 Sheep counting systems . . . . . . . . . . . . . . . . . . . . . . . . . 44 B.2 Dendrograms of Bhattacharya scores of colour words . . . . . . . . . 44List of Figures 1 Indo-European language tree . . . . . . . . . . . . . . . . . . . . . . 4 2 Indo-European migrations2. . . . . . . . . . . . . . . . . . . . . . . 5 3 Sheep dialects in Britain . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Work-flow of comparison of languages . . . . . . . . . . . . . . . . . 11 5 Work-flow of relationship analysis . . . . . . . . . . . . . . . . . . . . 12 6 Table illustrating Grimm’s Law chain shift . . . . . . . . . . . . . . . 15 7 Statistics of sheep counting systems . . . . . . . . . . . . . . . . . . . 18 8 Density plot of sheep counting systems . . . . . . . . . . . . . . . . . 19 9 Dendrogram of sheep counting systems . . . . . . . . . . . . . . . . . 20 10 Dendrogram of sheep counting systems with the best Silhouette cut . 21 11 Purity of clusters of sheep counting systems . . . . . . . . . . . . . . 22 12 Linguistic and geographical distances of sheep counting systems . . . 23 13 Statistics of “ColoursAll” . . . . . . . . . . . . . . . . . . . . . . . . . 24 14 Statistics of Indo-European languages (colour words) . . . . . . . . . 25 15 Statistics of Germanic languages (colour words) . . . . . . . . . . . . 26 16 Statistics of Romance languages (colour words) . . . . . . . . . . . . 27 17 Density plot of Germanic languages (colour words) . . . . . . . . . . 28 18 Density plot of Germanic languages (colour words) . . . . . . . . . . 29 19 Dendrogram of all languages (colour words) . . . . . . . . . . . . . . 30 20 Chronological dispersal of the Austronesian people . . . . . . . . . . 31 21 Dendrogram of Indo-European languages (colour words) . . . . . . . 32 22 Dendrogram of Germanic languages (colour words) . . . . . . . . . . 33 23 Dendrogram of Romance languages (colour words) . . . . . . . . . . 34 24 Cluster plot of sheep counting all to all comparison (k=10) . . . . . 44 25 Cluster plot of sheep counting all to all comparison (k=10) . . . . . 45 26 Dendrogram of Bhattacharya scores of “ColoursAll” . . . . . . . . . . 46 27 DendrogramofBhattacharyascoresofIndo-Europeanlanguages(colours) 47 28 Dendrogram of Bhattacharya scores of Germanic languages (colours) 48 29 Dendrogram of Bhattacharya scores of Romance languages (colours) 49 30 Dendrogram of Bhattacharya scores of Germanic and Romance lan- guages (colours) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50List of Tables 1 Table of Brittonic languages . . . . . . . . . . . . . . . . . . . . . . . 4 2 Table of phonetic encoding . . . . . . . . . . . . . . . . . . . . . . . 7 3 Table of Language files . . . . . . . . . . . . . . . . . . . . . . . . . . 8Abstract This research project aimed to overcome the challenge of analysing human language relationships, facilitate the grouping of languages and formation of genealogical relationship between them by developing automated comparison techniques. Techniqueswerebasedonthephoneticrepresentationofcertainkey words and concept. Example word sets included numbers 1-10 (curated), large database of numbers 1-10 and sheep counting numbers 1-10 (other sources), colours (curated), basic words (curated). To enable comparison within the sets the measure of Edit distance was calculated based on Levenshtein distance metric. This metric between two strings is the minimum number of single-character edits, operations including: insertions, deletions or substitutions. To explore which words exhibit more or less variation, which words are more preserved and examine how languages could be grouped based on linguistic distances within sets, several data analyt- ics techniques were involved. Those included density evaluation, hierarchical clustering, silhouette, mean, standard deviation and Bhattacharya coefficient calculations. These techniques lead to the development of a workflow which was later implemented by combining Unix shell scripts, a developed R package and SWI Prolog. This proved to be computationally efficient and permitted the fast exploration of large language sets and their analysis. 1 Introduction Theneedtouncoverpresumedunderlyinglinguisticevolutionaryprinciplesandanal- yse correlation between world’s languages has entailed this research. For centuries people have been speculating about the origins of language, however this subject is still obscure. Non-automated linguistic analysis of language relationships has been complicated and very time-consuming. Consequently, this research aims to apply a computational approach to compare human languages. It is based on the phonetic representation of certain key words and concept. This comparison of word similarity aims to facilitate the grouping of languages and the analysis of the formation of genealogical relationship between languages. This report contains a thorough description of the proposed methods, developed techniques and discussion of the results. During this projects several collections of words were gathered and examined, including colour words and numbers. The meth- ods included edit distance, phonetic substitution table, hierarchical clustering with a cut and other analysis methods. They all aimed to provide an insight regarding both technical data summary and its visual representation. 12 Background 2.1 Human languages For centuries, people have speculated over the origins of language and its early devel- opment. It is believed that language first appeared among Homo Sapiens somewhere between 50,000 and 150,000 years ago19. However, the origins of human language are very obscure. To begin with, it is still unknown if the human language originated from one original and universal Proto-Language. Alfredo Trombetti made the first scientific attempt to establish the reality of monogenesis in languages. His investigation con- cluded that it was spoken between 100,000 and 200,000 years ago, or close to the first emergence of Homo Sapiens22. However it was never accepted comprehensively. The concept of Proto-Language is purely hypothetical and not amenable to analysis in historical linguistics. Furthermore, there are multiple theories of how language evolved. These could be separated into two distinctly different groups. Firstly, some researchers claim that language evolved as a result of other evo- lutionary processes, essentially making it a by-product of evolution, selection for other abilities or as a consequence of yet unknown laws of growth and form. This theory is clearly established in Noam Chomsky10and Stephen Jay Gould’s work13. Both scientists hypothesize that language evolved together with the human brain, or with the evolution of cognitive structures. They were used for tool making, informa- tion processing, learning and were also beneficial for complex communication. This conforms with the theory that as our brains became larger, our cognitive functions increased. Secondly, another widely held theory is that language came about as an evolu- tionary adaptation, which is when a population undergoes a change in process over time to survive better. Scientists Steven Pinker and Paul Bloom in “Natural Lan- guage and Natural Selection”20theorize that a series of calls or gestures evolved over time into combinations, resulting in complex communication. Today there are 7,111 distinct languages spoken worldwide according to the 2019 Ethnologue language database. Many circumstances such as the spread of old civ- ilizations, geographical features, and history determine the number of languages spoken in a particular region. Nearly two thirds of languages are from Asia and Africa. The Asian continent has the largest number of spoken languages - 2,303. Africa follows closely with 2,140 languages spoken across continent. However, given the population of certain areas and colonial expansion in recent centuries, 86 percent of people use languages from Europe and Asia. It is estimated that there is around 4.2 billion speakers of Asian languages and around 1.75 billion speakers of European languages. Moreover, Pacific languages have approximately 1,000 speakers each on average, 2but altogether, they represent more than a third of our world’s languages. Papua New Guinea is the most linguistically diverse country in the world. This is possibly due to the effect of its geography imposing isolation on communities. It has over 840 languages spoken, with twelve of them lacking many speakers. It is followed by Indonesia, which has 709 languages spoken across the country. 2.1.1 Indo-European languages and Kurgan Hypothesis Indo-European languages is a language family that represents most of the modern languages of Europe, as well as specific languages of Asia. Indo-European language family consist of several hundreds of related languages and dialects. Consequently, it was an interest of the linguists to explore the origins of the Indo-European language family. In the mid-1950s, Marija Gimbutas, a Lithuanian-American archaeologist and anthropologist, combined her substantial background in linguistic paleontology with archaeologicalevidencetoformulatetheKurganhypothesis12. Thishypothesisisthe mostwidelyacceptedproposaltoidentifythehomelandofProto-Indo-European(PIE) (ancient common ancestor of the Indo-European languages) speakers and to explain the rapid and extensive spread of Indo-European languages throughout Europe and Asia17 18. The Kurgan hypothesis proposes that the most likely speakers of the Proto-Indo-European language were people of a Kurgan culture in the Pontic steppe, by the north side of the Black Sea. It also divides the Kurgan culture into four suc- cessive stages (I, II, III, IV) and identifies three waves of expansions (I, II, III). In addition, the model suggest that the Indo-European migration was happening from 4000 to 1000 BC. See figure 2 for visual illustration of Indo-European migration. Today there are approximately 445 living Indo-European languages, which are spoken by 3.2 billion people, according to Ethnologue. They are divided into the following groups: Albanian, Armenian, Baltic, Slavic, Celtic, Germanic, Hellenic, Indo-Iranian and Italic (Romance) 116. 2.1.2 Brittonic languages Brittonic or British Celtic languages derive from the Common Brittonic language, spoken throughout Great Britain south of the Firth of Forth during the Iron Age and Roman period. They are classified as Indo-European Celtic languages9. The family tree of Brittonic languages is showed in Table 1. Common Brittonic is ancestral to Western and Southwestern Brittonic. Consequently, Cumbric and Welsh, which is spoken in Wales, derived from Western Brittonic. Cornish and Breton, spoken in Cornwall and Brittany, respectively, originated from Southwestern side. Today Welsh, Cornish and Breton are still in use. However, it is worth to point out that Cornish is a language revived by second-language learners due to the last native speakers dying in the late 18th century. Some people claimed that the Cornish language is an important part of their identity, culture and heritage, and a revival 3Proto-Indo-Eu ropean Irish ScottishBreton Manx WelshCeltic Latin Iberian PortugueseSpanish GalicianGalician- Portu gueseItalian Romania n Old French Middle Fren ch FrenchLangue d’OïlIberianGallo- Gallic Catala nOccitanOccitanOscanItalic Eastern RomanceGreekClassical GreekHellenic Germanic West GermanicIcelan dicNorth Germanic Old East Norse Swedish Danish Norw egianNorseOld West Modern Engli shMiddle E nglishFrisianAnglo- OldOld E nglish FrisianOld Dutch Middle Du tch Dutch Afrikaan sOld High German GermanMiddle High Yiddish Germa nBalto- Slavic Baltic SlavicEastSerbia nSlavic SlavicWestLatvian Polish Russia nUkrainianLithuan ian CzechCroatian Prepared by Jack Lynch. Edited 2 2 Feb . 2014. Jack.Ly nch@rutg ers.ed uHindust aniMarathi Gujarat i Hindi UrduPunjabiIndic Irania n Old PersianAvestan FarsiKurdishMiddle PersianSanskritIndo-Iranian Armenia n ArmenianOld Armen ianAlbanian Albanian East Germanic Gothic FrisianSouth Slavic BulgarianFigure 1: Indo-European language tree16 began in the early 20th century. Cornish is currently a recognised minority language under the European Charter for Regional or Minority Languages. Table 1: Table of Brittonic languages Common Brittonic Western Brittonic Sothwestern estern Brittonic Cumbric Welsh Cornish Breton 2.1.3 Sheep Counting System Brittonic Celtic language is an ancestor to the number names used for sheep count- ing11 3. Until the Industrial Revolution, the use of traditional number systems was common among shepherds, especially in the fells of the Lake District. The sheep- counting system was referred to as Yan Tan Tethera. It was spread across Northern England and in other parts of Britain in earlier times. The number names varied 4Figure 2: Indo-European migrations2 according to dialect, geography, and other factors. They also preserved interesting indications of how languages evolved over time. The word “yan” or “yen” meaning “one”, in some northern English dialects repre- sents a regular development in Northern English15. During the development the Old English long vowel / A:/ <¯ a> was broken into /ie/, /ia/ and so on. This explains the shift to “yan” and “ane” from the Old English ¯ an, which is itself derived from the Proto-Germanic “*ainaz”14. In addition, the counting system demonstrates a clear connection with counting on the fingers. Particularly after numbers reach 10, as the best known examples are formed according to this structure: 1 and 10, 2 and 10, up to 15, and then 1 and 15, 2 and 15, up to 20. The count variability would end at 20. It might be due to the fact, that the shepherds, on reaching 20, would transfer a pebble or marble from one pocket to another, so as to keep a tally of the number of scores. 3 Aims and Objectives 3.1 Overall Aim The aim of this research was to develop computational methods to compare human languages based on the phonetic form of single words (i.e. not exploiting grammar). This comparison of word similarity aims to facilitate the grouping of languages, the identification of the the presumed underlying linguistic evolutionary principles and the analysis of the formation of genealogical relationship between languages. 53.2 Specific Objectives 1. Devise a way to encode the phonetic representation of words, using: (a) an in-house encoding, (b) an IPA (International Phonetic Alphabet). 2. Develop methods to analyze the comparative relationships between languages using: descriptiveandinferentialstatistics, clustering, visualisationofthedata, and analysis of the results. 3. Implement a repeatable process for running the analysis methods with new data. 4. Analyse the correlation between geographical distance and language similarity (linguistic distance), and investigate if it explains the evolutionary distance. 5. Examine which words exhibit more or less variation and the likely causes of it. 6. Explore which words are preserved better across the same language group and possible reasons behind it. 7. Explore which language group preserves particular words more in comparison to others and potential reasons behind it. 8. Determine if certain language groups are correct and exploit the possibility of forming new ones. 4 Data 4.1 Language files Language file or database is a set of languages, each of which is associated with an ordered list of words. All lists of words for a particular data set have the same length. For example: numbers(romani,[iek,dui,trin,shtar,panj,shov,efta,oksto,ena,desh]). numbers(english,[wun,too,three,foor,five,siks,seven,eit,nine,ten]). numbers(french,[un,de,troi,katre,sink,sis,set,wuit,neuf,dis]). Words and languages are encoded in this format for later use of Prolog. In Prolog each “numbers” line is a fact, which has 2 arguments; the first is the language name and the second is a list (indicated in between square brackets) of words. Words can be written down in their original form or encoded phonetically (as shown in the example). Where synonyms for a word are known, then the word itself is represented by a list of the synonym words. In the example below, Lithuanian, Russian and Italian have two words for the English ‘blue’: 6words(english,[black,white,red,yellow,blue,green]). words(lithuanian,[juoda,balta,raudona,geltona,[melyna,zhydra],zhalia]). words(russian,[chornyj,belyj,krasnyj,zholtyj,[sinij,goluboj],zeljonyj]). words(italian,[nero,bianco,rosso,giallo,[blu,azzurro],verde]). The main focus of this research was exploring words phonetically. Consequently, special encoding was used. It consisted of encoding phonemes by using only one letter; incorporating capital letters for encoding different sounds (See table 2). Table 2: Table of phonetic encoding Symbol Meaning c ts x ks C ch as in charity k as in cat T th S sh G dzh K kh Z zh D dz HSpanish/Portuguese sound of “j” A, I, O, U long vowels Table 3 summarises the language files that are obtained at the moment. 4.2 Sheep 4.2.1 Sheep counting words Sheep counting numbers were extracted from “Yan Tan Tethera”3page on Wikipedia and placed in a Prolog database. Furthermore, data was encoded phonetically using the set of rules provided by Prof. David Gilbert. In the given source, number sets ranged from 1-3 to 1-20 for different dialects. The initial step was to reduce the size of the data to sets of numbers 1-10. This way aiming: (a) to have Prolog syntax without errors (avoided “-”, “ ” as they were common symbols after numbers reached 10); (b) to avoid the effects of different methods of forming and writing down num- bers higher than 10. (Usually they were formed from numbers 1-10 and a 7Table 3: Table of Language files NameNumber of languagesNumber of words per languageDescription Numbers Small Collec- tion92 10 Numbers 1 to 10 Numbers Big Collection 3880 10 Numbers 1 to 10 Sheep Counting Num- bers54 10 Numbers 1 to 10 Basic Words 42 13Concept words: sun, moon, rain, water, fire, man, woman, mother, father, child, yes, no, blood Colours 23 6Main colours: black, white, red, yellow, blue, green Basic Words IPA 3 12Concept words as in “Basic Words” Numbers IPA 3 10 Numbers 1 to 10 Colours IPA 3 6 Colours as in “Colours” base. However, they were written in a different order, making the comparison inefficient.) In addition, the Wharfedale dialect was removed since only numbers 1-3 were provided; the Weardale dialect was eliminated as it had a counting system with base 5. Consequently, the final version of sheep counting numbers database consisted of 23 observations (dialects) with numbers 1-10. 4.2.2 Geographical data In order to enable the analysis of linguistic and geographical distance relationship, a geographical distance database was created. It was done by firstly creating a personalized Google Map with 23 pins, noting the places of different dialects (they were located approximately in the middle of the area) (Figure: 3). Subsequently, pairwise distances were calculated between all of them (taking walking distance) and added to the database for further use. 8Figure3: SheepdialectsinBritain. Amapwith23pins, notingtheplacesofdifferent dialects 4.3 Colours Colour words were extracted from “Colour words in many languages”5page on Om- niglot, collected from people and dictionaries. In addition, data was encoded pho- netically using the set of rules provided by Prof. David Gilbert. The latest version of the database consisted of 42 different languages, each con- taining 6 colours: black, white, red, yellow, blue, green. For the purposes of analysis the following groups were created: (a) All languages - “ColoursAll” (42 languages) (b) Indo-European languages - “ColoursIE” (39 languages) (c) Germanic languages - “ColoursPGermanic” (10 languages) (d) Romance languages - “ColoursPRomance” (11 languages) (e) Germanic and Romance languages - “ColoursPG_R” (21 languages) 94.4 IPA “Automatic Phonemic Transcriber”1was used to create 3 IPA encoded databases: (a) “BasicWords” - words in their original form were taken from Prof. David Gilbert’s database for basic words (including: sun, moon, rain, water, fire, man, woman, mother, father, child, yes, no, blood). (b) “Numbers” -numbersfrom1-10intheiroriginalformweretakenfromProf.David Gilbert’s small database of numbers. (c) “Colours” - words were taken from the above mentioned database (including words: black, white, red, yellow, blue, green). Eachoftheabovementioneddatabasesconsistedof3languages: English, Danishand German (these were the languages the Automatic Phonemic Transcriber provided) all encoded in IPA. As the research progressed, the difficulty of obtaining IPA encoding for different languages was faced. This study could not find a cross-linguistic IPA dictionary that included more than 3 languages. Consequently, the question of its existence was raised. 5 Methodology There are two main processes to be carried out. The first process (Figure: 4) aims to analyse a databases of words; explore which words exhibit more or less variation, which words are more preserved; examine how languages could be grouped based on linguistic distances of words. Itbeginswiththecalculationofpairwiselinguisticdistancesforthegivendatabase of words. A Phonetic Substitution Table is used to assign weights during the calcu- lation and could possibly be modified. The result is a new distance table which is analysed in the following ways: Performing “densityP” function. The outcome is density plots for every word of a database. Performing Hierarchical clustering. After, the “Best cut” is determined, which is either the best Silhouette value after calculation of all possible cases, or a forced number K which is a number of words per language in the language file Calculating Bhattacharya coefficients. Performing “mean_SD” function. 10The second process (Figure: 5) targets to investigate the relationship between two sets of distance data. In this research, it was applied to analyse the relationship between linguistic and geographical distances. It starts with producing two pairwise distance tables: one of them is calculated geographical distances, another one is calculated linguistic distances. Then the data from both tables is combined into a data frame for regression analysis in R. The outcome is an object of the class “lm” (result of R function “lm” being used), that is used for data analysis, and a scatter plot with a regression line for visual analysis. Both processes have been automated, see Section 6.5. Database of words Calculation of Linguistic distances Distance table “densityP” Hierarchical Clustering Density Plots Hierarchical clusters K = N Calculation of Silhouette values “Best cut” Dendrogram Silhouette Plot Cluster Plot Dendrogram + cut “tscore” “mean_SD” Normalized table Data frame of mean, SD and mean*SD of every subset Plots “bhatt” Data frame of Bhattacharya Coefficients of all pairs subset subset subset subset; all words; all to all N is the number of words Character weight substitution table Figure 4: Work-flow of comparison of languages 11 Database of words Calculation of Linguistic distances Distance table List of places Calculation of Geographical distances Distance table Combining data Data frame Regression analysis Object of class “lm” Scatter plot + regression line Figure 5: Work-flow of relationship analysis 6 Methods 6.1 Edit Distance For the purposes of this research Edit distance (a measure in computer science and computational linguistics for determining the similarity between 2 strings) was cal- culated based on Levenshtein distance metric. This metric between two strings is the minimum number of single-character edits, operations including: insertions, deletions or substitutions. The Levenshtein distance between two strings a,b (of length jajandjbj 12respectively) is given by leva;b(jaj;jbj)where leva;b(i; j) =8 >>>>< >>>>:max (i; j) If min (i; j)=0 min8 >< >:leva;b= (i1; j) + 1 leva;b= (i; j1) + 1 ; otherwise leva;b= (i1; j1) + 1 (ai 6=bj) where 1(ai 6=bj)is the indicator function equal to 0 when ai=bjand equal to 1 otherwise. A normalised edit distance between two strings can be computed by lev_norm a;b=leva;b max (jaj;jbj) Edit distance was implemented by Prof. David Gilbert using dynamic program- ming in SWI Prolog23. The program was used to compare two words with the same meaning from different languages. When pairwise comparing two words where either one or both comprise synonyms, all the alternatives for each word one one language are compared with the corresponding (set) of words in the other language, and the closest match is selected. In addition, all to all comparisons were made, i.e. edit distance was calculated for words having different meaning as well. Finally, the edit distance for two languages represented by two lists of equal length of corresponding words was computed by taking the average of the edit distance for each (correspond- ing) pair of words. An example of pairwise alignments is for the pair of words overa-hofa, where 3 alignments are produced with the use of gap penalty = 1and substitution penalties f$v= 0:2,e$o= 0:2and all other mismatches 1: [[-,h],[o,o],[v,f],[e,-],[r,-],[a,a]] [[o,-],[v,h],[e,o],[r,f],[a,a]] [[o,h],[v,-],[e,o],[r,f],[a,a]] each with the raw edit distance of 3.2, and the normalised edit distance of 3:2 max (joveraj;jhofaj)=3:2 5= 0:64 For the sake of clarity we can write the first alignment for example as - o v e r a h o f - - a where only 3 letters are directly aligned. 136.2 Phonetic Substitution Table In order to give a specified weight for different operations (insertion, deletion and substitution) Phonetic Substitution Table was created by incorporating Grimm’s law21and extending it in-house. Grimm’s Law, principle of relationships in Indo-European languages, describes a process of the regular shifting of consonants in groups. It consist of 3 phases in terms of chain shift7. 1. Proto-Indo-European voiceless stops change into voiceless fricatives. 2. Proto-Indo-European voiced stops become voiceless stops. 3. Proto-Indo-European voiced aspirated stops become voiced stops or fricatives. This is an abstract representation of the chain shift: bh > b > p > F dh > d > t > T gh > g > k > x gwh > gw > kw > xw Figure 6 illustrates how further consonant shifting following Grimm’s law affected words from different languages4. Phonetic substitution table was extended in-house by adding more shifts. In addition, it was also written in the way to work with the special encoding described in 4.1 section. Find the full table “editable” in Appendix A. Another phonetic substitution table, called “editableGaby”, was made (See Ap- pendix A). It was extended by adding pairs like “dzh” and “zh”; “dzh” and “ch”; “kh” and “g”; as well as “H”(sound of e.g. spannish/portuguese “j”) with “kh”, “g”, “k”, “h”. In addition, some of the weights were changed for certain pairs for experimental purposes. 6.3 Hierarchical Clustering 6.3.1 Using the OC program The OC program6is general purpose hierarchical cluster analysis program. It out- puts a list of the clusters and optionally draws a dendrogram in PostScript. It requires complete upper diagonal distance or similarity matrix as an input. 14 Proto-Indo-European Meaning Non-Germanic (unshifted) cognates Change Proto-Germanic Germanic (shifted) examples *pṓds "foot" Ancient Greek: πούς, ποδός (poús, podós), Latin: pēs, pedis, Sanskrit: pāda, Russian: под (pod) "under; floor", Lithuanian: pėda, Latvian: pēda, Persian: ﺎﭘ (pa) *p > f [ɸ] *fōt- English: foot, West Frisian: foet, German: Fuß, Gothic: fōtus, Icelandic, Faroese: fótur, Danish: fod, Norwegian, Swedish: fot *trit(y)ós "third" Ancient Greek: τρίτος (tritos), Latin: tertius, Welsh: trydydd, Sanskrit: treta, Russian: третий (tretij), Serbo-croatian: трећи (tretji), Lithuanian: trečias, Albanian: tretë *t > þ [θ] *þridjô English: third, Old Frisian: thredda, Old Saxon: thriddio, Gothic: þridja, Icelandic: þriðji, Danish, Swedish: tredje *ḱwón- ~ *ḱun- "dog" Ancient Greek: κύων (kýōn), Latin: canis, Welsh: ci (pl. cwn), Persian: ﮓﺳ (sag) *k > h [x] *hundaz English: hound, Dutch: hond, German: Hund, Gothic: hunds, Icelandic, Faroese: hundur, Danish, Norwegian, Swedish: hund *kʷód "what" Latin: quod, Irish: cad, Sanskrit: kád, Russian: ко- (ko-), Lithuanian: kas, Serbo-croatian(kajkavian dialect): кај (``kaj``) *kʷ > hw [xʷ] *hwat English: what, Gothic: ƕa("hwa"), Icelandic: hvað, Faroese: hvat, Danish: hvad, Norwegian: hva Figure 6: Table illustrating Grimm’s Law chain shift 156.3.2 Using R Hierarchical clustering in R was performed by incorporating clustering together with Silhouette value calculation and cut performance. Inordertofulfillagglomerativehierarchicalclusteringmoreefficiently, wecreated a set of functions in R: 1. “sMatrix” - Makes a symmetric matrix from a specified column. The function takes a specifically formatted data frame as an input and returns a new data frame. Having a symmetric matrix is necessary for “silhouetteV” and “hcutVi- sual” functions. 2. “silhouetteV” -CalculatesSilhouettevalueswith“k” valuevaryingfrom2ton-1 (n being the number of different languages/number of rows/number of columns in a data frame). The function takes a symmetric distance matrix as an input and returns a new data frame containing all Silhouette values. 3. “hcutVisual” - Performs hierarchical clustering and makes a cut with the given K value. Makes Silhouette plot, Cluster plot and dendrogram. Returns a “hcut” object from which cluster assignment, silhouette information, etc. can be extracted. It is important to note that K-Means clustering was not performed as the algo- rithm is meant to operate over a data matrix, not a distance matrix. 6.4 Further analysis with R Another set of functions was created to analyse collected data further. They target to ease the comparison of the mean, standard deviation, Bhattacharya coefficient within the words or language groups. Including: 1. “mean_SD” - Calculates mean, standard deviation, product of the mean and the SD multiplication for every column of the input. Visualises all three values for each column and places it in one plot, which is returned. 2. “densityP” - Makes a density plot for every column of the input and puts it in one plot, which is returned. 3. “tscore” - Calculates t-score for every value in the given data frame. (T-score is a standard score Z shifted and scaled to have a mean of 50 and a standard deviation of 10) 4. “bhatt” - Calculates Bhattacharya coefficient (the probability of the two dis- tributions being the same) for every pair of columns in the data frame. The function returns a new data frame. 166.5 Process automation In order to optimise and perform analysis in the most time-efficient manner processes of comparing languages were automated. It was done by creating two shell scripts and an R script for each of them. The first shell script named “oc2r_hist.sh” was made to perform hierarchical clustering with the best silhouette value cut. This script takes a language database as an input and performs pairwise distance calculation. It then calls “hClustering.R” R script, which reads in the produced OC file, performs hierarchical clustering and calculates all possible silhouette values. Finally, it makes a cut with the number of clusters, which provides the highest silhouette value. To enable this process the R script was written by incorporating the functions described in section 6.3.2. The outcome of this program is a table of clusters, a dendrogram, clusters’ and silhouette plots. The second shell script called “wordset_make_analyse.sh” was made to perform calculations of mean, standard deviation, Bhattacharya scores and produce density plots. This script takes a language database as an input and performs pairwise distance calculations for each word of the database. It then calls “rAnalysis.R” R script, whichreadsintheproducedOCfileandperformsfurthercalculations. Firstly, it calculates mean, standard deviation and the product of both of each word and outputs a histogram and a table of scores. Secondly, it produces density plots of each word. Finally,itconvertsscoresintoT-ScoresandcalculatesBhattacharyacoefficient for every possible pair of words. It then outputs a table of scores. To enable this process the R script was written by incorporating the functions described in section 6.4. Finally, both of the scripts were combined to minimise user participation. 7 Results 7.1 Sheep The sheep counting database was evaluated in the following ways: Obtaining average pairwise linguistic distance, pairwise linguistic distance of subsets (different words), Performing all to all comparison (where linguistic distance is calculated be- tween words with different meaning, as well as with the same), Collectinggeographicaldataandcomparingrelationshipbetweenlinguisticand geographical distances. Upon generation of the above mentioned data, the methods defined in 6 section were used. 177.1.1 Analysis of average and subset linguistic distance After applying functions “mean_SD” (Figure: 7) and “densityP” (Figure: 8) to the linguistic distances of every word (numbers 1 to 10) in R, the following observations were made. First of all, the most preserved number across all dialects was “10” with distance mean 0.109 and standard deviation 0.129. Numbers “1”, “2”, “3”, “4” had comparatively small distances, which might be the result of being used more frequently. On the other hand, number “6” showed more dissimilarities between dialects than other numbers. The mean score was 0.567 and standard deviation - 0.234. The product scores of mean and standard deviation helped to evaluate both at the same time. Moreover, density plots showed significant fluctuation and tented to have a few peaks. But in general, conformed with the statistics provided by “mean_SD”. Figure 7: Mean, SD and mean*SD of every number of sheep counting systems 7.2 Hierarchical clustering Hierarchical clustering was performed with the best Silhouette value cut (Figure 10). The Silhouette value suggested making 9 clusters. In this grouping, the most interesting observation was that Welsh, Breton and Cornish languages were placed together. It conforms with the fact that all 3 languages descended directly from the Common Brittonic language spoken throughout Britain before the English language became dominant. 18Figure 8: Density plots of each number of sheep counting systems 7.2.1 All to all comparison analysis To enable analysis of clusters of all to all comparison, hierarchical clustering was performed. This was done by two different approaches: calculating a silhouette value and choosing the number of clusters accordingly; forcing a function to make 10 clusters due to having numbers from 1 to 10 in the sheep counting database. By using function “silhouetteV” silhouette values were calculated for all possible kvalues. The returned data frame indicated the best number of clusters being 70 (see Appendix B.1 for dendrogram and cluster plot). The suggested clusters were not distinguished with very high clarity in terms of numbers 1-10 perfectly, but they were comparatively good. A pattern that numbers, which had lower mean and standard deviation scores, would result in purer clusters was noticed. Clusters of numbers “1”, “2”, “3”, “4”, “5” and “10” were not as mixed as “6”, “7”, “8”, “9”. Anotherwayoflookingatalltoallcomparisondatawasbyproducing10clusters. It was done by using “hcutVisual” and “cPurity” function (see Appendix B.1 cluster 19welshbretoncornishwestCountryDorsettongswaledaleteesdalelakesrathmellbowlandwensleydalederbyshirenidderdaleeskdalelincolnshirederbyshireDaleswestmorlandborrowdalescotsconistonwiltskirkbyLonsdalesouthWestEngland 0 0.502745Figure 9: Dendrogram of linguistic distances of 23 dialects plot). The results showed high impurities of clusters (Figure: 11). Two out of ten clusters were pure, both containing number “5”. Another relatively pure cluster was composed of number “10” and two entries of number “2”. The rest consisted of up to 7 different numbers. This shows that sheep counting numbers in different dialects are too different to form 10 clusters containing each number. However, considering the possibility that dialects were grouped and clustering was performed to the smaller groups, they would have reasonably pure clusters. Exploring this grouping options could be a subject for further work. 20welsh breton cornish rathmell bowland wensleydale derbyshire nidderdale coniston wilts kirkbyLonsdale southWestEngland lakes eskdale lincolnshire derbyshireDales westmorland borrowdale scots westCountryDorset tong swaledale teesdale0100200300HeightCluster DendrogramFigure 10: Dendrogram of sheep counting systems with the best Silhouette cut 21onextwoxthreex fourxfivexsixx sevenxeightxninex tenx num1 23 45 67 89 10Cluster PurityFigure 11: Purity of hierarchical clusters of sheep counting systems all to all com- parison. Clusters numbered according to K= 10. Colour key indicates number words. 7.2.2 Linguistic and Geographical distance relationship In order to investigate the correlation between linguistic and geographical distance, “lm” functionwasperformedandascatterplotwascreated. Theregressionlineinthe scatter plot suggested that the relationship existed. However, the R-squared value, extracted from the “lm” object, was equal to 0.131. This indicated that relationship existed, but was not significant. One assumption made was that Cornish, Breton and Welsh dialects might have had a weakening effect on the relationship, since they had large linguistic distances compared to other dialects. However this assumption could not be validated as the correlation was less significant after eliminating them. This highlights that although these dialects had large linguistic distance scores, they also had big geographical distances that do not contradict the relationship. In addition, comparison was done between linguistic distance and Log 10(GeographicalDistance ). This resulted in an even weaker relationship with R-squared being 0.097. 220300600900 0.0 0.2 0.4 0.6 lingDistgeoDistFigure12: Relationshipbetweenlinguisticandgeographicaldistancesofsheepcount- ing systems 7.3 Colours The Colours database was evaluated three different ways: getting average pairwise linguistic distance, subset pairwise linguistic distance for every word and performing all to all comparison to all groups (All languages, Indo-European, Germanic, Ro- mance, Germanic and Romance languages). After the above mentioned data was generated, the previously defined methods were applied. 7.3.1 Mean and Standard Deviation When examining the data calculated for “ColoursAll” none of the colours showed a clear tendency to be more preserved than others (Figure: 13). All colours had large distances and comparatively small standard deviation when compared with other groups. Small standard deviation was most likely the result of most of the distances being large. Indo-European language group scores were similar to “ColoursAll”, exhibiting slightly larger standard deviation (Figure: 14). Conclusion could be drawn that words for color “Red” are more similar in this group. The mean score of linguistic distances was 0.61, and SD was equal to 0.178, when average mean was 0.642 and SD 0.212. However, no colour stood out distinctly. Germanic and Romance language groups revealed more significant results. Ger- 23Figure 13: Mean, SD and mean*SD of every colour of all languages manic languages preserved the colour “Green” considerably (Figure: 15). The mean and SD was 0.168 and 0.129, when on average mean was reaching 0.333 and SD 0.171. In addition, the colour “Blue” had favorable scores as well - mean was 0.209 and SD was 0.106. Furthermore, Romance languages demonstrated slightly higher means and standard deviations, on average reaching 0.45 and 0.256 (Figure: 16). Similarly to Germanic, the most preserved colour word in Romance languages was “Green” with a mean of 0.296 and SD of 0.214. It was followed by words for “Black” and then for “Blue”, both being quite similar. 7.3.2 Density Plots Density plots of all languages and Indo-European languages were similar: both hav- ing multiple peaks with the most density around scores of 0.75 (big linguistic dis- tance). Moreover, Germanic languages density distribution consisted of two peaks for words “White”, “Blue” and “Green” (Figure: 17). This could possibly be the result of certain weighting in the Phonetic Substitution Table or indicate possible further grouping of languages. The color “Black” had more normal distribution and smoother bell shape compared to others. Furthermore, Romance languages also ob- tained density plots with two peaks for words “White”, “Yellow”, “Blue” (Figure: 18). In contrast, “Black”, “Red” and “Green” distributions were quite smooth. In order to experiment how the Phonetic Substitution Table affects the linguistic distances, “densityP” function was applied to the linguistic distances calculated with “GabyTable” substitution table. The aim was to eliminate the two peaks in the 24Figure 14: Mean, SD and mean*SD of every colour of Indo-European languages Germanic language group for word “Green”. In Germanic languages word for green tended to begin with either “gr” or “khr” (encoded as “Kr”) - both sounding similar phonetically. However, in the original substitution table, a weight for changing “K”(kh) to “g” (and the other way around) did not exist. Consequently, a new table was implemented with this substitution. This change resulted in notably smaller linguistic distances - the mean for the word “Green” was 0.099. However, it did not solve the occurrence of two peaks. The density of “Green” again had two main peaks, but differently distributed compared to the previous case. 7.3.3 Bhattacharya Coefficients Bhattacharya coefficients were calculated within each group for different pairs of colours. This helped to evaluate which colours were closer in distribution. In addi- tion, hierarchical clustering was done with Bhattacharya coefficients (find the den- drograms in the Appendix B.2). However, the potential meaning behind the results was not fully examined. AnotherpotentialuseofBhattacharyacoefficientsistheirapplicationtothesame word from a different language group. As a result, the preservation of particular words can be analysed across language groups, enabling to compare and evaluate potential reasons behind it. 25Figure 15: Mean, SD and mean*SD of every colour of Indo-European languages 7.3.4 Hierarchical Clustering Hierarchical clustering with the best Silhouette value cut was performed in R for every group of formed language groups: all languages, Indo-European, Romance, Germanic, and both Germanic and Romance together. It is important to note that the results of the language group “Romance and Germanic” will not be discussed as it was used more for testing purposes and as expected resulted in a K=2 cut. After making the cut, one cluster consisted of Romance languages and another consisted of Germanic languages. To begin with, clustering of all languages showed some interesting results that complied with the grouping of the languages (find the dendrogram in Figure: 19). The suggested cut by Silhouette value was 23. Some of the clusters were more a coincidence than the actual similarity of languages and did not correspond with the existing language grouping. Despite that, most of the clusters resulted in the actual language groups, or languages closely related. To begin with, Baltic Romani, Punjabi’ and Urdu were placed in the same cluster. Even though Baltic Romani is far away from South Asia geographically, it is believed to have originated from this area. Xhosa and Zulu formed another cluster both being the languages of the Nguni branch and spoken in South Africa. Hawaiian, Malagasy and Maori languages were grouped together and they all belong to Austronesian ethnolinguistic group8(see figure 20). Sinhala (language of Sinhalese people, who make up the largest ethnic group in Sri Lanka), Dhivehi (spoken in Maldives) and Maldivian languages fell in the same group after the cut. They all are spread across islands in the Indian Ocean. 26Figure 16: Mean, SD and mean*SD of every colour of Indo-European languages Estonian and Finnish both being representatives of the Uralic language family were the same cluster. Moreover, clusters of Indo-European languages were quite pure as well (groups are visible in the dendrogram of all languages, however for clarity see figure 21). There were four larger groups that stood out. First of all, the group of Germanic languages was produced accurately. It consisted of Faroese, Icelandic, German, Luxemburgish, Yiddish, English, Norwegian, Swedish, Afrikaans and Dutch. All of these languages are considered to be in the branch of Germanic languages. Another clusterwasSlaviclanguages, whichconsistedofCroatian, Polish, Russian, Slovenian, Czech, Slovak and Lithuanian. Lithuanian and Latvian, according some sources, are considered to be in a separate branch, known as Baltic languages. On the other hand, in other sources they are regarded as Slavic languages. In this case, in terms of colour words Lithuanian was appointed to the Slavic languages, whereas Latvian formed a cluster on its own. In relation to Romance languages, these were divided into two groups. The first one was made of Ladino (language that derived from medieval Spanish), Spanish(Castilian), Galician and Portuguese, forming a group of the Western Romance languages. The second one consisted of Sicilian, Italian, Neapolitan, Catalan and Romanian and could be called a group of Mediterranean Romance languages. Furthermore, clustering results of the Germanic languages file (Figure: 22) show high relation with geographical prevalence of the languages and language develop- ment history. German, Luxembourgish (has similarities with other varieties of High German languages) and Yiddish (a High German-based language) were all in the 27Figure 17: Density plots of each colour of Germanic languages same cluster. Also, Afrikaans and Dutch were placed in the same group, and it is known that Afrikaans derived from Dutch vernacular of South Holland in the course of 18th century. Other clusters included Faroese and Icelandic, Swedish and Nor- wegian, as well as English forming a cluser on its own. Finally, when looking at the clusters of Romance languages file (Figure 23) it is evident that one cluster, consisting of Ladino, Spanish, Galician and Portuguese, remained the same as in “ColoursAll”, “ColoursIE”. Another cluster that was formed from Romance languages in these databases was broken down into 3 clusters during separate clustering of Ro- mance languages. Romanian and Catalan formed clusters on their own and Italian, Neopolitan and Sicilian were members of another cluster. These three languages were close geographically. 7.4 IPA Hierarchical Clustering was performed to all three IPA databases and compared with the results of hierarchical clustering of in-house phonetically encoded databases (they were created by taking subsets of German, English and Danish languages from “Basic Words”, “Numbers Small Collection” and “Colours” databases). The first characteristic noticed was that both IPA and non-IPA databases had the same groupingoflanguages. Thisshowsevidencetowardssubstantiatedphoneticencoding done in-house. Another noted tendency was that pairwise linguistic distance scores tented to be higher for IPA databases. This might be due to some graphemes being written with a few letters in IPA databases, while phonetic encoding done in-house expressed graphemes as one symbol. 28Figure 18: Density plots of each colour of Romance languages Potential further work would be generating an IPA-designated Phonetic Sub- stitution table (so far clustering has been done with “editable”) and running the routines with the new weight table. Also, complementing the IPA databases with more languages would be an important step towards receiving more accurate results. 7.5 Small Numbers 7.5.1 All to all comparison Analysis was carried out in two ways. First of all, hierarchical clustering was per- formed with the best silhouette value cut. For this data set best silhouette value was 0.48 and it suggested making 329 clusters. Clusters did not exhibit high pu- rity. However, the ones that did quite clearly corresponded to unique subgroups of language families. Anotherwayoflookingatalltoallcomparisondatawasbyproducing10clusters. The anticipated outcome was members being distinguished by numbers, forming 10 clean clusters. However, all the clusters were very impure and consisting multiple different numbers. This might be due to different languages having phonetically similar words for different words, in this case. All to all pairwise comparison could be an advantageous tool when used for lan- guage family branches or smaller, but related subsets. It could validate if languages belong to a certain group. 29faroese icelandic german luxembourgish yiddish english norwegian swedish afrikaans dutch ladino spanish galician portuguese french sicilian italian neapolitan catalan romanian croatian polish russian lithuanian slovenian czech slovak romani_baltic punjabi urdu sinhala dhivehi maldivian arabic maltese kurdish welsh hungarian latvian hebrew armenian persian xhosa zulu japanese quechua guarani mandarin albanian basque latin filipino malay estonian finnish hawaiian malagasy maori greek swahili lingala somali0.00.51.01.52.0HeightCluster DendrogramFigure 19: Dendrogram of hierarchical clustering with Silhouette value cut for all languages (colour words) 8 Conclusions Thisprojecthasaimedtodevelopcomputationalmethodstoanalyseandunderstand connections between human languages. The project included collecting words from different languages in order to form new databases, forming rules for phonetic encoding of words and adjusting phonetic substitution table. Several computational methods of calculating pairwise distance between two words were taken, includingaverage, subsetand all toall words distance calculation. It was done by incorporating edit distance and phonetic substitution table, and implementing it in SWI Prolog. This was followed by detailed analysis of distance scores, which was conducted by the specific automated routines and developed R functions. They enabled performing hierarchical clustering with a cut either according to silhouette value, or to specified K value. They provided summary of mean, standard deviation and other statistics, like Bhattacharya scores. All these 30Figure 20: Chronological dispersal of the Austronesian people techniques delivered a thorough analysis of data and the automation of processes ensured they were used efficiently. The resulting outcome of analysis of old sheep counting systems in different English dialects was the observation that numbers “1”,“2”,“3”,“4” and “10” were more uniform within different dialects than others, posing that they might have been the most frequently used ones. Analysis of all to all comparison did not provide pure clusters and shows that sheep counting numbers in different dialects are too different to form 10 clusters containing each number. This suggests that dialects should be grouped into subsets. Furthermore, hierarchical clustering with the best silhouette cut suggested the potential 9 groups, which consist members with the most similar counting words. Surprisingly, it was not entirely based on location. This corresponded with the difficulty of finding relationship between geographic and linguistic distance, the conducted tests showed it was insignificant. Analysis of colour words revealed that within Indo-European languages words for colour red were moderately more preserved. Both Germanic and Romance language groups tended to have considerably more uniform words for green and blue colours. In addition, Romance language group preserved colour black reasonably well. Analy- sis of linguistic distances distribution showed multiple peaks within words for various language groups, suggesting that further language grouping could be done. Further- more, the resulting outcome of hierarchical clustering with silhouette cut was known and officially accepted language families. Most of the clusters were subgroups of existing language families. Some of them suggested different sub-grouping according to colour words (e.g. Lithuanian was appointed to Slavic languages, while Latvian formed cluster on its own). IPA databases resulted in the same relationships between languages as non-IPA phonetically encoded databases. However, to fully explore the potential of IPA- encoded databases they ought to be expanded and a customized weights table should be created. In conclusion, this project resulted in creation of several felicitous computational 31faroese icelandic german luxembourgish yiddish english norwegian swedish afrikaans dutch ladino spanish galician portuguese latin sicilian italian neapolitan catalan romanian latvian croatian polish russian lithuanian slovenian czech slovak punjabi urdu dhivehi sinhala kurdish armenian persian welsh french albanian greek0.00.51.01.52.0HeightCluster DendrogramFigure 21: Dendrogram of hierarchical clustering with Silhouette value cut for Indo- European languages (colour words) techniques to explore many languages and their correlation all at once. 9 Further Work One of the areas where further work could be performed is thorough analysis of numbers both Small and Big Collection databases, Basic words database. In addition, analysis routines could be enhanced by adding Bhattacharya scores, calculated in a different manner. In other words, potentially beneficial use of Bhat- tacharyacoefficientswouldapplyingthemtothesamewordfromadifferentlanguage group. As a result, the preservation of particular words could be analysed across language groups, enabling to compare and evaluate potential reasons behind it. 32faroese icelandic german luxembourgish yiddish english norwegian swedish afrikaans dutch0.00.20.40.6HeightCluster DendrogramFigure 22: Dendrogram of hierarchical clustering with Silhouette value cut for Ger- manic languages (colour words) Moreover, regarding IPA-encoded data potential further work would be gener- ating a customized IPA Phonetic Substitution table. Also, an important step to- wards receiving more accurate and interesting results would be augmenting the IPA databases with more languages. Finally, classifying languages in language databases and automatically analysing purity of clusters would be a step forward towards fully automated and consistent process. Consequently, a list of 118 languages containing their language families and branches has been created. It could be incorporated with existing language databases. 33ladino spanish galician portuguese french latin sicilian italian neapolitan catalan romanian0.000.250.500.75HeightCluster DendrogramFigure 23: Dendrogram of hierarchical clustering with Silhouette value cut for Ro- mance languages (colour words) 3410 Summary of contributions My personal contributions during this undergraduate research assistantship include: Data Collection. Created a Sheep counting numbers database. Made geographical data database and a map of dialects. Collectedcolourwordsfrom42differentlanguagesandmadeadatabase. Made thefollowingsubsets: Indo-European, Germanic, Romance, RomanceandGer- manic. Created numbers, colours and basic words databases in IPA encoding. Made a list of 118 languages, their language families and branches. Transforming data using phonetics. Transformed sheep counting numbers, colours (including Indo-European, Germanic, Romance, Romance and Germanic subsets) databases using a specified phonetic encoding. Mean, SD and density analysis. Analysed mean, SD and density of sheep numbers, colours (including all subsets). Produced tables and plots. T-Scores and Bhattacharya calculations. Calculated T-Scores and Bhat- tacharya coefficients for sheep numbers, colours (including all subsets); Made den- drograms from Bhattacharya scores. Hierarchical clustering. Performed hierarchical clustering for sheep numbers, colours (all subsets), IPA (all three). Created dendrograms. Performed hierarchical clustering with the best silhouette cut value for sheep numbers all to all, colours (all subsets), small numbers all to all. Made den- drograms, Silhouette plots, Cluster plots. Performed hierarchical clustering with k=10 cut for numbers all to all, colours (all subsets), small numbers all to all. Made dendrograms, Silhouette plots, Cluster plots. 35Code development. CreatedapackageinR“CompLinguistics”,whichconsistedoffunctions: “mean_SD”, “densityP”, “sMatrix”, “tscore”, “bhatt”, “silhouetteV”, “hcutVisual”. Produced R script that automates the processes of file reading, generating a certain format data frame, performing hierarchical clustering with the best silhouette value cut. In addition, created another R script, which performed calculations of mean, standard deviation, Bhattacharya scores and analysis of distribution. Several shellscrips. “editableGaby” phonetic substitution table. Acknowledgements Gabija Mikulyte was supported by an undergraduate research grant from the De- partment of Computer Science at Brunel University London. 36Bibliography [1]Automatic phonemic transcriber . http://tom.brondsted.dk/text2phoneme/. (Accessed on 07/20/2019). [2]Indo-european migrations . Indo-European migrations. (Accessed on 07/25/2019). 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Distin ,Cultural Evolution , Cambridge University Press, 2011. [12]M. Gimbutas ,The Prehistory of Eastern Europe , no. pt. 1 in American School of Prehistoric Research. Bulletin, Peabody Museum, 1956. [13]S. Gould, S. Gold, and T. Gould ,The Structure of Evolutionary Theory , Harvard University Press, 2002. [14]B. Griffiths ,A Dictionary of North East Dialect , Northumbria University Press, 2005. [15]D. Leith ,A Social History of English , A Social History of English, Routledge, 1997. 37[16]J. Lynch ,Indo-european language tree . http://andromeda.rutgers.edu/jlynch/Lan- guageTree.pdf, February 2014. (Accessed on 07/18/2019). [17]J. Mallory ,In Search of the Indo-Europeans: Language, Archaeology, and Myth, Thames and Hudson, 1991. [18]J. Marler ,The beginnings of patriarchy in europe: Reflections on the kurgan theory of marija gimbutas , The Rules of Mars: Readings on the Origins, History and Impact of Patriarchy. Manchester: Knowledge, Ideas and Trends, (2005), pp. 53–75. [19]M. S. Perreault, Charles ,Dating the origin of language using phonemic diversity, 7,4 (2012). [20]S. Pinker and P. Bloom ,Natural language and natural selection , Behavioral and Brain Sciences, 13 (1990), p. 707–727. [21]C. U. Press ,The Columbia Encyclopedia , Columbia University Press, 6th ed., June 2000. [22]A. Trombetti ,L’unità d’origine del linguaggio , Libreria Treves di Luigi Bel- trami, 1905. [23]J. Wielemaker, T. Schrijvers, M. Triska, and T. Lager ,SWI-Prolog , Theory and Practice of Logic Programming, 12 (2012), pp. 67–96. 38A Phonetic Substitution tables A.1 Editable This table was mostly used for calculations of pairwise linguistic distances. Symbol “%” indicates comments. %Substitution costs table t(S1,S2,D):- S1=S2 -> D=0 ; % no cost if same character ( t1(S1,S2,D) -> true ; ( t1(S2,S1,D) -> true ; % try S1-S2 otherwise S2-S1, in t1/3 table D=1)). % else cost=1 if not in t1/3 table % old simplistic table %t(S1,S2,0):- S1 = S2. %t(S1,S2,1):- S1 \== S2. % a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z %Grimm’s law %close consonants t1(b,p,D):- tweight(consonant1,D). % b ->p t1(d,t,D):- tweight(consonant1,D). % d -> t t1(g,k,D):- tweight(consonant1,D). % g -> k t1(p,f,D):- tweight(consonant1,D). % p -> f t1(t,’T’,D):- tweight(consonant1,D). % t -> th t1(k,’C’,D):- tweight(consonant1,D). % k -> ch t1(’C’,h,D):- tweight(consonant1,D). % ch -> h t1(b,f,D):- tweight(consonant1x2,D). % b -> p -> f t1(d,’T’,D):- tweight(consonant1x2,D). % d -> t -> th t1(g,’C’,D):- tweight(consonant1x2,D). % g -> k -> ch t1(g,h,D) :- tweight(consonant1x3,D). % g -> k -> ch -> h t1(f,v,D):- tweight(consonant1,D). t1(g,j,D):- tweight(consonant1,D). t1(s,z,D):- tweight(consonant1,D). t1(v,w,D):- tweight(consonant1,D). t1(f,w,D):- tweight(consonant1x2,D). % f -> v -> w t1(’F’,w,D):- tweight(consonant1x2,D). % F -> v -> w % from numberslist10 t1(f,’F’,0). % F from ph in original t1(’S’,’š’,0). % ’S’ from sh in original t1(’C’,’č’,0). % ’S’ from sh in original t1(’T’,’ j’,0). % ’S’ from th in original % other close consonants t1(’š’,s,D):- tweight(consonant1,D). % sh <-> s t1(’S’,s,D):- tweight(consonant1,D). % sh <-> s t1(’C’,’S’,D):- tweight(consonant1,D). % ch <-> sh 39t1(’C’,’š’,D):- tweight(consonant1,D). % ch <-> sh t1(’č’,’S’,D):- tweight(consonant1,D). % ch <-> sh t1(’č’,’š’,D):- tweight(consonant1,D). % ch <-> sh t1(’K’,k,D):- tweight(consonant1,D). % kh <-> k t1(’G’,k,D):- tweight(consonant1,D). % gh <-> k t1(’G’,g,D):- tweight(consonant1,D). % gh <-> g t1(’K’,’G’,D):- tweight(consonant1,D). % kh <->gh t1(’Z’,z,D):- tweight(consonant1,D). % zh <-> z t1(c,s,D):- tweight(consonant1,D). % ts <-> s t1(x,k,D):- tweight(consonant1,D). % ks <-> k t1(’D’,d,D):-tweight(consonant1,D). % dh <-> d % vowels %t1(S1,S2,0.2):- Vowels=[a,e,i,o,u,y], member(S1,Vowels), member(S2,Vowels). t1(a,Y,V):- (Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(e,Y,V):- (Y=a;Y=’A’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(i,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(o,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(u,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=y;Y=’Y’), tweight(vowel,V). t1(y,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’), tweight(vowel,V). % same vowel t1(A1,A2,0):- t_a(A1), t_a(A2). t1(E1,E2,0):- t_e(E1), t_e(E2). t1(I1,I2,0):- t_i(I1), t_i(I2). t1(O1,O2,0):- t_o(O1), t_o(O2). t1(U1,U2,0):- t_u(U1), t_u(U2). t1(Y1,Y2,0):- t_y(Y1), t_y(Y2). % close vowels t1(X,Y,V):- tvowel(X), tvowel(Y), tweight(vowel,V). % long vowels t1(’A’,Y,V):- (Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’E’,Y,V):- (Y=’A’;Y=a;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’I’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’O’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’U’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’Y’;Y=y), tweight(vowel,V). t1(’Y’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u), tweight(vowel,V). %long-short vowels t1(’A’,a,Z):- tweight(longvowel,Z). t1(’E’,e,Z):- tweight(longvowel,Z). t1(’I’,i,Z):- tweight(longvowel,Z). t1(’O’,o,Z):- tweight(longvowel,Z). t1(’U’,u,Z):- tweight(longvowel,Z). t1(’Y’,y,Z):- tweight(longvowel,Z). 40%long consonants t1(’M’,m,Z):- tweight(longconsonant,Z). t1(’N’,n,Z):- tweight(longconsonant,Z). % weight table tweight(vowel,0.2). tweight(longvowel,0.1). tweight(consonant1,0.2). tweight(consonant1x2,0.4). tweight(consonant1x3,0.8). tweight(longconsonant,0.05). tvowel(V):- t_a(V); t_e(V); t_i(V); t_o(V); t_u(V); t_y(V). A.2 editableGaby This table was created based on Editable illustrated before. Comments and “!!” symbol indicates where changes were made. % Substitution costs table t(S1,S2,D):- S1=S2 -> D=0 ; % no cost if same character ( t1(S1,S2,D) -> true ; ( t1(S2,S1,D) -> true ; % try S1-S2 otherwise S2-S1, in t1/3 table D=1)). % else cost=1 if not in t1/3 table % old simplistic table %t(S1,S2,0):- S1 = S2. %t(S1,S2,1):- S1 \== S2. % a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z /* Phonetic encoding c - ts x - ks C - ch as in charity k - as in cat T - th S - sh G - dzh %!! K - kh Z - zh D - dz H - spanish/portuguese sound of ’j’ %!! F - ph A,I,O,U,Y - long vowels */ 41% Grimm’s law %close consonants t1(b,p,D):- tweight(consonant1,D). % b ->p t1(d,t,D):- tweight(consonant1,D). % d -> t t1(g,k,D):- tweight(consonant1,D). % g -> k t1(p,f,D):- tweight(consonant1,D). % p -> f t1(t,’T’,D):- tweight(consonant1,D). % t -> th t1(k,’C’,D):- tweight(consonant1x2,D). % k -> ch !! t1(’C’,h,D):- tweight(consonant1x2,D). % ch -> h !! t1(b,f,D):- tweight(consonant1x2,D). % b -> p -> f t1(d,’T’,D):- tweight(consonant1x2,D). % d -> t -> th t1(g,’C’,D):- tweight(consonant1x2,D). % g -> ch t1(g,h,D) :- tweight(consonant1x1,D). %g->k & g->h & k->h same, ch further !! t1(f,v,D):- tweight(consonant1,D). t1(g,j,D):- tweight(consonant1,D). %!! t1(s,z,D):- tweight(consonant1,D). t1(v,w,D):- tweight(consonant1,D). t1(f,w,D):- tweight(consonant1x2,D). % f -> v -> w t1(’F’,w,D):- tweight(consonant1x2,D). % F -> v -> w % from numberslist10 t1(f,’F’,0). % F from ph in original t1(’S’,’š’,0). % ’S’ from sh in original t1(’C’,’č’,0). % ’S’ from sh in original t1(’T’,’ j’,0). % ’S’ from th in original % other close consonents t1(’š’,s,D):- tweight(consonant1,D). % sh <-> s t1(’S’,s,D):- tweight(consonant1,D). % sh <-> s t1(’C’,’S’,D):- tweight(consonant1,D). % ch <-> sh t1(’C’,’š’,D):- tweight(consonant1,D). % ch <-> sh t1(’č’,’S’,D):- tweight(consonant1,D). % ch <-> sh t1(’č’,’š’,D):- tweight(consonant1,D). % ch <-> sh t1(’K’,k,D):- tweight(consonant1,D). % kh <-> k t1(’K’,g,D):- tweight(consonant1,D). % kh <-> g t1(’G’,’Z’,D):- tweight(consonant1,D). % dzh <-> zh !! t1(’G’,’C’,D):- tweight(consonant1,D). % dzh <-> ch !! t1(’K’,’G’,D):- tweight(consonant1,D). % kh <->gh t1(’Z’,z,D):- tweight(consonant1,D). % zh <-> z t1(’Z’,s,D):- tweight(consonant1x2,D). % zh -> z -> s !! t1(c,s,D):- tweight(consonant1,D). % ts <-> s t1(x,k,D):- tweight(consonant1,D). % ks <-> k t1(’D’,d,D):-tweight(consonant1,D). % dh <-> d t1(’K’,g,D):-tweight(consonant1,D). % kh -> g %!! t1(’H’,’K’,D):-tweight(consonant1,D). %!! t1(’H’,g,D):-tweight(consonant1,D). %!! t1(’H’,k,D):-tweight(consonant1,D). %!! t1(’H’,h,D):-tweight(consonant1,D). %!! 42% vowels %t1(S1,S2,0.2):- Vowels=[a,e,i,o,u,y], member(S1,Vowels), member(S2,Vowels). t1(a,Y,V):- (Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(e,Y,V):- (Y=a;Y=’A’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(i,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(o,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V). t1(u,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=y;Y=’Y’), tweight(vowel,V). t1(y,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’), tweight(vowel,V). % same vowel t1(A1,A2,0):- t_a(A1), t_a(A2). t1(E1,E2,0):- t_e(E1), t_e(E2). t1(I1,I2,0):- t_i(I1), t_i(I2). t1(O1,O2,0):- t_o(O1), t_o(O2). t1(U1,U2,0):- t_u(U1), t_u(U2). t1(Y1,Y2,0):- t_y(Y1), t_y(Y2). % close vowels t1(X,Y,V):- tvowel(X), tvowel(Y), tweight(vowel,V). % long vowels t1(’A’,Y,V):- (Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’E’,Y,V):- (Y=’A’;Y=a;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’I’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’O’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V). t1(’U’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’Y’;Y=y), tweight(vowel,V). t1(’Y’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u), tweight(vowel,V). %long-short vowels t1(’A’,a,Z):- tweight(longvowel,Z). t1(’E’,e,Z):- tweight(longvowel,Z). t1(’I’,i,Z):- tweight(longvowel,Z). t1(’O’,o,Z):- tweight(longvowel,Z). t1(’U’,u,Z):- tweight(longvowel,Z). t1(’Y’,y,Z):- tweight(longvowel,Z). %long consonants t1(’M’,m,Z):- tweight(longconsonant,Z). t1(’N’,n,Z):- tweight(longconsonant,Z). % weight table tweight(vowel,0.2). tweight(longvowel,0.1). tweight(consonant1,0.2). tweight(consonant1x2,0.4). tweight(consonant1x3,0.8). 43 ! 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"# $% &' ()* +, − / 01 23 456 789 :; <= >?@A B C DE F10_borrowdale10_bowland 10_breton 10_coniston10_cornish 10_derbyshire 10_derbyshireDales10_eskdale 10_kirkbyLonsdale10_lakes 10_lincolnshire10_nidderdale10_rathmell 10_scots10_southWestEngland10_swaledale10_teesdale10_tong10_welsh10_wensleydale 10_westCountryDorset10_westmorland10_wilts 1_borrowdale1_bowland1_breton 1_coniston1_cornish 1_derbyshire 1_derbyshireDales1_eskdale 1_kirkbyLonsdale1_lakes 1_lincolnshire1_nidderdale 1_rathmell1_scots1_southWestEngland 1_swaledale1_teesdale1_tong1_welsh 1_wensleydale1_westCountryDorset 1_westmorland1_wilts 2_borrowdale 2_bowland2_breton 2_coniston2_cornish 2_derbyshire2_derbyshireDales 2_eskdale 2_kirkbyLonsdale2_lakes 2_lincolnshire 2_nidderdale2_rathmell2_scots 2_southWestEngland2_swaledale 2_teesdale2_tong2_welsh 2_wensleydale2_westCountryDorset 2_westmorland 2_wilts3_borrowdale3_bowland 3_breton3_coniston 3_cornish3_derbyshire 3_derbyshireDales3_eskdale3_kirkbyLonsdale3_lakes 3_lincolnshire3_nidderdale 3_rathmell3_scots 3_southWestEngland3_swaledale3_teesdale3_tong 3_welsh3_wensleydale 3_westCountryDorset3_westmorland3_wilts4_borrowdale4_bowland 4_breton4_coniston 4_cornish4_derbyshire 4_derbyshireDales4_eskdale 4_kirkbyLonsdale4_lakes4_lincolnshire4_nidderdale4_rathmell 4_scots 4_southWestEngland4_swaledale4_teesdale4_tong4_welsh4_wensleydale4_westCountryDorset 4_westmorland4_wilts5_borrowdale 5_bowland5_breton 5_coniston5_cornish 5_derbyshire5_derbyshireDales5_eskdale 5_kirkbyLonsdale5_lakes5_lincolnshire 5_nidderdale5_rathmell5_scots 5_southWestEngland 5_swaledale5_teesdale 5_tong5_welsh 5_wensleydale 5_westCountryDorset5_westmorland 5_wilts 6_borrowdale 6_bowland6_breton 6_coniston6_cornish 6_derbyshire6_derbyshireDales6_eskdale 6_kirkbyLonsdale6_lakes 6_lincolnshire 6_nidderdale6_rathmell6_scots 6_southWestEngland6_swaledale6_teesdale 6_tong6_welsh 6_wensleydale6_westCountryDorset6_westmorland6_wilts7_borrowdale 7_bowland7_breton7_coniston 7_cornish 7_derbyshire7_derbyshireDales7_eskdale 7_kirkbyLonsdale7_lakes 7_lincolnshire 7_nidderdale7_rathmell7_scots 7_southWestEngland7_swaledale 7_teesdale7_tong 7_welsh7_wensleydale 7_westCountryDorset7_westmorland7_wilts8_borrowdale8_bowland 8_breton8_coniston 8_cornish 8_derbyshire 8_derbyshireDales8_eskdale 8_kirkbyLonsdale8_lakes 8_lincolnshire8_nidderdale 8_rathmell8_scots 8_southWestEngland8_swaledale8_teesdale 8_tong 8_welsh8_wensleydale 8_westCountryDorset8_westmorland 8_wilts 9_borrowdale9_bowland9_breton 9_coniston9_cornish 9_derbyshire 9_derbyshireDales9_eskdale 9_kirkbyLonsdale9_lakes9_lincolnshire9_nidderdale 9_rathmell 9_scots 9_southWestEngland9_swaledale9_teesdale 9_tong9_welsh 9_wensleydale 9_westCountryDorset9_westmorland9_wilts −10−5051015 −15 −10 −5 0 5 10 15 Dim1 (40%)Dim2 (17.5%)cluster a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a !!a ""a ##a $$a%%a &&a ''a ((a ))a **a ++a ,,a −−a a //a 00a 11a 22a 33a 44a 55a 66a77a 88a 99a ::a ;;a <<a ==a >>a ??a @@a AAa BBa CCa DDa EEa FFa1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1819 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3637 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 5455 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70Cluster plotFigure 24: Cluster plot of all to all sheep counting systems comparison with best silhouette value cut tweight(longconsonant,0.05). tvowel(V):- t_a(V); t_e(V); t_i(V); t_o(V); t_u(V); t_y(V). B Dendrograms and Cluster plots B.1 Sheep counting systems Figures 24 and 25. B.2 Dendrograms of Bhattacharya scores of colour words Figures 26, 27, 28, 29 and 30. 4410_borrowdale10_bowland 10_breton 10_coniston10_cornish 10_derbyshire 10_derbyshireDales10_eskdale 10_kirkbyLonsdale10_lakes 10_lincolnshire10_nidderdale10_rathmell 10_scots10_southWestEngland10_swaledale10_teesdale10_tong10_welsh10_wensleydale 10_westCountryDorset10_westmorland10_wilts 1_borrowdale1_bowland1_breton 1_coniston1_cornish 1_derbyshire 1_derbyshireDales1_eskdale 1_kirkbyLonsdale1_lakes 1_lincolnshire1_nidderdale 1_rathmell1_scots1_southWestEngland 1_swaledale1_teesdale1_tong1_welsh 1_wensleydale1_westCountryDorset 1_westmorland1_wilts 2_borrowdale 2_bowland2_breton 2_coniston2_cornish 2_derbyshire2_derbyshireDales 2_eskdale 2_kirkbyLonsdale2_lakes 2_lincolnshire 2_nidderdale2_rathmell2_scots 2_southWestEngland2_swaledale 2_teesdale2_tong2_welsh 2_wensleydale2_westCountryDorset 2_westmorland 2_wilts3_borrowdale3_bowland 3_breton3_coniston 3_cornish3_derbyshire 3_derbyshireDales3_eskdale3_kirkbyLonsdale3_lakes 3_lincolnshire3_nidderdale 3_rathmell3_scots 3_southWestEngland3_swaledale3_teesdale3_tong 3_welsh3_wensleydale 3_westCountryDorset3_westmorland 3_wilts4_borrowdale4_bowland 4_breton4_coniston 4_cornish4_derbyshire 4_derbyshireDales4_eskdale 4_kirkbyLonsdale4_lakes4_lincolnshire4_nidderdale4_rathmell 4_scots 4_southWestEngland4_swaledale4_teesdale4_tong4_welsh4_wensleydale4_westCountryDorset 4_westmorland4_wilts5_borrowdale 5_bowland5_breton 5_coniston5_cornish 5_derbyshire5_derbyshireDales5_eskdale 5_kirkbyLonsdale5_lakes5_lincolnshire 5_nidderdale5_rathmell5_scots 5_southWestEngland 5_swaledale5_teesdale 5_tong5_welsh 5_wensleydale 5_westCountryDorset5_westmorland 5_wilts 6_borrowdale 6_bowland6_breton 6_coniston6_cornish 6_derbyshire6_derbyshireDales6_eskdale 6_kirkbyLonsdale6_lakes 6_lincolnshire 6_nidderdale6_rathmell6_scots 6_southWestEngland6_swaledale6_teesdale 6_tong6_welsh 6_wensleydale6_westCountryDorset6_westmorland6_wilts7_borrowdale 7_bowland7_breton7_coniston 7_cornish 7_derbyshire7_derbyshireDales7_eskdale 7_kirkbyLonsdale7_lakes 7_lincolnshire 7_nidderdale7_rathmell7_scots 7_southWestEngland7_swaledale 7_teesdale7_tong 7_welsh7_wensleydale 7_westCountryDorset7_westmorland7_wilts8_borrowdale8_bowland 8_breton8_coniston 8_cornish 8_derbyshire 8_derbyshireDales8_eskdale 8_kirkbyLonsdale8_lakes 8_lincolnshire8_nidderdale 8_rathmell8_scots 8_southWestEngland8_swaledale8_teesdale 8_tong 8_welsh8_wensleydale 8_westCountryDorset8_westmorland 8_wilts 9_borrowdale9_bowland9_breton 9_coniston9_cornish 9_derbyshire 9_derbyshireDales9_eskdale 9_kirkbyLonsdale9_lakes9_lincolnshire9_nidderdale 9_rathmell 9_scots 9_southWestEngland9_swaledale9_teesdale 9_tong9_welsh 9_wensleydale 9_westCountryDorset9_westmorland9_wilts −10−5051015 −15 −10 −5 0 5 10 15 Dim1 (40%)Dim2 (17.5%)cluster a a a a a a a a a a1 2 3 4 5 6 7 8 9 10Cluster plotFigure 25: Cluster plot of all to all sheep counting systems comparison with K=10 cut 45redyellowwhitegreenblackblue 0.820431 0.950568Figure 26: Dendrogram of Bhattacharya scores of “ColoursAll” 46greenbluewhiteyellowblackred 0.525667 0.866715Figure27: DendrogramofBhattacharyascoresofIndo-Europeanlanguages(colours) 47greenbluewhiteyellowblackred 0.525667 0.866715Figure 28: Dendrogram of Bhattacharya scores of Germanic languages (colours) 48yellowblueblackgreenwhitered 0.670457 0.93171Figure 29: Dendrogram of Bhattacharya scores of Romance languages (colours) 49bluewhitegreenblackredyellow 0.682005 0.936915Figure 30: Dendrogram of Bhattacharya scores of Germanic and Romance languages (colours) 50

1902.04605v1.Ultra_low_damping_in_lift_off_structured_yttrium_iron_garnet_thin_films.pdf

1 This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Applied Physics Letters 111 (19), 192404 (2017) and may be found at https://aip.scitation.org/doi/abs/10.1063/1.5002004 Ultra -low damping in lift-off structured y ttrium iron garnet thin films A. Krysztofik ,1 L. E. Coy,2 P. Kuświk ,1,3 K. Załęski,2 H. Głowiński ,1 and J. Dubowik1 1Institute of Molecular Physics, Polish Academy of Sciences, PL -60-179 Poznań, Poland 2NanoBioMedical Centre, Adam Mickiewicz University, PL -61-614 Poznań, Poland 3Centre for Advanced Technology, Adam Mickiewicz University, PL -61-614 Poznań, Poland Electronic mail: adam.krysztofik@ifmpan.poznan.pl , hubert .glowinski @ifmpan.poznan.pl We show that using maskless photolithography and the lift-off technique patterned yttrium iron garnet thin films possessing ultra -low Gilbert damping can be accomplished . The films of the 70 nm thickness we re grown on (001)-oriented gadolinium gallium garne t by means of pulsed laser deposition and exhibit high crystalline quality, low surface roughness and effective magnetization of 127 emu/cm3. The Gilbert damping parameter is as low as 5×10−4. The obtained structures have well-defined sharp edges which along with good structural and magnetic film properties, pave a path in the fabrication of high -quality magnonic circuits as well as oxide -based spintronic devices. Yttrium iron garnet (Y 3Fe5O12, YIG) has become an intensively studied material in recent years due to exceptionally low damping of magnetization precession and electrical insulation enabling its application in research on spin -wave propagation1–3, spin-wave based logic devices4–6, spin pumping7, and thermally -driven spin caloritronics8. These applications inevitably entail film structurization in order to construct complex integrated devices . However, the fabrication of high -quality thin YIG films requires deposition temperatures over 500 C6,9–18 leading to top -down lithographical approach that is ion-beam etching of a previously deposited plain film where as patterned resist layer serves as a mask. Consequently, this metho d introdu ces crystallographic defects , imperfections to surface structure and, in the case of YIG films, causes significant increase of the damping parameter .19–21 Moreover, it does not ensure well-defined structure edges for insulators , which play a crucial role in devices utilizing 2 edge spin waves22, Goos -Hänchen spin wave shifts23,24 or standing spin waves modes25. On the contrary, t he bottom -up structurization deals with th ese issues since it allows for the film grow th in the select ed, patterned areas followed by a removal of the resist layer along with redundant film during lift-off process. Additionally, it reduces the patterning procedure by one step , that is ion etching , and imposes room -temperature deposition which both are particularly important whenever low fabrication budget is required. In this letter we report on ultra -low damping in the bottom -up structured YIG film by means of direct writing photolithography technique. In our case, t he method allows for structure patterning with 0.6 µm resolution across full writing area . In order to not preclude the lift -off process, the pulsed laser deposition (PLD) was conducted at room temperature and since such as -deposited films are amorphous19,27 the ex-situ annealing was performed for recrystallization. Note that post -deposition annealing of YIG films is commonly carried out regardless the substrate temperature during film deposition6,12,13,28,29. As a reference we investigated a plain film which was grown in the same deposition process and underwent the same fabrication procedure except for patterning. Henceforth, we will refer to the structured and the plain film as Sample 1 an d Sample 2, respectively . We anticipate that such a procedure may be of potential for fabrication of other magnetic oxide structures useful in spintronics. Structural characteriza tion of both samples was performed by means of X-Ray Diffraction (XRD). Atomic force microscopy (AFM) was applied to investigate surface morphology and the quality of structure edges. SQUID magnetometry provided information on the saturation magnetization and magnetocrystalline anisotropy field . Using a coplanar waveguide connected to a vector network analyzer , broadband ferromagnetic resonance (VNA -FMR) was performed to determine Gilbert damping parameter and anisotropy fields . All the experiments were co nducted at the room temperature. The procedure of samples preparation was as follows. The (001) -oriented gadolinium gallium garnet substrates were ultrasonicated in acetone, trichloroethylene and isopropanol to remove surface impurities. After a 1 minute o f hot plate baking for water evaporation, a positive photoresist was spin - coated onto the substrate (Sample 1). Using maskless photolithography an array of 500 μm x 500 μm squares separated over 500 μm was patterned and the exposed areas were developed. Detailed parameters of photolithography process can be found in Ref.26. We chose rather large size of the squares to provide a high signal -to-noise ratio in the latter measurements. Thereafter, plasma etching was performed to remove a residual resist. We would like to emphasize the importance of this step in the fabrication p rocedure as the resist residues may locally affect crystalline structure of a YIG film causing an undesirable increase of overall magnetization damping. Both substrates were then placed in a high vacuum chamber of 9×10-8 mbar base pressure and a film was d eposited from a stoichiometric ceramic YIG target under 2×10-4 mbar partial pressure of oxygen. We used a Nd:YAG laser (λ = 355 nm) for the ablation with pulse rate of 2 Hz which yielded 1 nm/min growth rate. The target -to-3 substrate distance was approximat ely 50 mm. After the deposition the l ift-off process for the Sample 1 was performed using sonication in acetone to obtain the expected structures. Subsequently, both samples were annealed in a tube furnace under oxygen atmosphere (p ≈ 1 bar) for 30 minutes at 850°C. The heating and cooling rates were about 50 C/min and 10 C/min, respectively. FIG. 1. (a) XRD θ−2θ plot near the (004) reflection of structured ( Sample 1 ) and plain ( Sample 2 ) YIG film. Blue arrows show clear Laue reflections of the plain film. Insets show schematic illustration of the structured and plain film used in this study. (b) Height profile (z(x)) taken from the structured sample (left axis), right shows the differential of the p rofile, clearly showing the slope change. Inset shows 3D map of the structure’s edge. The structure of YIG films was determined by the X -ray diffraction. Although the as-deposited films were amorphous, with the annealing treatment they inherited the lattice orientation of the GGG substrate and recrystallized along [ 001] direction. Figure 1 (a) presents diffraction curves taken in the vicinity of ( 004) Bragg reflection. The ( 004) reflection position of structured YIG well coincide s with the reflection of the plain film. The 2 θ=28.70 9 corresponds to the cubic lattice constant of 12.428 Å. A comparison of this value with lattice parameter of a bulk YIG (12.376 Å) suggest distortion of unit 4 cells due to slight nonstoichiometry.16,30 Both samples exhibit distinct Laue oscillation s depicted by the blue arrows, indicating film uniformity and high crystalline order , although the structured film showed lower intensity due to the lower mass of the film . From the oscillation period we estimated film thickness of 73 nm in agreement with the nominal thickness and the value determined using AFM for Sample 1 ( Fig. 1 (b)). By measuring the diffraction in the expanded angle range w e also confirmed that no additional phases like Y 2O3 or Fe 2O3 appeared. The surface morphology of the structured film was investigated by means of AFM. In Fig. 1 (b) profile of a square’s edge is shown. It should be highlighted that no edge irregularities has formed during lift -off process. The horizontal distance between GGG substrate and the surface of YIG film is equal to 170 nm as marked in Fig. 1 (b) by the shaded area. A fitting with Gaussian function to the derivative of height profile yields the full width at half maximum of 61 nm. This points to the well - defined struct ure edges achieved with bottom -up structurization. Both samples have smooth and uniform surface s. The comparable values of root mean square (RMS) roughness (0.306 nm for Sample 1 and 0.310 nm for Sample 2) indicate that bottom -up structurization process did not leave any resist residues. Note that a roughness of a bare GGG substrate before deposition was 0.281 nm, therefore, the surface roughness of YIG is increased merely by 10%. FIG. 2. Hysteresis loops of structured (Sample 1) and plain (Sample 2) YIG films measured by SQUID magnetometry along [100] direction at the room temperature . Figure 2 shows magnetization reversal curves measured along [ 100] direction. For each hysteresis loop a paramagnetic contribution arising for the GGG substrates was subtracted. The saturation magnetization 𝑀𝑠 was equal to 117 emu/cm3 and 118.5 emu/cm3 for Sample 1 and 2, respectively . Both hysteresis loops demonstrate in -plane anisotropy. For the (001) -oriented YIG the [ 100] direction is a “hard” in -plane axis and the magnetization saturates at 𝐻𝑎 = 65 Oe. This value we identify as -100 -75 -50 -25 0 25 50 75 100-1.0-0.50.00.51.0 Sample 1 Sample 2M / MS Magnetic Field (Oe)5 magnetocrystalline anisotropy field. The VNA -FMR measurements shown in Fig. 3 (a) confirm these results. Using Kitte l dispersion relation, i.e. frequency 𝑓 dependence of resonance magnetic field 𝐻: 𝑓=𝛾 2𝜋√(𝐻+𝐻𝑎cos 4𝜑)(𝐻+1 4𝐻𝑎(3+cos 4𝜑)+4𝜋𝑀𝑒𝑓𝑓), (1) 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑢, (2) we derived 𝐻𝑎 and the effective magnetization 𝑀𝑒𝑓𝑓, both comparable to the values determined using SQUID and close to the values of a bulk YIG (see Table I .). Here, the azimuthal angle 𝜑 defines the in-plane orientation of the magnetization direction with respect to the [100] axis of YIG and 𝛾 is the gyromagnetic ratio ( 1.77×107𝐺−1𝑠−1). To better compare the values of 𝐻𝑎 between samples and to determine if the results are influenced by additional anisotropic contribution arising from the squares’ shape in the structured film we performed angular resolved resonance measurements (inset in Fig. 3(a)) . The fitting according to Eq. (1) gives |𝐻𝑎| equal to 69.5±0.6 for Sample 1 and 69.74±0.28 for Sample 2 in agreement with the values derived from 𝑓(𝐻) dependence and better accuracy. Hence, we conclude that the structurization did not affect the in -plane anisotropy. The deviations of the derived 𝑀𝑠 and 𝐻𝑎 from bulk values can be explained in the framework of Fe vacancy model developed for YIG films as a result of nonstoichiometry.13,30 For the experimentally determined 𝑀𝑠 and 𝐻𝑎 the model yields the chemical unit Y 3Fe4.6O11.4 which closely approximates to the composition of a stoichiometric YIG Y 3Fe5O12. TABLE I. Key parameters reported for PLD and LPE YIG films. AFM SQUID VNA -FMR Film thickness RMS rough - ness (nm) Ms (emu/cm3) Ha (Oe) Field orientation Meff (emu/cm3) |Ha| (Oe) Hu (Oe) α (× 10-4) ΔH 0 (Oe) Sample 1 70 nm 0.306 117±1 65±5 (100): (110): (001): 125±1 126±1 129±2 64±1 63±1 − -101±18 -113±18 -151±28 5.53±0.13 5.24±0.12 5.19±0.64 1.45±0.09 2.86±0.09 2.61±0.34 Sample 2 70 nm 0.310 118.5±2 65±5 (100): (110): (001): 124±1 127±1 131±2 62±1 65±1 − -69±28 -107±28 -157±36 5.05±0.07 5.09±0.09 5.02±0.18 0.97±0.05 1.28±0.06 1.48±0.09 LPE-YIG31 106 nm 0.3 143 − (112): − − − 1.2 0.75 LPE-YIG30 120 μm − 139±2 − (111): 133±2 85±6 76±1 0.3 − Although the saturation magnetization of the films is decreased by 15% with respect to the bulk value we can expect similar spin wave dynamics since magnon propagation does not solely depend on 𝑀𝑠 but on the effective magnetization or equivalently, on the uniaxial anisotropy field 𝐻𝑢.12 Substitution of 𝑀𝑠 into Eq. (2) gives average values of 𝐻𝑢 equal to -122 Oe and -111 Oe for Sample 1 and 2, respectively (to determine 𝐻𝑢 from the out -of-plane FMR measurements when H || [001] we 6 used the 𝑓=𝛾 2𝜋(𝐻+𝐻𝑎−4𝜋𝑀𝑒𝑓𝑓) dependence13 to fit the data and assumed the value of 𝐻𝑎 from angular measurements ). As 𝑀𝑒𝑓𝑓𝑆𝑎𝑚𝑝𝑙𝑒 1,2≈𝑀𝑒𝑓𝑓𝑏𝑢𝑙𝑘, it follows that the low value of 𝑀𝑠 in room - temperature deposited thin films is “compensated ” by uniaxial anisotropy field. Note that for bulk YIG saturation magnetization is diminished by 𝐻𝑢/4𝜋 giving a lower value of 𝑀𝑒𝑓𝑓 while for Sample 1 and 2, 𝑀𝑠 is augmented by 𝐻𝑢/4𝜋 giving a higher value of 𝑀𝑒𝑓𝑓 (Table I .). The negative sign of uniaxial anisotropy field is typical for PLD -grown YIG films and originates from preferential distribution of Fe vacancies between different si tes of YIG’s octahedral sublattice.30 This point s to the growth -induced anisotropy mechanism while the stress -induced contribution is of ≈10 Oe29 and, as it can be estimated according to Ref.32, the transition layer at the substrate -film interface due to Gd, Ga, Y ions diffusion is ca. 1.5 nm thick for the 30 min of annealing treatment. We argue that the growth - induced anisotropy due to ordering of the magnetic ions is related to the growth condition which in our study is specific. Namely, it is crystallization of an amorphous material. Gilbert damping parameter 𝛼 was obtained by fitting dependence of linewidth 𝛥𝐻 (full width at half maximum ) on frequency 𝑓 as shown in Fig. 3 (b): 𝛥𝐻 =4𝜋𝛼 𝛾𝑓+𝛥𝐻0, (3) where 𝛥𝐻0 is a zero -frequency linewidth broadening . The 𝛼 parameter of both samples is nearly the same , 5.32×10−4 for Sample 1 and 5.05×10−4 for Sample 2 on average (see Table I.) . It proves that bottom -up patterning does not compromise magnetization damping. The value of 𝛥𝐻0 contribution is around 1.5 Oe although small variations of 𝛥𝐻0 on 𝜑 can be noticed. Additional comments on angular dependencies of 𝛥𝐻 can be found in the supplementary material. The derived values of 𝛼 remain one order of magnitude smaller than for soft ferromagnets like Ni 80Fe2033, CoFeB34 or Finemet35, and are comparable to values reported for YIG film s deposited at hi gh temperatures (from 1×10−4 up to 9×10−4).6,9,11,14,15,17,18 It should be also highlighted that 𝛼 constant is significantly increased in comparison to the bulk YIG made by means of Liquid Phase Epitaxy (LPE) . However, recently reported LPE-YIG films of nanometer thickness , suffer from the increased damping as well (Table I.) due to impurity elements present in the high -temperature solutions used in LPE technique31. As PLD method allow s for a good contamination control , we attribute the increase as a result of slight nonstoichiometry determined above with Fe vacancy model .30 Optimization of growth conditions , which further improve the film composition may resolve this issue and allow to cross the 𝛼=1×10−4 limit. We also report that additional annealing of the samples (for 2h) did not influence damping nor it improved the value of 𝐻𝑎 or 𝑀𝑒𝑓𝑓 (within 5% accuracy). 7 FIG. 3. (a) Kittel dispersion relation s of the structured (Sample 1) and plain (Sample 2) YIG film. The i nset shows angular dependence of resonance field revealing perfect fourfold anisotropy for both samples . (b) Linewidth dependence on frequency fitted with Eq. (3). The inset shows resonance absorptions peaks with very similar width (5.3 Oe for Sample 1 and 4.7 Oe for Sample 2 at 10 GHz ). Small differences of the resonance field originate from different values of 4𝜋𝑀𝑒𝑓𝑓. In conclusion , the lift-off patterned YIG films possessing low damping have been presented. Although the structurization procedure required deposition at room temperature , the 𝛼 parameter does not diverge from those reported for YIG thin films grown at temperatures above 500 C. Using the plain, reference film fabricated along with the structured one, we have shown that structurization does not significantly affect structural nor magnetic properties of the films, i.e. out-of-plane lattice constant, surface roughness, saturation magnetization, anisotropy fields and damping. The structures obtain ed with bottom -up structurization indeed possess sharp , well-defined edges . In particular, o ur findings will help in the development of magnonic and spintronic devices utilizing film boundary effects and low damping of magnetization precession . 8 Supplementary Material See supplementary material for the angular dependence of resonance linewidth . The research received funding from the European Union Horizon 2020 research and innovation progra mme under the Marie Skłodowska -Curie grant agreement No 644348 (MagIC). We would like to thank Andrzej Musiał for the assistance during film annealing. 1 H. Yu, O. d’Allivy Kelly, V. Cros, R. Bernard, P. Bortolotti, A. Anane, F. Brandl, R. Huber, I. Stasinopoulos, and D. Grundler, Sci. Rep. 4, 6848 (2015). 2 S. Maendl, I. Stasinopoulos, and D. Grundler, Appl. Phys. Lett. 111, 12403 (2017). 3 A. V. Sadovnikov, C.S. Davies, V. V. Kruglyak, D. V. Romanenko, S. V. Grishin, E.N. Beginin, Y.P. Sharaevskii, and S.A. Nikitov, Phys. Rev. B 96, 60401 (2017). 4 A.A. Nikitin, A.B. Ustinov, A.A. Semenov, A. V. Chumak, A.A. 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1009.3922v1.Diffusive_properties_of_persistent_walks_on_cubic_lattices_with_application_to_periodic_Lorentz_gases.pdf

Diffusive properties of persistent walks on cubic lattices with application to periodic Lorentz gases Thomas Gilberty, Huu Chuong Nguyen y, 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, hnguyen@ulb.ac.be, dps@fciencias.unam.mx Abstract. We calculate the diffusion coefficients of persistent random walks on cubic and hypercubic lattices, where the direction of a walker at a given step depends on the memory of one or two previous steps. These results are then applied to study a billiard model, namely a three-dimensional periodic Lorentz gas. The geometry of the model is studied in order to find the regimes in which it exhibits normal diffusion. In this regime, we calculate numerically the transition probabilities between cells to compare the persistent random-walk approximation with simulation results for the diffusion coefficient. Submitted to: J. Phys. A: Math. Theor. 1. Introduction Problems dealing with the persistence of motion of tracer particles – that is, the tendency to continue or not in the same direction at a scattering event – are encountered in many areas of physics; see e.g. [1] and references therein. We are specifically interested in the effect of persistence for the motion of random walkers on regular lattices. The diffusive properties of persistent random walks on two-dimensional regular lattices were the subject of a previous paper by two of the present authors [2]. There, we presented a theory making use of the symmetries of such lattices to derive the transport coefficients of walks with a two-step memory. In the first part of the present paper, we extend this theory to hyper-cubic lattices in arbitrary dimensions, which is possible by describing the geometry of the lattices in a suitable way. Persistence effects naturally arise in the context of deterministic diffusion [3, 4, 5, 6], which is concerned with the interplay between dynamical properties at the microscopic scale and transport properties at the macroscopic scale. A variety of different techniques are now available, which rely on the chaotic properties of model systems to describe their macroscopic properties [7], [8, chap. 25]. In particular, periodic Lorentz gases and related models, such as multi-baker maps, are simple deterministic dynamical systems with strong chaotic properties which also exhibit diffusive regimes. Although the transport coefficients of these models can be expressed formally in terms of the microscopic dynamical properties, actually computing them is usually difficult, with the exception of some of the simplest toy models [9, 10, 11].arXiv:1009.3922v1 [cond-mat.stat-mech] 20 Sep 2010Diffusive properties of persistent walks on cubic lattices 2 One reason for this is that memory effects can remain important, in spite of the chaotic character of the underlying dynamics. The diffusive properties of these models therefore provide ideal applications of the formalism presented in this paper. An example of this was illustrated in reference [12], for a class of two-dimensional periodic billiard tables. Extending these results, in this paper we apply the formalism to model the diffusive properties of higher-dimensional periodic Lorentz gases. The diffusive properties of the three-dimensional periodic Lorentz gas, which consists of the free motion of independent tracer particles in a cubic array of spherical obstacles, are interesting in their own right. In two spatial dimensions, the existence of diffusive regimes in such systems has been rigorously established [13, 14]. It relies on the finite-horizon property, which requires that the system admits no ballistic trajectories, i.e. those which never collide with any obstacle. In this case, it is possible to change scales from microscopic to macroscopic, reducing the complicated motion of tracer particles at the microscopic level to a diffusive equation at the macroscopic level. When the horizon is infinite on the other hand, there is rather a weakly superdiffusive process, with mean-squared displacement growing like tlogt[15, 16], as recently shown rigorously in [17]. The necessity of finite horizon to have normal diffusion in two dimensions led to the idea that this was also necessary in three dimensions – see, for example, reference [18]. Recently, however, it was argued by one of the present authors [19] that in higher-dimensional billiards, normal diffusion, by which we mean an asymptotically linear growth in time of the mean- squared displacement, may arise even in the absence of finite horizon. In fact, three different types of horizon can be identified in the three-dimensional periodic Lorentz gas. The key observation is that it is only “planar” gaps – those with infinite extension in two dimensions – which induce anomalous diffusion. If there are only “cylindrical” gaps, whose extension is limited to a single dimension, then the available space in which particles can move ballistically is limited. This leads to a decay of correlations which is fast enough to give normal diffusion at the level of the mean-squared displacement, although higher moments of the displacement distribution may be non-Gaussian [19]. The paper is organized as follows. Section 2 describes the computation of the transport coefficient of walks on hypercubic lattices with one and two-step memories. In the second part of this paper, we apply this formalism to the diffusive regimes of the three-dimensional periodic Lorentz gas. In section 3, we give a detailed description of the three-dimensional periodic Lorentz gas introduced in reference [19], in particular delimiting the regimes with qualitatively different behaviour in parameter space. We then apply the results on persistent random walks to the diffusive regimes of this model in section 4. Conclusions are drawn in section 5. 2. Persistent random walks on cubic lattices In this section, we describe a way to incorporate the specific geometry of cubic and hyper- cubic lattices in the framework presented in reference [2] for calculating diffusion coefficients for persistent random walks on lattices. We start by considering the motion of independent walkers on a regular cubic lattice in three dimensions. Given their initial position r0at time t=0, the walkers’ trajectories are specified by the sequence fv0;:::;vngof their successive displacements. Here we consider dynamics in discrete time, so that the time sequences are simply assumed to be incremented by identical time steps tas the walkers move from site to site. In the sequel we will loosely refer to the displacement vectors as velocity vectors; they are in fact dimensionless unit vectors.Diffusive properties of persistent walks on cubic lattices 3 The sequence of successive displacements is determined by the underlying dynamics, whether deterministic or stochastic. At the coarse level of description of the lattice dynamics, this is interpreted as a persistent type of random walk, where some memory effects are accounted for: the probability that the nth step is taken in the direction vndepends on the past history vn1;vn2;:::. The quantity of interest here 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 autocorrelations 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, here d=3, and `is the lattice spacing. The (dimensionless) velocity autocorrelations are computed as averages h
iover the equilibrium distribution, denoted by m, of the underlying process, so that the problem reduces to computing the quantities hv0
vni=å v0;:::;vnv0
vnm(fv0;:::;vng): (2.2) Following the approach of reference [2], we wish to compute the terms in this sum, and hence the corresponding diffusion coefficient (2.1), for three different types of random walks, namely those with zero-step, single-step and two-step memories. These cases all involve factorisations of the measure m(fv0;:::;vng)into 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 schemes we outline below allow to write equation (2.2) as a sum of powers of matrices, so that (2.1) boils down to a geometric series, which can then be resummed to obtain an expression for the diffusion coefficient that is readily computable given the probabilities that characterise the allowed transitions in the process. 2.1. Description of geometry of cubic lattices It is first necessary to find a succinct description of the geometry of the cubic lattices that we wish to study. The six directions of the three-dimensional cubic lattice and corresponding displacement vectors are specified in terms of the unit vectors eiof a Cartesian coordinate system asei,i=1;2;3. The crucial property required for the application of our method is that all of these unit vectors can be obtained by repeated application of a single transformation G, which generates the cyclic group GfGiGi;i=0;:::; 5g: (2.3) One possible choice of Ggives the following group elements: G1=G4=G=0 @0 01 1 0 0 0 1 01 A; G2=G5=G2=0 @01 0 0 01 1 0 01 A; G3=G0=G3=0 @1 0 0 01 0 0 011 A:(2.4)Diffusive properties of persistent walks on cubic lattices 4 Figure 1 displays the six possible directions of a walker on this lattice, numbered according to repeated iterations by G. Thus a walker with incoming direction e1, indicated by the arrow, can be deflected to any of the six directions Gie1,i=0;:::; 5, corresponding respectively to e1,e2,e3,e1,e2, ande3. 012 3 4 5 Figure 1. The possible directions of motion on a cubic lattice, labelled from 0 to 5 relative to the incoming direction shown by the arrow. These directions are obtained by successive applications of the transformation Ggiven in equation (2.4). A similar transformation Gcan easily be identified for a walk on a d-dimensional hyper- cubic lattice: G=0 BBBBB@0 0


01 1 0


0 0 0 1


0 0 ......... 0 0


1 01 CCCCCA; (2.5) which maps the unit vectors onto the 2 d-cycle e17!e27!


7! ed7!e17!


7! ed. 2.2. No-Memory Approximation (NMA) We now proceed to calculate the diffusion coefficient (2.1) for random walks with different memory lengths. The simplest case is that of a Bernoulli process for the velocity trials, so that the walkers have no memory of their history as they proceed to their next position. The probability measure mthus factorises: m(fv0;:::;vng) =n Õ i=0p(vi): (2.6) Given that the lattice is rotation invariant and that pis uniform, the velocity autocorrelation (2.2) vanishes: hv0
vni=dn;0: (2.7)Diffusive properties of persistent walks on cubic lattices 5 The diffusion coefficient of the random walk without memory is then given by DNMA=`2 2dt: (2.8) 2.3. 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): (2.9) Here, P(v0jv)denotes the one-step conditional probability that the walker moves with displacement v0, given that it made a displacement vat the previous step. Considering for definiteness the three-dimensional lattice and using the elements of the group G, we express each velocity vector vkin terms of the first one, v0, asvk=Gikv0, where each ik2f0;:::; 5g. Substituting this into the expression for the velocity autocorrelation hv0
vni, equation (2.2), we obtain, using the factorisation (2.9), å v0;:::;vnv0
vnn Õ i=1P(vijvi1)p(v0) =6 å i0;:::;in=1v0
Ginv0min;in1


mi1;i0pi0: (2.10) In this expression, min;in1P(Ginv0jGin1v0) (2.11) are the elements of the stochastic matrix Mof the Markov chain associated to the persistent random walk, and pip(ei)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. The same notations were used in [2] and will be used throughout this article. The terms involving Min (2.10) constitute the matrix product of ncopies of M. Furthermore, since the invariant distribution is uniform over the lattice directions, we can choose an arbitrary direction for v0, and hence write hv0
vni=v0
v0m(n) 1;1+v0
Gv0m(n) 2;1+


+v0
G5v0m(n) 6;1; =m(n) 1;1m(n) 4;1(2.12) where m(n) i;jdenote the elements of Mn. The actual value of the diffusion coefficient depends on the probabilities P(Gjvjv), which are parameters of the model, subject to the constraints åjP(Gjvjv) =1. To simplify the notation, we assume rotational invariance of the process, i.e. independence with respect to the value of v, and we denote the conditional probabilities of these walks by PjP(Gjvjv), where j=0;:::; 5. The transition matrix Mgiven by (2.11) is thus the cyclic matrix M=0 BBBBBB@P0P1P2P3P4P5 P5P0P1P2P3P4 P4P5P0P1P2P3 P3P4P5P0P1P2 P2P3P4P5P0P1 P1P2P3P4P5P01 CCCCCCA: (2.13)Diffusive properties of persistent walks on cubic lattices 6 The matrix Mnshares the same property of cyclicity, so that it also has only six distinct entries. It is thus possible to proceed along the lines described in [2] and obtain the recurrence relation 0 B@m(n) 1;1m(n) 4;1 m(n) 2;1m(n) 5;1 m(n) 3;1m(n) 6;11 CA=0 @P0P3P1P4P2P5 P5P2P0P3P1P4 P4P1P5P2P0P31 A0 B@m(n1) 1;1m(n1) 4;1 m(n1) 2;1m(n1) 5;1 m(n1) 3;1m(n1) 6;11 CA; =0 @P0P3P1P4P2P5 P5P2P0P3P1P4 P4P1P5P2P0P31 An10 @P0P3 P1P4 P2P51 A: (2.14) [Note that the left-hand side of this equation was chosen to reduce the size of the matrix involved and to calculate the element required in (2.12).] As a consequence, we can write for the velocity autocorrelation (2.12) hv0
vni= 1 0 0
0 @P0P3P1P4P2P5 P5P2P0P3P1P4 P4P1P5P2P0P31 An10 @P0P3 P1P4 P2P51 A; (2.15) and thus obtain the expression of the diffusion coefficient (2.1) as D1SMA DNMA=2 641+2 1 0 00 @1+P3P0P4P1 P5P2 P2P51+P3P0P4P1 P1P4 P2P51+P3P01 A10 @P0P3 P1P4 P2P51 A3 75; (2.16) by using the result that å¥ n=0An= (IA)1, where Iis the identity matrix, for a square matrix Awhose eigenvalues are all strictly less than 1 in modulus, This result easily generalises to a hyper-cubic lattice in any dimension d. Note also that for a symmetric process, in which P1=P4andP2=P5, we recover the diffusion coefficient D1SMA =DNMA1+P0P3 1P0+P3; (2.17) in agreement with the result stated in [2]. 2.4. Two-Step Memory Approximation (2-SMA) Let us now suppose 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): (2.18) The velocity autocorrelation (2.2) function is then hv0
vni=å fvn;:::;v0gv0
vnn Õ i=2P(vijvi1;vi2)p(v0;v1): (2.19) Since the probability transitions P(vijvi1;vi2)have symmetries similar to those used in reference [2], the computation of equation (2.19) reduces to an expression very similar to that found there for walks on one- and two-dimensional lattices. The details of the derivation are a bit more involved than the one-step memory persistent walks, so we will limit ourselves to stating the results.Diffusive properties of persistent walks on cubic lattices 7 Letting z=2ddenote the coordination number of the lattice, and writing‡ Pj;k P(GzkGzjvjGzjv;v), which is the conditional probability of making a displacement v given that the two preceding displacements were successively GjGkvandGkv, we define the zzmatrix K(f)0 BBB@P00 P10


Pz1;0 fP01 fP11


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


fz1Pz1;z11 CCCA: (2.20) The argument fin this expression is a complex number such that fz=1. In the case of two-dimensional lattices, only two of these roots are relevant, corresponding to the complex exponential of the smallest angle between two lattice vectors, f=exp(2ip=z). For hyper- cubic lattices in arbitrary dimensions, however, we must consider a priori all the zpossible roots of unity, fjexp(2ipj=z),j=0;:::; z1. A direct calculation of (2.19) shows that the velocity autocorrelation takes the form hv0
vni= 1


1" z1 å j=0ajK(fj)n1diag(1;fj;:::;fz1 j)#0 B@p1 ... pz1 CA; (2.21) where diag (1;fj;:::;fz1 j)denotes the matrix with elements listed on the main diagonal and 0 elsewhere. For the three-dimensional cubic lattice, the coefficients ajare found to be a0=a2=a4=0; a1=a3=a5=2;(2.22) which compares to a1=a3=2 and a0=a2=0 in the case of the two-dimensional square lattice [2]. In the case of a d-dimensional hyper-cubic lattice, this generalises to a2j=0;j=0;:::; d1; a2j+1=2;j=0;:::; d1;(2.23) The diffusion coefficient of a two-step memory persistent random walk on a d- dimensional hyper-cubic lattice is thus D2SMA DNMA=1+4 1


1( d å j=1[IzK(f2j1)]1diag(1;f2j1;:::;fz1 2j1))0 B@p1 ... pz1 CA; (2.24) where Izdenotes the zzidentity matrix. 3. Three-dimensional periodic Lorentz gas Equations (2.8), (2.16) and (2.24) can be put to the test to probe the diffusive regimes of periodic Lorentz gases. The diffusive motion of the tracers results from the chaotic nature of the microscopic dynamics and the fast decay of correlations, which are in turn due to the convex nature of the obstacles. Taking into consideration the different diffusive regimes of these models, which, as we argued earlier, depend on the nature of their horizon, we ‡ This expression differs from that given in [2] due to a typographical error in that paper – they are really the same.Diffusive properties of persistent walks on cubic lattices 8 investigate how the microscopic dynamical properties of the system determine the diffusion coefficient. Machta and Zwanzig [20] addressed this issue in a particular limiting case, showing that, in the limit where the obstacles are so close together that a tracer will remain localised on ekach lattice site for a very long time (compared to the mean time separating two collision events), the process of diffusion on the Lorentz gas is well approximated by the dimensional prediction (2.8), where the lattice spacing `is the distance separating two neighbouring obstacles and tis the trapping time, which can be computed in terms of the geometry of the billiard as a simple consequence of ergodicity. That is to say, when the geometry of the billiard is such that two neighbouring disks nearly touch, the Lorentz gas is well approximated by a Bernoulli process, modeling the random hopping of tracers from cell to cell, with time- and length-scales specified according to the geometry of the billiard. Different approximation schemes have been proposed to go beyond this zeroth-order approximation and account for corrections to it [21, 22]; see, in particular, reference [23] for an overview. A consistent approach to understanding the effect of these corrections in two- dimensional diffusive billiards was described in [12]. The idea is to approximate the hopping process of tracer particles by persistent random walks with finite memory, and thus estimate the diffusion coefficient of the billiard by the two-dimensional lattice equivalents of the one- or two-step formulas (2.16) and (2.24). We discuss below the transposition of these results to the diffusive regimes of the three- dimensional periodic Lorentz gas. 3.1. Geometry of simple three-dimensional periodic Lorentz gas model We begin with a detailed description of the geometry and the different horizon regimes of the system studied in reference [19]; additional details are given in reference [24]. The model consists of a three-dimensional (3D) periodic Lorentz gas constructed out of cubic unit cells of side length `, having eight “outer” spheres of radius rout`at its corners and a single “inner” sphere of radius rin`at its centre – see figure 2. The infinitely-extended periodic structure formed in this way is symmetric under interchange of rinandrout; without loss of generality, we take routrin. Figure 2. Geometry of the obstacles in a single cell of the 3D periodic Lorentz gas model for rout=0:45`andrin=0:30`, in the cylindrical-horizon regime. This model seems to be the simplest one which allows a finite horizon, although this is possible only when the spheres are permitted to overlap. It is known that finite-horizonDiffusive properties of persistent walks on cubic lattices 9 periodic Lorentz gases with non-overlapping spheres in fact exist in any dimension [25], but we are not aware of any explicit constructions of such models, even in the case of three dimensions. Lattice of outer spheres The spheres of radius rout`form a simple cubic lattice. This lattice has the following properties: When rout<1=2, the spheres are disjoint. In this case, there are free planes [25] in the structure, that is, infinite planes which do not intersect any of the spheres, in particular there are free planes centered on the faces of the unit cell. In this case, we say that there is a planar horizon (PH). When routis small, there are additional planes at different diagonal angles, analogously to the two-dimensional infinite-horizon Lorentz gas [15, 16, 17]. When rout>1=2, the spheres overlap, thereby automatically blocking all planes. The overlaps (intersections) of the spheres partially cover the faces of the cubes, leaving a space in between which acts as an exit towards the adjacent cell. When rout1=p 2, the overlaps completely cover the faces of the unit cell, so that it is no longer possible to exit the cell. When routp 3=2, all of space is covered, and it is no longer possible to define a billiard dynamics. Conditions for normal diffusion: cylindrical horizon As shown in reference [19], the necessary and sufficient condition to have normal diffusion is that all free planes are blocked; if there are free planes, then the diffusion is weakly anomalous. The conditions to block all planes are as follows. All free planes are automatically blocked for rout1=2, when the rout-spheres overlap. If the rout-spheres do not overlap, then it is necessary to introduce the rin-sphere to block planes which are parallel to the faces of the unit cell. For this blocking to occur, we need rin1=2rout. Furthermore, we must also block diagonal planes at 45 degree angles, which requires thatrout1=(2p 2)orrin1=(2p 2). If all of these conditions are satisfied, then we no longer have free planes, but may have free cylinders (“cylindrical gaps”) in the structure; we then say that there is a cylindrical horizon (CH). Conditions for finite horizon Stronger statistical properties – e.g. faster decay of correlations – may be expected when there is a finite horizon [18, 19], i.e. where the length of free paths between collisions with obstacles is bounded above. To obtain this, not only all planar gaps, but also all cylindrical gaps must be blocked, i.e. all holes viewed from any direction must be blocked. To do so, the following conditions must be fulfilled: Therout-spheres must overlap, rout1=2. Furthermore, the projection of the rin- sphere on each face of the unit cell must cover the available exit space, as illustrated in figure 3(a). Letting dbe the maximum width of overlap of the resulting discs of radius routon a face of the unit cell, we have d2=r2 out1=4, and we need rin1=2dto block the space.Diffusive properties of persistent walks on cubic lattices 10 ρin=1 2−dd ρout (a) Face of unit cell d rin (b) Mid-plane of unit cell Figure 3. Geometry of the 3D periodic Lorentz gas. (a) Cross-section of the unit cell in one of its faces. The overlapping outer spheres of radius rout>1=2, give rise to four overlapping discs (shown in green); the maximum width of their overlap is denoted d. The central disc (red) shows the minimum radius rin=1=2dof the central sphere such that its projection covers the gap between the rout-discs on the face. (b) Geometry of the mid-plane of a unit cell for parameters giving a finite horizon. The outer discs are cross-sections of the overlaps of the outer rout-spheres, and have radius dequal to the overlap parameter in (a). The inner disc is the cross-section of the inner rin-sphere. Figure 4. Finite horizon can be achieved in a three-dimensional lattice, provided the spheres are allowed to overlap. Here the available space for diffusing particles is shown for parameter values rout=0:65`andrin=0:15`(in the FH1 region) in an unfolded channel. We must block cylindrical corridors which cross the structure at a 45 degree angle at the level of the mid-plane of a unit cell, which corresponds to the planar cross-section with most available space in the unit cell. The mid-plane has the geometry shown in figure 3(b), with four outer discs of radius d, and a central disc of radius rin; these discs are the intersection of the rout-overlaps and of the rin-sphere, respectively, with the mid- plane. Free diagonal trajectories in this plane at an angle of 45 degrees give rise to small cylindrical corridors. These will be blocked if there is no free line in the mid-plane. This blocking occurs provided either d1=(2p 2), i.e.routp 3=(2p 2, or if rin1=(2p 2),Diffusive properties of persistent walks on cubic lattices 11 thus giving rise to two distinct finite horizon regimes (FH1 and FH2), which are in fact disjoint. Figure 4 depicts the space available for tracer particles in a channel of three consecutive cells for a particular finite-horizon case. Localisation of trajectories Having fixed rout, it is also necessary to calculate the value of rinabove which the trajectories become localised (L) between neighbouring spheres, and are thus no longer able to diffuse. For rout<1=p 2, when there are still exits available on the faces of the cubic unit cell, this happens exactly when the discs in the mid-plane touch, i.e. when rin+d=1=p 2, so that the condition for localised trajectories becomes [19] rin1=p 2p r2out1=4. Condition to fill space Finally, we calculate when the spheres fill all space (denoted U, for undefined): When rout<1=p 2, this occurs when the rin-spheres are large enough that their intersection with each face of the cube, which is a disc of radiusq r2 in1=4, covers the exit on a face left open by the rout-spheres. This gives the condition r2 inr2 out+ 1=4p r2out1=4. When rout>1=p 2, the condition is that rinbe large enough to cover the space left by therout-spheres inside the unit cell. The condition can again be found by looking at the mid-plane, where there is most available space: the disc of radius rinmust cover the space left by the discs of radius d(which are cross-sections of the overlaps of the rout-spheres). This occurs when rin1=2p r2out1=2. Parameter space The complete parameter space of this model is shown in figure 5, exhibiting the regions in parameter space corresponding to the regimes of qualitatively different behaviour discussed above§. Note that if rin>p 3=2rout, then the rout- and rin- spheres overlap, and if rin>0:5 then neighbouring rin-spheres also overlap. These conditions are marked by the dotted lines in the figure. 4. Persistence in the diffusive regimes of the three-dimensional Lorentz gas In this section, we study the dependence of the diffusion coefficient on the geometrical parameters of the 3D periodic Lorentz gas model in the finite- (FH1) and cylindrical-horizon (CH) regimes, comparing the numerical results with the finite-memory approximations (2.8), (2.16) and (2.24). 4.1. Approximation by the NMA process The computation of the dimensional formula (2.8) relies on that of the residence time t. An exact formula is available for this quantity [27]: t=jQj j¶QjjS2j jB2j; (4.1) § A similar diagram of parameter space for a two-dimensional version of the model was given in reference [26]. However, the symmetry between routandrinwas overlooked there; see also reference [19].Diffusive properties of persistent walks on cubic lattices 12 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 rout0.00.10.20.30.40.50.60.70.8rin PHCH FH1FH2LU Figure 5. Parameter space of the three-dimensional periodic Lorentz gas as a function of the geometrical parameters routandrin. Solid lines divide regimes of qualitatively different behaviour, which are also shaded with different colours and labelled as follows: PH: planar horizon; CH: cylindrical horizon; FH1 and FH2: finite horizon; L: localised, non-diffusive motion; U: undefined (all space filled). Note that the FH regime is divided into two disjoint regions. The dashed lines mark the different conditions referred to in the text. The diagonal dotted line separates regions where the rin-spheres do (above) and do not (below) overlap the rout-spheres. The diagram is reflection-symmetric in the line rin=rout, but for clarity only the lower half is shown. wherejQjdenotes the volume of the billiard domain outside the obstacles, j¶Qjthe surface area of the available gaps separating neighbouring cells, jS2j=4pthe surface area of the unit sphere in three dimensions, and jB2j=pthe volume (area) of the unit disk in two dimensions, and we assume unit velocity. The explicit formulas giving the values of jQjandj¶Qjare rather lengthy and will not be given here; see reference [28]. The validity of equation (4.1) can be tested by comparison with numerical computation of the residence time, as shown in figure 6. Here, and in the remainder of the paper, we restrict attention to values of routclose to the limiting value 1 =p 2 and rinclose to 0, so that the geometry is that of a single, cubic unit cell. 4.2. Approximation by the 1SMA and 2SMA processes Single- and two-step memory processes can be derived as approximations, at the lattice level, to the dynamics of the Lorentz gas. This is done by computing numerically the statistics of tracer particles as they jump from cell to cell, so as to estimate the single- and two-step memory probability transitions. The results are shown in figure 7 for the single-step memory process, where the six transition probabilities Pi,i=0;:::; 5, are displayed as functions of the outer radius routforDiffusive properties of persistent walks on cubic lattices 13 ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóárin=0. órin=0.15 0.000.050.100.150.200.00.20.40.60.8 d=1 2-routt-1 Figure 6. Residence time t, equation (4.1), compared to direct numerical simulations. The results are shown for two values of the inner radius, rin=0 and rin=0:15, as functions of d1=p 2rout, which is the characteristic size of the gaps separating neighbouring cells. The curves are very similar since the volume of the inner sphere remains small. In this and the following results we take `=1. different values of the inner radius rin. For the two-step process, the computation of the transition probabilities Pi;jis shown in figure 8 for rin=0, that is in the absence of a sphere at the center of the cell. The six different panels each correspond to a given i=0;:::; 5. The same is shown in figure 9 for rin=0:15. áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ àààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0. 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routPk (a) áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ àààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.03 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routPk (b) áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ àààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.06 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routPk (c) áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ àààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.09 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routPk (d) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.12 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routPk (e) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routPk (f) Figure 7. Numerical computations of the probabilities P0;:::; P5of the single step memory process, appearing in (2.13). The six panels shown correspond to as many different values of rin, where the probabilities are shown as functions of d. The dashed line at Pk=1=6 indicates the value for a memoryless (NMA) walk. Here and in figures 8 and 9, the conventions are as follows: Empty squares (blue), P0; empty upward triangles (cyan), P1; empty downward triangles (green), P2; filled squares (red) P3; filled upward triangles (magenta), P4; filled downward triangles (brown), P5. In all cases we verify the symmetry P1=P2=P4=P5, which also remain close to 1 =6.Diffusive properties of persistent walks on cubic lattices 14 áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0. 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP0,j (a) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0. 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP1,j (b) áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0. 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP2,j (c) áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ àààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0. 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP3,j (d) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0. 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP4,j (e) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0. 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP5,j (f) Figure 8. Numerical computations of the 36 probabilities Pi;jwhich appear in (2.20), corresponding to a cell with no sphere at its center, i.e. the inner radius rin=0. The symmetries of the process are reflected by the similarities between figures 8(b), 8(c), 8(e) and 8(f). The colour coding is similar to figure 7. áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ àààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP0,j (a) áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP1,j (b) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP2,j (c) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP3,j (d) áááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP4,j (e) ááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ ààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15 0.000.050.100.150.200.00.10.20.30.4 d=1 2-routP5,j (f) Figure 9. Numerical computations of the probabilities 36 Pi;jwhich appear in (2.20), corresponding to inner radius rin=0:15. Here again the symmetries of the process are reflected by the similarities between figures 9(b), 9(c), 9(e) and 9(f).Diffusive properties of persistent walks on cubic lattices 15 4.3. Diffusion coefficient of the billiard rin=0. 0.000.050.100.150.201.01.21.41.61.8 d=1 2-rout6Dtl-2 (a) rin=0.03 0.000.050.100.150.201.01.21.41.61.8 d=1 2-rout6Dtl-2 (b) rin=0.06 0.000.050.100.150.201.01.21.41.61.8 d=1 2-rout6Dtl-2 (c) rin=0.09 0.000.050.100.150.201.01.21.41.61.8 d=1 2-rout6Dtl-2 (d) rin=0.12 0.000.050.100.150.201.01.21.41.61.8 d=1 2-rout6Dtl-2 (e) rin=0.15 0.000.050.100.150.201.01.21.41.61.8 d=1 2-rout6Dtl-2 (f) Figure 10. Diffusion coefficient normalised with respect to the dimensional prediction DNMA, equation (2.8), vs. gap size d=1=p 2rout, plotted for different values of the inner radius rin=0;:::; 0:15. The symbols (black) correspond to direct numerical computation of this quantity, the long dashed (green) lines to the single-step memory diffusion coefficient (2.16), and the solid (red) lines to the two-step memory diffusion coefficient (2.24). The vertical dotted lines indicate the separation between the finite and infinite horizon regimes. Having computed the probability transitions associated to the single and two-step memory processes, we can compute the invariant distribution Pand substitute the results into equations (2.16) and (2.24) to obtain values of the diffusion coefficients. These are compared to the diffusion coefficient of the billiard calculated from direct simulations in figure 10. We can draw several conclusions from the results shown in figure 10. Firstly, we remark that in the 3D model studied here there is relatively little back-scattering, i.e. motion inDiffusive properties of persistent walks on cubic lattices 16 which the particle reverses its direction between arriving and leaving a given cell. This gives an important contribution to the diffusion coefficient, and, in particular, corresponds to the fact that here we find that the diffusion coefficient is larger than the memoryless (NMA) approximation, while in reference [12] the diffusion coefficient tended to lie below the results of this approximation. Note, however, that this effect depends strongly on the particular model used. It is also interesting to note that in the finite-horizon regime, i.e. left of the dotted vertical lines in figures 10(b)-10(f), approximating the diffusion coefficient by the one-step memory process (2.16) is just as good as the two-step process (2.24). In the cylindrical-horizon regime, however, the two results are different; the single-step approximation gets poorer as rout decreases, whereas the two-step process yields more accurate estimates. This corresponds to the fact that correlations decay more slowly in the cylindrical-horizon regime [19], so that memory effects persist for longer. 5. Conclusions The cyclic structures of certain regular lattices underly symmetries of their statistical properties which can be exploited to greatly simplify their analysis. Examples are two- dimensional lattices such as the square, the honeycomb and the triangular lattice, which were studied in reference [2]. Other examples include, in higher dimensions, the hypercubic lattices studied in this paper. Having exhibited the cyclic structures of these lattices, we were able to extend our previous results to hypercubic lattices with suitable adaptations, in order to calculate the diffusion coefficients of persistent random walks with up to two steps of memory. Our method is especially useful to compute the correlations of persistent walks on such regular lattices. In particular, the velocity autocorrelations of a two-step persistent walk may be recast in terms of matrix powers, which can then easily be resummed to obtain a readily- computable expression for the diffusion coefficient. Among the many applications of persistent random walks, deterministic diffusive processes are ideal candidates to apply our method. The three-dimensional periodic Lorentz gas is particularly interesting as it exhibits two distinct types of diffusive regimes, one with finite horizon, where memory effects decay fast, and another with cylindrical horizon, where memory effects can remain important. In this latter case, the approximation of the diffusive process by a two-step memory walk proves much more accurate than the single-step process. We remark that the application of our formalism to the diffusive properties of Lorentz gases relies on the numerical computation of the transition probabilities corresponding to the persistent process with which we approximate the deterministic process. Since there are 30 transition probabilities for the two-step memory walk, their analytical calculation is a daunting task. It relies on knowledge of the statistics of trapped trajectories and involves contributions from different time scales. Nonetheless, this computation is formally possible, and is in principle much simpler than that of the actual diffusion coefficient. Acknowledgments This research benefited 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. DPSDiffusive properties of persistent walks on cubic lattices 17 acknowledges financial support from DGAPA-UNAM grant IN105209 and CONACYT grant CB101246. References [1] Haus J W and Kehr K W 1987 Diffusion in regular and disordered lattices Phys. Rep. 150263. [2] Gilbert T and Sanders D P 2010 Diffusion coefficients for multi-step persistent random walks on lattices J. Phys. A Math. Theor. 435001. [3] Geisel T and Nierwetberg J 1982 Onset of diffusion and universal scaling in chaotic systems Phys. Rev. Lett. 487. [4] Fujisaka H and Grossmann S 1982 Chaos-induced diffusion in nonlinear discrete dynamics Z. Phys. B 48 261. [5] Schell M, Fraser S and Kapral R 1983 Subharmonic bifurcation in the sine map: An infinite hierarchy of cusp bistabilities Phys. Rev. A 28373. [6] Grassberger P 1983 New mechanism for deterministic diffusion Phys. Rev. A 283666. [7] Gaspard P 1998 Chaos, Scattering and Statistical Mechanics (Cambridge: Cambridge University Press). [8] Cvitanovi ´c P and Artuso R 2010 Chapter “Deterministic diffusion” in Cvitanovi ´c P, Artuso R, Mainieri R, Tanner G, and Vattay G Chaos: Classical and Quantum ChaosBook.org/version13 (Copenhagen: Niels Bohr Institute) [9] Dorfman J R 1999 An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge: Cambridge University Press). [10] Klages R and Dorfman J R 1995 Simple maps with fractal diffusion coefficients Phys. Rev. Lett. 74387. [11] Klages R and Dorfman J R 1999 Simple deterministic dynamical systems with fractal diffusion coefficients Phys. Rev. E 595361. [12] Gilbert T and Sanders D P 2009 Persistence effects in deterministic diffusion Phys. Rev. E ,8041121. [13] Bunimovich L A and Sinai Ya G 1980 Markov partitions for dispersed billiards Commun. Math. Phys. 78 247. [14] Bunimovich L A and Sinai Ya G 1981 Statistical properties of Lorentz gas with periodic configuration of scatterers Commun. Math. Phys. 78479. [15] Zacherl A, Geisel T, Nierwetberg J, and Radons G 1986 Power spectra for anomalous diffusion in the extended Sinai billiard Phys. Lett. A 114317. [16] Bleher P M 1992 Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon J. Stat. Phys. 66315. [17] Szasz D and Varj ´u T 2007 Limit laws and recurrence for the planar Lorentz process with infinite horizon J. Stat. Phys. 12959 2007. [18] Chernov N 1994 Statistical properties of the periodic Lorentz gas. Multidimensional case J. Stat. Phys. 7411. [19] Sanders D P 2008 Normal diffusion in crystal structures and higher-dimensional billiard models with gaps Phys. Rev. E 78060101. [20] Machta J and Zwanzig R 1983 Diffusion in a periodic Lorentz gas Phys. Rev. Lett. 501959. [21] Klages R and Dellago C 2000 Density-dependent diffusion in the periodic Lorentz gas J. Stat. Phys. 101145. [22] Klages R and Korabel N 2002 Understanding deterministic diffusion by correlated random walks J. Phys. A Math. Gen. 354823. [23] Klages R 2007 Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics (Singapore: World Scientific). [24] Sanders D P 2008 Deterministic Diffusion in Periodic Billiard Models (PhD thesis, University of Warwick, 2005) arXiv preprint arXiv:0808.2252 . [25] Henk M and Zong C 2000 Segments in ball packings Mathematika ,4731. [26] Garrido P L 1997 Kolmogorov–Sinai entropy, Lyapunov exponents, and mean free time in billiard systems J. Stat. Phys. 88807. [27] Chernov N 1997 Entropy, Lyapunov exponents, and mean free path for billiards J. Stat. Phys. ,881. [28] Nguyen H C 2010 Etude du comportement diffusif d’un billard chaotique `a trois dimensions et son approximation par une marche al ´eatoire persistante Th`ese de Master, Universit ´e Libre de Bruxelles .

2106.09077v1.Sharp_upper_and_lower_bounds_of_the_attractor_dimension_for_3D_damped_Euler_Bardina_equations.pdf

arXiv:2106.09077v1 [math.AP] 16 Jun 2021SHARP UPPER AND LOWER BOUNDS OF THE ATTRACTOR DIMENSION FOR 3D DAMPED EULER–BARDINA EQUATIONS ALEXEI ILYIN1, ANNA KOSTIANKO3,4, AND SERGEY ZELIK1,2,3 Abstract. The dependence of the fractal dimension of global attrac- tors for the damped 3D Euler–Bardina equations on the regularizat ion parameterα>0 and Ekman damping coefficient γ >0 is studied. We presentexplicitupper boundsforthisdimensionforthecaseofthe whole space, periodic boundary conditions, and the case of bounded dom ain with Dirichlet boundary conditions. The sharpness of these estimat es whenα→0 andγ→0 (which corresponds in the limit to the classi- cal Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus. Contents 1. Introduction 2 2. A priori estimates, well-posedness and dissipativity 5 3. Asymptotic compactness and attractors 8 4. Upper bounds for the fractal dimension 12 5. Sharp lower bound on T314 5.1. Squire’s transformation 16 5.2. Instability analysis on T217 5.3. 3D lower bound 20 Appendix A. Collective Sobolev inequalities for H1-orthonormal families 21 A.1. The case of the whole space and a domain with Dirichlet BC 21 A.2. The case of periodic BC: Estimates for the lattice sums 2 4 Appendix B. A pointwise estimate for the nonlinear term 27 References 28 2000Mathematics Subject Classification. 35B40, 35B45, 35L70. Key words and phrases. Regularized Euler equations, Bardina model, unbounded do- mains, attractors, fractal dimension, Kolmogorov flows. This work was supported by Moscow Center for Fundamental and A pplied Mathe- matics, Agreement with the Ministry of Science and Higher Education of the Russian Federation, No. 075-15-2019-1623 and by the Russian Science Fo undation grant No.19- 71-30004 (sections 2-4). The second author was partially suppor ted by the Leverhulme grant No. RPG-2021-072 (United Kingdom). 12 A. ILYIN, A. KOSTIANKO, AND S. ZELIK 1.Introduction Being the central mathematical model in hydrodynamics, the Navier-Stokes andEulerequationspermanentlyremaininthefocusofbotht heanalysisofPDEs and thetheory of infinite dimensional dynamical systems and their attractors, see [2,8,13,15,16,25,26,43,44,45]andthereferencestherei nformoredetails. Most studied is the 2D case where reasonable results on the global well-posedness and regularity of solutions as well as the results on the existen ce of global attractors and their dimension are available. However, the global well -posedness in the 3D case remains a mystery and even listed by the Clay institute o f mathematics as one of the Millennium problems. This mystery inspires a comp rehensive study of various modifications/regularizations of the initial Navi er-Stokes/Euler equations (suchasLeray- αmodel,hyperviscousNavier-Stokes equations, regulariza tionsvia p-Laplacian, etc.), many of which have a strong physical back ground and are of independent interest, see e.g. [14, 19, 26, 34, 37] and the re ferences therein. In the present paper we shall be dealing with the following re gularized damped Euler system:/braceleftbigg ∂tu+(¯u,∇x)¯u+γu+∇xp=g, div¯u= 0, u(0) =u0.(1.1) with forcing gand Ekman damping term γu,γ >0. The dampingterm γumakes the system dissipative and is important in various geophysi cal models [39]. Here and below ¯uis a smoothed (filtered) vector field related with the initial velocity fielduas the solution of the Stokes problem u= ¯u−α∆x¯u+∇xq,div¯u= 0, (1.2) whereα>0 is a given small parameter. In other words, ¯u= (1−αA)−1u, whereA:= Π∆ xis the Stokes operator and Π is the Helmholtz–Leray projecti on to divergent free vector fields in the corresponding domain. System (1.1), (1.2) (at least in the conservative case γ= 0) is often referred to as the simplified Bardina subgrid scale model of turbulence, see [4, 5, 24] for the derivation of the model and further discussion, so in this pa per we shall be calling (1.1) the damped Euler–Bardina equations. We also mention t hat rewriting (1.1) in terms of the variable ¯ ugives ∂t¯u−α∂t∆x¯u+(¯u,∇x)¯u+γ¯u+∇xp=αγ∆x¯u+g (1.3) whichisadampedversionoftheso-called Navier–Stokes–Vo ight equationsarising in the theory of viscoelastic fluids, see [23, 38] for the deta ils. Our main interest in the present paper is to study the dimensi on of global attractors for system (1.1) in 2D and 3D paying main attentio n to the most complicated 3D case. Note that, unlike the classical Euler e quations, Bardina- Euler equations can be interpreted as an ODE with bounded non lineariry in the proper Hilbert space, so no problems with well-posedness ar ise, see [5] and alsoDAMPED 3D EULER–BARDINA EQUATIONS 3 section §2 below, so the main aim of our study is to get as sharp as possib le bounds for the corresponding global attractors. Each case d= 2 andd= 3 in turn is studied in three different settings as far as the bounda ry conditions are concerned. More precisely, the system is studied (1) on the torus Ω = Td= [0,2π]d. In this case the standard zero mean condition is imposed on u, ¯uandg; (2) in the whole space Ω = Rd; (3) in a bounded domain Ω ⊂Rdwith Dirichlet boundary conditions for ¯ u. We denote by Ws,p(Ω) the standard Sobolev space of distributions whose deriv a- tives up to order sbelong to the Lebesgue space Lp(Ω). In the Hilbert case p= 2 we will write Hs(Ω) instead of Ws,2(Ω). In order to work with velocity vector fields, we denote by Hs=Hs(Ω) the subspace of [ Hs(Ω)]dconsisting of diver- gence free vector fields. In the case of Ω ⊂Rdwe assume in addition that vector fields from Hssatisfy Dirichlet boundary conditions and in the case of per iodic boundary conditions Ω = Tdwe assume that these vector fields have zero mean. We also recall that equation (1.1) possesses the standard en ergy identity 1 2d dt/parenleftig /ba∇dbl¯u/ba∇dbl2 L2(Ω)+α/ba∇dbl∇x¯u/ba∇dbl2 L2(Ω)/parenrightig +γ/parenleftig /ba∇dbl¯u/ba∇dbl2 L2(Ω)+α/ba∇dbl∇x¯u/ba∇dbl2 L2(Ω)/parenrightig = (g,¯u), where(u,v) isthestandardinnerproductin[ L2(Ω)]d. Forthisreason itisnatural to consider problem (1.1) in the phase space H1with norm /ba∇dbl¯u/ba∇dbl2 α:=/ba∇dbl¯u/ba∇dbl2 L2+α/ba∇dbl∇x¯u/ba∇dbl2 L2. Ourfirstmain result is thefollowing theorem which gives an e xplicit upperbound for the fractal dimension of the attractor in the 3D case. Theorem 1.1. Letd= 3, letΩbe as described above, and let g∈[L2(Ω)]3(in the periodic case we assume also that ghas zero mean). Then the solution semigroup S(t)associated with equation (1.1)possesses a global attractor A⋐H1with finite fractal dimension satisfying the following inequali ty: dimFA≤1 12π/ba∇dblg/ba∇dbl2 L2 α5/2γ4. (1.4) The analogue of this estimate for the 2D case reads dimFA≤1 16π/ba∇dblg/ba∇dbl2 L2 α2γ4(1.5) with the following improvement for the case when Ω =T2orΩ =R2: dimFA≤1 8π/ba∇dblcurlg/ba∇dbl2 L2 αγ4(1.6) due to estimates related with the vorticity equation.4 A. ILYIN, A. KOSTIANKO, AND S. ZELIK Since the general case γ >0 is reduced to the particular one with γ= 1 by scalingt→γ−1t,u→γ−2u,g→γ−2g, the most interesting in estimates (1.4), (1.5) and (1.6) is the dependence of the RHS on α. For the viscous case of equations (1.1) ∂tu+(¯u,∇x)¯u+∇xq=ν∆xu+g the following estimate is proved in [5]: dimFA≤C/ba∇dblg/ba∇dbl6/5 L2 ν12/5α18/5 for the case Ω = T3. We see that even in the case ν= 1 this estimate gives essentially worse dependence on αthan our estimate (1.4). The upper bounds for3DNavier-Stokes-Voight equation obtainedin[23]give evenworsedependence on the parameter α(likeα−6). Estimates (1.5) and (1.6) have been proved for Ω =T2in a recent paper [22]. The sharpness of these estimates in th e limit asα→0 was also established there for the case of the 2D Kolmogorov flows. However, to the best of our knowledge, no lower bounds for the dimension of the attractor of the Euler–Bardina equations in 3D are availabl e in the literature. Our second main result covers this gap. Namely, we consider t he 3D Kol- mogorov flows on the torus Ω = T3for equations (1.1) generated by the family of the right-hand sides parameterized by an integer paramet ers∈N: g=gs= g1=γ2λ(s)sin(sx3), g2= 0, g3= 0,(1.7) wheres∼α−1/2andλ(s) is a specially chosen amplitude, see §5. Then, perform- ing an accurate instability analysis for the linearization of equation (1.1) on the corresponding Kolmogorov flow (in the spirit of [33], see als o [20, 21, 32]), we get the following result. Theorem 1.2. LetΩ =T3and letγ >0, andα >0. Then in the limit α→0the integer parameter sand the amplitude λ(s)can be chosen so that the corresponding forcing g=gsof the form (1.7)produces the global attractor A=As, whose dimension satisfies the following lower bound: dimFA≥c/ba∇dblg/ba∇dbl2 L2 α5/2γ4, (1.8) wherec>0is an absolute effectively computable constant. Estimate (1.8) shows that our upper bound(1.4) is optimal. A gain, to the best of our knowledge, this is the first optimal two-sided estimat e for the attractor dimension in a 3D hydrodynamical problem. In this connection we recall thecelebrated upperboundin [1 1] for the attractor dimension of the classical Navier–Stokes system on the 2D to rus, which is stillDAMPED 3D EULER–BARDINA EQUATIONS 5 logarithmically larger than the corresponding lower bound in [32]. On the other hand,addingtothesystemanarbitraryfixeddampingmakes it possibletoobtain the estimate for the attractor dimension that is optimal in t he vanishing viscosity limit [21]. Wefinallyobservethattheobtainedlowerestimatesforthea ttractordimension grow asα→0 in both 2D and 3D cases (and even are optimal for the case of tori), so one may expect that the limit attractor A0(which corresponds to the case of non-modified damped Euler equation) is infinite dimen sional. Indeed, the existence of the attractor A0in the proper phase space is well-known in 2D at least ifg∈W1,∞, see [9] and references therein and we expect that some weaker version of the limit attractor A0can be also constructed in 3D using the trajectory approach, see [8], and the concept of dissipativ e solutions for 3D Euler introduced by P. Lions, see [31]. However, the situation wit h the dimension is much more delicate since the obtained lower bounds for the in stability index on Kolmogorov’s flows are optimal for intermediate values of αonly and do not provide any reasonable bounds for the limit case α= 0. Thus, the question of finite or infinite-dimensionality of the limit attractor rem ains completely open even in the 2D case. The paper is organized as follows. The key estimates for the s olutions of problem (1.1) are derived in §2. Global well-posedness and dissipativity are also discussed there. The existence of a global attractor Ais verified in §4. To make the proof independent of the choice of a (bounded or unbounde d) domain Ω, we use the so called energy method for establishing the asympto tic compactness of the associated semigroup. The upper bounds for its dimension are obtained in §5 via the volume con- traction method [2, 10, 44]. The essential role in getting op timal bounds for the global Lyapunov exponents is played by the collective Sobol ev inequalities for H1-orthonormal families proved in Appendix A based on the idea s of [27]. Their role is somewhat similar to the role of the Lieb–Thirring ine qualities [28, 29] in the dimension estimates of the attractors of the classical N avier–Stokes equations [2, 44]. The corresponding inequality in the 2D case has also been used in [22]. Finally, the sharplower boundsof the dimension for the case Ω =T3are obtained in§5 by adapting/extending the ideas of [22, 33] to the 3D case. 2.A priori estimates, well-posedness and dissipativity We start with the standard energy estimate, which looks the s ame in the 2D and 3D cases as well as for the three types of boundary conditi ons. Proposition 2.1. Letube a sufficiently regular solution of equation (1.1). Then the following dissipative energy estimate holds: /ba∇dbl¯u(t)/ba∇dbl2 α≤ /ba∇dbl¯u(0)/ba∇dbl2 αe−γt+1 γ2/ba∇dblg/ba∇dbl2 L2, (2.1)6 A. ILYIN, A. KOSTIANKO, AND S. ZELIK where /ba∇dbl¯u/ba∇dbl2 α:=/ba∇dbl¯u/ba∇dbl2 L2+α/ba∇dbl∇x¯u/ba∇dbl2 L2. (2.2) Proof.Indeed, multiplying equation (1.1) by ¯ u, integrating over Ω and using the relation between uand ¯uas well as the standard fact that the inertial term vanishes after the integration, we arrive at d dt/parenleftbig /ba∇dbl¯u/ba∇dbl2 L2+α/ba∇dbl∇x¯u/ba∇dbl2 L2/parenrightbig +2γ/parenleftbig /ba∇dbl¯u/ba∇dbl2 L2+α/ba∇dbl∇x¯u/ba∇dbl2 L2/parenrightbig = 2(g,¯u)≤ ≤2/ba∇dblg/ba∇dblL2/ba∇dbl¯u/ba∇dblL2≤γ/ba∇dbl¯u/ba∇dbl2 L2+1 γ/ba∇dblg/ba∇dbl2 L2.(2.3) Applying the Gronwall inequality, we get the desired estima te (2.1) and complete the proof. /square The next corollary is crucial for our upper bounds for the att ractor dimension. Corollary 2.2. Letube a sufficiently smooth solution of problem (1.1). Then the following estimate holds: limsup t→∞1 t/integraldisplayt 0/ba∇dbl∇xu(s)/ba∇dblL2ds≤1 γ√ 2α/ba∇dblg/ba∇dblL2. (2.4) Proof.Indeed, integrating estimate (2.3) over t, taking thelimit t→ ∞and using the fact that /ba∇dblu(t)/ba∇dbl2 αremains bounded (due to estimate (2.1), we arrive at limsup t→∞1 t/integraldisplayt 0/ba∇dbl∇xu(s)/ba∇dbl2 L2ds≤1 2αγ2/ba∇dblg/ba∇dbl2 L2. Using after that the H¨ older inequality 1 t/integraldisplayt 0/ba∇dbl∇xu(s)/ba∇dblL2ds≤/parenleftbigg1 t/integraldisplayt 0/ba∇dbl∇xu(s)/ba∇dbl2 L2dx/parenrightbigg1/2 , we get the desired result and finish the proof of the corollary . /square We now turn to the two dimensional case without boundary. In t his case, more accurate estimates are available due to the possibilit y to use the vorticity equation. Indeed, applying curl to (1.1) and setting ω= curlu, we obtain the vorticity equation for ω: ∂tω+(¯u,∇x)¯ω+γω= curlg, ω= (1−α∆x)¯ω. (2.5) The estimates for the solution on the torus T2were derived in [22]. Although for R2they are formally the same, we reproduce them for the sake of c ompleteness. Proposition 2.3. Letube a sufficiently smooth solution of (1.1), whereΩ =T2 orR2and letω:= curluand¯ω:= curl¯u. Then, the following dissipative estimate holds: /ba∇dbl¯ω(t)/ba∇dbl2 α≤ /ba∇dbl¯ω(0)/ba∇dbl2 αe−γt+1 γ2/ba∇dblcurlg/ba∇dbl2 L2. (2.6)DAMPED 3D EULER–BARDINA EQUATIONS 7 Proof.Taking the scalar product of equation (2.5) with ¯ ω, we see that the non- linear term vanishes and using that (ω,¯ω) =/ba∇dbl¯ω/ba∇dbl2 L2+α/ba∇dbl∇x¯ω/ba∇dbl2 L2, (2.7) we obtain 1 2d dt/parenleftbig /ba∇dbl¯ω/ba∇dbl2 L2+α/ba∇dbl∇x¯ω/ba∇dbl2 L2/parenrightbig +γ/parenleftbig /ba∇dbl¯ω/ba∇dbl2 L2+α/ba∇dbl∇x¯ω/ba∇dbl2 L2/parenrightbig = (curlg,¯ω)≤ ≤ /ba∇dblcurlg/ba∇dblL2/ba∇dbl¯ω/ba∇dblL2≤1 2γ/ba∇dblcurlg/ba∇dbl2 L2+γ 2/ba∇dbl¯ω/ba∇dbl2 L2.(2.8) This gives the desired estimate (2.6) by the Gronwall inequa lity and finishes the proof of the proposition. /square Analogously to Corollary 2.2, we get the following estimate . Corollary 2.4. LetΩ =T2orR2and letube a sufficiently smooth solution of problem (1.1). Then the following estimate holds: limsup t→∞1 t/integraldisplay1 0/ba∇dbl∇x¯u(s)/ba∇dblL2ds≤1 γmin/braceleftbigg /ba∇dblcurlg/ba∇dblL2,/ba∇dblg/ba∇dblL2√ 2α/bracerightbigg .(2.9) Indeed, the second inequality was already proved in Corolla ry 2.2 and the first one is an immediate corollary of (2.6) and the fact that /ba∇dbl∇¯u/ba∇dblL2=/ba∇dbl¯ω/ba∇dblL2. Let us conclude this section by discussing the well-posedne ss of problem (1.1) and justification of the estimates obtained above. We will co nsider below only the 3D case (the 2D case is analogous and even slightly simple r). We also note from the very beginning that equation (1.1) can b e rewritten in the form of an ODE in a Hilbert space with bounded nonlinear iry. Indeed, applying the Helmholtz–Leray projection Π to both sides of ( 1.1) together with the operator Aα:= (1−αA)−1, whereA= Π∆xis the Stokes operator in Ω, we arrive at ∂t¯u+γ¯u+B(¯u,¯u) =AαΠg,¯u/vextendsingle/vextendsingle t=0= ¯u0, (2.10) whereB(¯u,¯v) :=AαΠ((¯u,∇x)¯v). It is natural to consider this system in the phase space ¯ u∈H1(Ω) with norm (2.2). Then the nonlinear operator Bis bounded from H1toH3/2: /ba∇dblB(¯u,¯v)/ba∇dblH3/2≤Cα/ba∇dbl¯u/ba∇dblα/ba∇dbl¯v/ba∇dblα, (2.11) whereCαdependsonlyon α. Indeed,if ¯ u,¯v∈H1, thenbytheSobolevembedding theorem ¯u,¯v∈L6(Ω) and (¯u,∇x)¯v∈L3/2(Ω) by H¨ older’s inequality. Together with the (L3/2→W2,3/2)-boundedness of the operator (1 −αA)−1, we get that B(¯u,¯v)∈W2,3/2(Ω). Finally, the Sobolev embedding W2,3/2⊂H3/2proves estimate (2.11).8 A. ILYIN, A. KOSTIANKO, AND S. ZELIK Thus,B(¯u,¯u) isaregularizing operatorin H1andequation (2.10)is anODE in H1with bounded nonlineariry. Therefore the local existence a nd uniqueness of a solution as well as (an infinite)differentiability of the corr espondinglocal solution semigroup are straightforward corollaries of the Banach co ntraction principle or the implicit function theorem, see e.g. [18] for the details . Thus, to get the global well-posedness and dissipativity we only need to ver ify the proper a priori estimate. Since this has already been done in Proposition 2. 1, we have proved the following theorem. Theorem 2.5. Let¯u0∈H1(Ω),g∈[L2(Ω)]d(in the case of periodic BC we also assume that ghas zero mean). Then there exists a unique global solution ¯u∈C([0,∞),H1)of problem (2.10)(which is simultaneously the unique solution of(1.1)). Moreover, the function t→ /ba∇dbl¯u(t)/ba∇dbl2 L2+α/ba∇dbl∇x¯u(t)/ba∇dbl2 L2 is absolutely continuous and the following energy identity holds: 1 2d dt/parenleftbig /ba∇dbl¯u(t)/ba∇dbl2 L2+α/ba∇dbl∇x¯u(t)/ba∇dbl2 L2/parenrightbig + +γ/parenleftbig /ba∇dbl¯u(t)/ba∇dbl2 L2+α/ba∇dbl∇x¯u(t)/ba∇dbl2 L2/parenrightbig = (g,¯u).(2.12) In particular, the dissipative estimate (2.1)holds for any solution uof classu∈ C([0,∞),H1). Corollary 2.6. Let the assumptions of Theorem 2.5 holds. Then equation (2.10) generates a dissipative solution semigroup S(t)¯u0:= ¯u(t), t≥0 (2.13) in the phase space H1(Ω). Moreover, S(t)isC∞-differentiable for every fixed t. Indeed, the existence of the semigroup is an immediate corol lary of the well- posedness proved in the theorem and the differentiability fol lows from the ODE structure of (2.10) and the fact that the map ¯ u→B(¯u,¯u) isC∞-smooth as a map from H1toH1. 3.Asymptotic compactness and attractors In this section we construct a global attractor for the solut ion semigroup S(t) generated by problem (1.1). We start with recalling the defin ition of a weak and strong global attractor, see [2, 8] for more details. We will mainly consider below the most complicated case Ω = R3since in the case of a bounded domain the asymptotic compactness is an immediate corollary of thefac t thatB(¯u,¯u)∈H3/2 ifu∈H1, see Remark 3.5. Definition 3.1. A set Aw⊂H1is a weak global attractor of the semigroup S(t) if 1)Awis a compact set in H1with weak topology;DAMPED 3D EULER–BARDINA EQUATIONS 9 2)Awis strictly invariant, i.e., S(t)Aw=Aw; 3)Awattracts the images of all bounded sets in the weak topology o fH1, i.e. for every bounded set B⊂H1and every neighbourhood O(Aw) of the attractor in the weak topology, there exists T=T(O,B) such that S(t)B⊂ O(Aw) for allt≥T. Analogously, Asis a strong attractor if it is compact in the strong topology o f H1, is strictly invariant and attracts the images of bounded se ts in the strong topology as well. Obviously Aw=As if both attractors exist. We will use the following criterion for verifying the existe nce of an attractor, see [2, 44] for the details. Proposition 3.2. Let the operators operators S(t)be continuous in the weak topology for every fixed tand let the semigroup S(t)possess a bounded absorbing setB. The latter means that for every bounded B⊂H1there exists T=T(B) such that S(t)B⊂ Bfor allt≥T. Then there exists a weak global attractor Awof the semigroup S(t)which is gen- erated by all complete (defined for all t∈R) bounded solutions of problem (2.10): Aw=K/vextendsingle/vextendsingle t=0, (3.1) whereK:={¯u∈Cb(R,H1),¯usolves(2.10)}. Let, in addition, S(t)be asymptotically compact on B. The latter means that for every sequence ¯un 0∈ Band every sequence tn→ ∞, the sequence {S(tn)¯un 0}∞ n=1 is precompact in the strong topology of H1. Then Awis also a strong global attractor for the semigroup S(t). We start with verifying the existence of a weak attractor. Proposition 3.3. Let the assumptions of Theorem 2.5 hold. Then the solution semigroupS(t)generated by equation (1.1)possesses a weak global attractor Aw in the phase space H1. Proof.The existence of a bounded absorbing set Bis an immediate corollary of the dissipative estimate (2.1). We may take B:={¯u∈H1,/ba∇dbl¯u/ba∇dbl2 L2+α/ba∇dbl∇x¯u/ba∇dbl2 L2≤2 γ2/ba∇dblg/ba∇dbl2 L2}. Thus, we only need to check the weak continuity. Let ¯ un 0∈ Bbe a sequence of the initial data weakly converging to ¯ u0: ¯un 0⇁¯u0inH1. Denote by ¯ un(t) :=S(t)¯un 010 A. ILYIN, A. KOSTIANKO, AND S. ZELIK the corresponding solutions. We need to check that for every fixedT, ¯un(T)⇁ ¯u(T) inH1, where ¯u(t) :=S(T)¯u0. To see this we recall that ¯ unis bounded uniformly with respect to nin L∞(0,T;H1) due to estimate (2.1). Moreover, from equation (2.10) we se e also that∂t¯unis uniformly bounded in the same space. Thus, passing to a sub se- quence, if necessary, we may assume that ¯ un(t)⇁ v(t) for every t∈[0,T] and ∂t¯un⇁∂tvinL2(0,T;H1)forsomefunction v(t)suchthatv,∂tv∈L∞(0,T;H1). So, it remains to verify that v(t) =S(t)¯u0by passing to the limit in equations (2.10) for functions ¯ un. This passing to the limit is obvious for linear terms, so we on ly need to prove the convergence of the nonlinear term B(¯un,¯un). In turn, this is the same as to prove that, in the sense of distributions, (¯un,∇x)¯un= div(¯un⊗¯un)⇁div(v⊗v) = (v,∇x)v. The last statement will be proved if we check that ¯un⊗¯un⇁v⊗vinL2((0,T)×Ω). (3.2) To verify (3.2), we recall that the sequence ¯ un⊗¯unis uniformly bounded in L2due to dissipative estimate (2.1) and the embedding H1((0,T)×Ω)⊂L4. Moreover, since the embedding H1((0,T)×R3)⊂L2((0,T);L2 loc(Ω)) is compact, we have the strong convergence ¯ un→vinL2((0,T);L2 loc(Ω)) and, therefore, the convergence ¯ un→valmost everywhere. Since the sequence ¯ un⊗¯unis uniformly bounded in L2((0,T)×Ω), we may assume without loss of generality that it is weakly convergent to some ψ∈L2((0,T)×Ω). Along with the established convergence almost everywhere this implies that ψ=v⊗v, see e.g. [30], and proves (3.2). Thus, we have proved that vsolves the equation (2.10) and by the uniqueness v(t) = ¯u(t). This finishes the proof of weak continuity of the operators S(t) and the existence of a weak global attractor now follows from Pro position 3.2. The theorem is proved. /square We are now ready to verify the existence of a strong global att ractor. Proposition 3.4. Let the assumptions of Theorem 2.5 hold. Then the solution semigroupS(t)generated by equation (2.10)possesses a strong global attractor A=Asin the phase space H1. Proof.According to Proposition 3.2, we only need to verify the asym ptotic com- pactness of S(t) onB. We will use the so-called energy method for this purpose, see [3, 36] for more details. Let{¯un 0} ⊂ B, lettn→ ∞be arbitrary and let ¯ un(t) :=S(tn)¯un 0. Define also ¯vn(t) := ¯un(t+tn). Then these functions are defined on the time intervals t∈[−tn,∞) and, due to the existence of a weak global attractor, withou t loss of generality, we may assume that ¯ v(t)⇁¯u(t) inH1for allt∈Rto some completeDAMPED 3D EULER–BARDINA EQUATIONS 11 trajectory ¯u∈ K. In particular, ¯vn(0) =S(tn)¯un 0⇁¯u(0) (3.3) and we only need to check that this convergence is strong. It is convenient to use the equivalent norm (2.2) in the space H1. Then, the strong convergence in (3.3) will be proved if we verify that /ba∇dbl¯vn(0)/ba∇dbl2 α→ /ba∇dbl¯u(0)/ba∇dbl2 α. (3.4) To see this we integrate the energy identity (2.12) for ¯ vn(t) in time and get /ba∇dbl¯vn(0)/ba∇dbl2 α=/ba∇dbl¯un 0/ba∇dbl2 αe−2γtn+/integraldisplay0 −tne2γs(g,¯vn(s))ds. (3.5) Passing to the limit n→ ∞in this relation and using the weak convergence of ¯vnto ¯uand uniform boundedness of ¯ vnand the initial data ¯ un 0, we conclude that lim n→∞/ba∇dbl¯vn(0)/ba∇dbl2 α=/integraldisplay0 −∞e2γs(g,¯u(s))ds. (3.6) On the other hand, integrating the energy identity for the li mit solution ¯ uin time, we arrive at /ba∇dbl¯u(0)/ba∇dbl2 α=/integraldisplay0 −∞e2γs(g,¯u(s))ds. (3.7) Equalities(3.6)and(3.7)imply(3.4), thereforetheconve rgencein(3.3)isactually strong. Thus, the desired asymptotic compactness is proved and the proposition is also proved. /square Remark 3.5. Since the operator B(¯u,¯u) is regularizing, one can easily increase the regularity of the global attractor Ausing the decomposition of the semigroup into the decaying linear part and the regularizing nonlinea r part (see [17]): S(t) :=L(t)+K(t), wherev(t) =L(t)¯u0solves ∂tv+γv= 0, v/vextendsingle/vextendsingle t=0= ¯u0 andw(t) :=K(t)¯u0satisfies ∂tw+γw+B(¯u,¯u) =AαΠg, w/vextendsingle/vextendsingle t=0= 0. Combining this decomposition with bootstrappingargument s, we may check that theregularity oftheattractor Aisrestricted bytheregularity of gonlyanditwill beC∞-smooth ifg∈H∞(R3). Moreover, using the proper weighted estimates, see [35], we may get the estimates on the rate of decay for solu tions belonging to the attractor as |x| → ∞in terms of the decay rate of gwhich clarify the reason whyAis compact. However, all these estimates do not seem very hel pful for estimation of the attractor dimension (since they grow rapi dly with respect to γ,α→0) and therefore we will not go into more details here.12 A. ILYIN, A. KOSTIANKO, AND S. ZELIK 4.Upper bounds for the fractal dimension In this section we derive upper bounds for the fractal dimens ion of the at- tractor A. As usual for the Navier–Stokes type equations, these bound s will be obtained by means of the volume contraction method, see [2, 1 0, 44] and the references therein. On the analytical side, the Lieb–Thirr ing inequalities for L2- orthonormal families [28, 29] which are an indispensable to ol for the dimension estimates of the attractors for the Navier–Stokes equation s are replaced in our case by the collective Sobolev inequalities for H1-orthonormal families and are proved in the Appendix A. Furthermore, since system (1.1) in the 2D case has already be en studied in [22] (for the case Ω = T2), we will concentrate here on the 3D case only. Theorem 4.1. Suppose that Ωis either the 3D torus T3, or a bounded domain Ω⊂R3(endowed with Dirichlet BC), or the whole space Ω =R3. Letg∈ [L2(Ω)]d(in the case of T3we also assume that ghas zero mean). Then the global attractor Acorresponding to the regularized damped Euler system (1.1) has finite fractal dimension satisfying the following estim ate: dimFA≤1 12π/ba∇dblg/ba∇dbl2 L2 α5/2γ4. (4.1) Proof.The solution semigroup S(t) :H1→H1is smooth with respect to the initial data (see Corollary 2.6), so we only need to estimate then-traces for the linearization of equation (2.10) over trajectories on the a ttractor. This lineariza- tion of (1.1) reads:/braceleftigg ∂t¯θ=−γ¯θ−B(¯u(t),¯θ)−B(¯θ,¯u(t)) =:Lu(t)¯θ, div¯θ= 0,¯θ/vextendsingle/vextendsingle t=0=¯θ0∈H1(Ω),(4.2) whereB(¯u,¯v) :=AαΠ((¯u,∇x)¯v). In order to utilize the well-known cancelation property ((¯u,∇x)¯θ,¯θ)≡0 for the inertial term in the Navier-Stokes equations, it is n atural to endow the spaceH1with the scalar product (¯θ,¯ξ)α= (¯θ,¯ξ)+α(∇x¯θ,∇x¯ξ) = ((1−αA)¯θ,¯ξ) (4.3) associated with the norm (2.2). Then, using that Π Aα=Aαand Π¯θ=¯θ, we get the cancelation (B(¯u,¯θ),¯θ)α=/parenleftbig AαΠ(¯u,∇x)¯θ,(1−α∆x)¯θ/parenrightbig = =/parenleftbig AαΠ(¯u,∇x)¯θ,(1−αΠ∆x)¯θ/parenrightbig =/parenleftbig Π(¯u,∇x)¯θ,¯θ/parenrightbig = ((¯u,∇x)¯θ,¯θ)≡0 of the most singular term B(¯u,¯θ) and, therefore, only the more regular term B(¯θ,¯u) will impact the trace estimates.DAMPED 3D EULER–BARDINA EQUATIONS 13 Following the general strategy, see e.g. [44], the n-dimensional volume contrac- tion factors ωn(A) (=the sums of the first nglobal Lyapunov exponents) which control the dimension can be estimated from above by the foll owing numbers: q(n) := limsup t→∞sup u(t)∈Asup {¯θj}n j=11 t/integraldisplayt 0n/summationdisplay j=1(Lu(τ)¯θj,¯θj)αdτ, where the first (inner) supremum is taken over all orthonorma l families {¯θj}n j=1 with respect to the scalar product ( ·,·)αinH1: (¯θi,¯θj)α=δij,divθj= 0, (4.4) and the second (middle) supremum is over all trajectories u(t) on the attractor A. Then, using the cancellation mentioned above together wit h the pointwise estimate (B.1) proved in Appendix B, we get n/summationdisplay j=1(Lu(t)¯θj,¯θj)α=−n/summationdisplay j=1γ/ba∇dbl¯θj/ba∇dbl2 α−n/summationdisplay j=1((¯θj,∇x)¯u,¯θj)≤ ≤ −γn+/radicalbigg 2 3/integraldisplay Ωρ(x)|∇x¯u(t,x)|dx≤ −γn+/radicalbigg 2 3/ba∇dbl∇x¯u(t)/ba∇dblL2/ba∇dblρ/ba∇dblL2,(4.5) where ρ(x) =n/summationdisplay j=1|¯θj(x)|2. We now use estimate (A.8) from Appendix A /ba∇dblρ/ba∇dblL2≤1 2√πn1/2 α3/4(4.6) and obtainn/summationdisplay j=1(Lu(t)¯θj,¯θj)α≤ −γn+1√ 6πn1/2 α3/4/ba∇dbl∇x¯u(t)/ba∇dblL2. (4.7) Finally, using (2.4), we arrive at q(n)≤ −γn+1 2√ 3πn1/2 α5/4/ba∇dblg/ba∇dblL2 γ. It only remains to recall that, according to the general theo ry,ωn(A)≤q(n) and any number n∗for whichωn∗(A)≤0 andωn(A)<0 forn > n∗is an upper bound both for the Hausdorff [2, 44] and the fractal [6, 7] dime nsion of the global attractor A. This gives the desired estimate dimFA≤1 12π/ba∇dblg/ba∇dbl2 L2 α5/2γ4 and finishes the proof of the theorem. /square14 A. ILYIN, A. KOSTIANKO, AND S. ZELIK Remark 4.2. Estimates (1.5) and (1.6) for T2(and the fact that it is sharp) were proved in [22]. The upper bound for R2is exactly the same once we now know (A.8) for R2. For a bounded domain we only need to replace in the proof in [22] the estimates of the solutions on the attractor by (2. 4). Alternatively, one can go through the proof of Theorem 4.1 and replace the 3D cons tants by their 2D counterparts accordingly. 5.Sharp lower bound on T3 The aim of this section is to show that estimate (4.1) for syst em (1.1) on T3= [0,2π]3is sharp in the limit as α→0. We consider a family of right-hand sides g=gs= g1=γλ(s)sinsx3, g2= 0, g3= 0,(5.1) depending only on x3and parameterized by s∈N,s≫1. The amplitude functionλ(s) will be specified in the course of the proof. Corresponding t o the familygsis the family of stationary solutions of (1.1) /vector u0(x3) = u0(x3) =λ(s)sinsx3, 0, 0,(5.2) withp= 0. In fact, ¯/vector u0= (1−α∆x)−1/vector u0= (¯u0,0,0)T also depends only on x3and therefore ( ¯/vector u0,∇x)¯/vector u0= 0. We now consider system (1.1) linearized on the stationary so lution (5.2) ∂tw+ ¯u0∂¯w ∂x1+ ¯w3∂¯u0 ∂x3e1+γw+∇xq= 0, divw= 0,(5.3) wheree1= (1,0,0)Tand ¯w= (1−α∆x)−1w. The standing assumption is/integraldisplay T3w(x,t)dx= 0. (5.4) We shall look for the solution of the linear problem (5.3) in t he form w(x,t) = w1(x3) w2(x3) w3(x3) ei(ax1+bx2−act), q(x,t) =q(x3)ei(ax1+bx2−act),(5.5)DAMPED 3D EULER–BARDINA EQUATIONS 15 wherea,b∈Zso thatwandqare 2π-periodic in each xi. If such a solution of (5.3) is found, then substituting (5.5) into (5.3) and set tingt= 0 we see that w(x,0) = w1(x3) w2(x3) w3(x3) ei(ax1+bx2) is a vector-valued eigenfunction of the stationary operato r L3(/vector u0)w= ¯u0∂¯w ∂x3¯u+ ¯w3∂¯u0 ∂x3e1+γw+∇xq (5.6) andiacis the corresponding eigenvalue. If Re( iac)<0, then the corresponding mode is unstable. We substitute (5.5) into (5.3) and obtain the system −γw1−ia(¯u0¯w1−cw1) =iaq+ ¯w3u′ 0 −γw2−ia(¯u0¯w2−cw2) =ibq, −γw3−ia(¯u0¯w3−cw3) =q′, iaw1+ibw2+w′ 3= 0,(5.7) where′=∂/∂x3. Lemma 5.1. There are no unstable solutions of equation (5.3) ∂tw=L3(/vector u0)w that can be written in the form (5.5)witha= 0. Proof.Leta= 0. Then the solutions of (5.3) are sought in the form w(x,t) = w1(x3) w2(x3) w3(x3) ei(bx2−ct), q(x) =q(x3)ei(bx2−ct), and (5.7) goes over to −γw1+icw1= ¯w3u′ 0 −γw2+icw2=ibq, −γw3+icw3=q′, ibw2+w′ 3= 0. Letb/\e}atio\slash= 0. Then w2=−w′ 3/(ib). Substituting this into the second equation and differentiating the third with respect to x3we obtain q′′=b2q, which gives that q= 0, since qis periodic. Since we are looking for unstable solutions, it follows that Re( ic)<0 and therefore −γ+ic/\e}atio\slash= 0. This gives that w2=w3= 0, and, finally, w1= 0. Ifa=b= 0, thenw′ 3= 0, andw3= 0 by periodicity and zero mean condition. This givesq= 0 andw1=w2= 0. The proof is complete. /square16 A. ILYIN, A. KOSTIANKO, AND S. ZELIK 5.1.Squire’s transformation. We now reduce the 3D instability analysis to the instability analysis of the transformed 2D problem. The key role is played by the Squire’s transformation (see [41], [12], [33]). Since we a looking for unstable solutions of (5.3), in view of Lemma 5.1 we may assume that a/\e}atio\slash= 0 in (5.7). Multiplying the first equation in (5.7) by aand the second by ba adding up the results, we obtain −/hatwideγ/hatwidew1−i/hatwidea(¯u0¯/hatwidew1−/hatwidec/hatwidew1) =i/hatwidea/hatwideq+¯/hatwidew3u′ 0, −/hatwideγ/hatwidew3−i/hatwidea(¯u0¯/hatwidew3−/hatwidec/hatwidew3) =/hatwideq′, i/hatwidea/hatwidew1+/hatwidew′ 3= 0,(5.8) where /hatwidea2=a2+b2,/hatwidew1=aw1+bw2 /hatwidea,/hatwidew3=w3, /hatwideγ=γ/hatwidea a,/hatwideq=q/hatwidea a,/hatwidec=c.(5.9) The solutions of this problem on the 2d torus T2 |/hatwidea|=x1∈[0,2π/|/hatwidea|], x3∈[0,2π] are sought in the form /hatwidew(x1,x3,t) =/parenleftigg /hatwidew1(x3) /hatwidew3(x3)/parenrightigg ei(/hatwideax1−/hatwidea/hatwidect),/hatwideq(x1,x3,t) =q(x3)ei(/hatwideax1−/hatwidea/hatwidect),(5.10) and if such a solution is found, then the vector function /hatwidew(x1,x3,0) =/parenleftigg /hatwidew1(x3) /hatwidew3(x3)/parenrightigg ei/hatwideax1(5.11) is a vector-valued eigenfunction with eigenvalue i/hatwidea/hatwidecof the stationary operator L2(/vector u0)/hatwidew=/hatwideγ/hatwidew+ ¯u0∂/hatwide¯w ∂x1+/hatwide¯w3∂¯u0 ∂x3e1+∇x/hatwideq,div/hatwidew= 0,(5.12) onT2 |/hatwidea|, where the stationary solution and the generating right-ha nd side are /vector u0(x3) =/braceleftigg u0(x3) =λ(s)sinsx3, 0,gs(x3) =/braceleftigg g1(x3) =λ(s)/hatwideγsinsx3, 0,(5.13) and where as before ¯ u0= ¯u0(x3) = (1−∆x)−1u0. To avoid unnecessary complications we assume in what follow s that/radicalbig a2+b2=/hatwidea>0, a>0, and formulate the main result on the Squire’s reduction of th e 3D instability analysis to the 2D case.DAMPED 3D EULER–BARDINA EQUATIONS 17 Lemma 5.2. Let/hatwidewin(5.11)be an unstable eigenfunction of the operator (5.12) on the torus T2 /hatwidea= [0,2π//hatwidea]×[0,2π]. Then for any pair of integers a,b∈Zwith a2+b2=/hatwidea2 there exist an unstable solution of system (5.7)onT3= [0,2π]3. Proof.Once the/hatwide·-variables are known, q,w3,candγare found from (5.9). It remains to find w1andw2. We consider the second equation in (5.7): Aw2:= (−γ+iac)w2−ia¯u0¯w2=ibq. Since/hatwidewis unstable, Re( iac)<0 and therefore Re( −γ+iac)<0. Suppose that Aw2= 0 for some w2. Taking the scalar product in (complex) L2(0,2π) withw2 and taking into account that the second term is purely imagin ary we obtain for the real part Re(−γ+iac)/ba∇dblw2/ba∇dbl2 L2= 0, which gives that w2= 0, andAhas a trivial kernel. In addition, Ais a Fredholm operator, since the second term is compact (smoothing). Hen ce it has a bounded inverse, and since qis known, we have found w2. Finally, w1= (/hatwidea/hatwidew1−bw2)/a. /square 5.2.Instability analysis on T2.We now have to recall the instability analysis for the 2D problem that was carried out in detail in our previo us work [22]. The problem was studied on the standard torus T2= [0,2π]2and we now denote the second coordinate by x3, so thatx1,x3are the coordinates on T2. The family of the forcing terms and the corresponding stationary solutio ns are as in (5.13) and the linearized stationary operator is precisely (5.12). Ap plying curl to (5.12) we obtain the equivalent scalar operator in terms of the vortic ity whose spectrum was studied in [22] Lsω:=J/parenleftbig (∆x−α∆2 x)−1ωs,(1−α∆x)−1ω/parenrightbig + +J/parenleftbig (∆x−α∆2 x)−1ω,(1−α∆x)−1ωs/parenrightbig +γω=−σω,(5.14) where J(a,b) =∇a·∇⊥b=∂x1a∂x3b−∂x3a∂x1b, and ωs= curl/vector u0=−λ(s)scossx2, ω= curl/hatwidew. The following result was proved in [22] (see Theorem 4.1 and C orollary 4.2.) Theorem 5.3. Given a large integer s>0let a fixed pair of integers t,rbelong to a bounded region A(δ)defined by conditions t2+r2<s2/3, t2+(−s+r)2>s2, t2+(s+r)2>s2, t≥δs,(5.15)18 A. ILYIN, A. KOSTIANKO, AND S. ZELIK where0<δ<1/√ 3. There exists an absolute constant c1such that for λ≥λ2(s,γ) =c1γ(1+αs2)2 s(5.16) in(5.13)the linear operator Lson the torus T2= [0,2π]2has a real negative (unstable) eigenvalue σ <0of multiplicity 2. The corresponding eigenfunctions are ω1(x1,x3) =∞/summationdisplay n=−∞at,sn+rcos(tx1+(sn+r)x3), ω2(x1,x3) =∞/summationdisplay n=−∞at,sn+rsin(tx1+(sn+r)x3).(5.17) We now observe that ω1andω2in (5.17) are the real and imaginary parts of the complex-valued eigenfunction ω1(x1,x3)+iω2(x1,x3) =/bracketleftigg∞/summationdisplay n=−∞at,sn+rei(sn+r)x3/bracketrightigg eitx1. Recovering the corresponding divergence free vector funct ion, that is, applying the operator ∇⊥ x∆−1 x, we obtain an unstable vector valued eigenfunction of the operatorL2(/vector u0) written in the required form (5.11): w(x1,x3) =/parenleftigg w1(x3) w3(x3)/parenrightigg eitx1. For the 3D instability analysis below we need to repeat the co nstruction of an unstable eigenmode on the torus T2 εwithx1∈[0,2π/ε],x2∈[0,2π], whereε>0 is arbitrary (not necessarily small). Proposition 5.4. Letrandt′:=tεbelong to the region A(δ): t′2+r2<s2/3, t′2+(−s+r)2>s2, t′2+(s+r)2>s2, t′≥δs.(5.18) Letλbe defined in (5.16)and letgsand/vector u0be the same as before but in two dimensions: gs(x3) = (γλ(s)sinsx3,0)T, /vector u0(x3) = (λ(s)sinsx3,0)T. Then there exists an unstable solution w(x1,x3) =/parenleftigg w1(x3) w3(x3)/parenrightigg eitεx1, x∈T2 ε. (5.19) of the form (5.11)of the operator (5.12)on the torus T2 ε. Proof.Following the proof of Theorem 4.1 in [22] we see that a word fo r word repetition of it shows that if t′=εt,rsatisfy (5.18), then the correspondingDAMPED 3D EULER–BARDINA EQUATIONS 19 operator (5.14) has an unstable (real negative) eigenvalue of multiplicity two with eigenfunctions ω1(x1,x3) =∞/summationdisplay n=−∞at,sn+rcos(tεx1+(sn+r)x3), ω2(x1,x3) =∞/summationdisplay n=−∞at,sn+rsin(tεx1+(sn+r)x3), from which we construct the required vector valued complex e igenfunction (5.19) as before. /square It is convenient for us to single out a small rectangle Din the (t′,r)-plane inside the region A(δ) defined by (5.18), see Fig.1: |r| ≤c2s,0<c3s≤t′=tε≤c4s. (5.20) Hereδ=δ∗∈(0,1/√ 3) is fixed, and all the constants ciare absolute constants, whose explicit values can easily be specified. δsr t′,tA(δ) D Figure 1. The region A(δ) and the rectangle D.20 A. ILYIN, A. KOSTIANKO, AND S. ZELIK 5.3.3D lower bound. We can now formulate the main results of this section. Theorem 5.5. We consider the linearized system on the 3D torus T3= [0,2π]3 with right-hand side gsand stationary solution /vector u0given in (5.1)and(5.2), where λ=λ3(s) =√ 2λ2(s,γ) =√ 2c1γ(1+αs2)2 s. (5.21) Then for each triple of integers a,b,rsatisfying c3s≤/hatwidea=/radicalbig a2+b2≤c4s,|r| ≤c2s, a≥ |b|,(5.22) there exists an unstable solution of the linearized operato r(5.3). Proof.We fixa,b,rsatisfying (5.22). Then in view of the first two inequalities in (5.22) the pair ( t′,r)∈D⊂A(δ), wheret′=/hatwidea·1 (so that we set t= 1). Applying Squire’s transformation we obtain a 2D linearized problem on the torus T2 /hatwideaof the form (5.12) with /hatwideγ=γ/hatwidea/a. Then in view of the third inequality in (5.22) we have λ=√ 2λ2(s,γ) =λ2(s,√ 2/hatwideγa//hatwidea)≥λ2(s,/hatwideγ). We now see from Proposition 5.4 that the 2D linearized proble m (5.12) has an unstable eigenvalue. Lemma 5.2 says, in turn, that so does the linearized problem (5.6) on the standard torus T3= [0,2π]3. /square The number of integers ( a,b,r) satisfying (5.22) is of order c5s3, where c5=1 4πc2(c2 4−c2 3). Theorem 5.6. Let the right-hand side in (1.1)begsdefined in (5.1)withλ(s) defined in (5.21). Then the dimension of the corresponding attractor A=As satisfies for an absolute constant c6the lower bound dimFA≥c6/ba∇dblgs/ba∇dbl2 L2 α5/2γ4. (5.23) Proof.We study our system in the limit α→0. Sincesis at our disposal we set s=1√α. Thenλin (5.21) and /ba∇dblgs/ba∇dbl2 L2in (5.1) become λ=c7γ√α,/ba∇dblgs/ba∇dbl2 L2=c8γ4α. We can finally write dimFA≥c5s3=c51 α3/2=c5α/ba∇dblgs/ba∇dbl2 L2 α5/2/ba∇dblgs/ba∇dbl2 L2=c5α/ba∇dblgs/ba∇dbl2 L2 α5/2c8αγ4, which gives (5.23) with c6=c5/c8. /squareDAMPED 3D EULER–BARDINA EQUATIONS 21 Remark 5.7. We recall the sharplower boundin [22] for the attractor dime nsion for system (1.1) on the 2D torus T2: dimFA≥cabs/ba∇dblcurlgs/ba∇dbl2 L2 αγ4. (5.24) Since forλ∼γ√αwe have /ba∇dblcurlgs/ba∇dbl2 L2∼γ4and/ba∇dblgs/ba∇dbl2 L2∼γ4α, it follows that estimate (5.24) can equivalently be written in terms of the o ther dimensionless number in (1.8) as follows dimFA≥c′ abs/ba∇dblgs/ba∇dbl2 L2 α2γ4. Appendix A.Collective Sobolev inequalities for H1-orthonormal families We prove here some spectral inequalities for orthonormal fa milies of functions which are the key technical tool in our derivation of sharp up per bounds for the attractor dimension. These inequalities are somehow co mplementary to the classical Lieb–Thirring inequalities for orthonormal sys tems inL2and in our case we estimate the proper norms of the same quantity ρ(x) =/summationtextn i=1|¯θi(x)|2, but for families{¯θi}n i=1that are orthonormal in H1with norm (2.2) depending on α. Our exposition utilizes the ideas from [27] as well as extend s the results of [22] to 3D case. We start with the case Ω = Rdor Ω⊂Rdwith Dirichlet boundary condi- tions (analogously to the classical Lieb–Thirring inequal ity, the case of Dirichlet boundary conditions follows from the case of whole space by z ero extension). A.1.The case of the whole space and a domain with Dirichlet BC. Theorem A.1. LetΩ⊆Rdbe an arbitrary domain. Let a family of vector functions {¯θi}n i=1∈H1(Ω)withdiv¯θi= 0be orthonormal with respect to the scalar product m2(¯θi,¯θj)L2+(∇¯θi,∇¯θj)L2=m2(¯θi,¯θj)L2+(curl¯θi,curl¯θj)L2=δij,(A.1) Then the function ρ(x) :=/summationtextn j=1|¯θj(x)|2satisfies /ba∇dblρ/ba∇dblL2≤1 2√πn1/2 m, d= 2, /ba∇dblρ/ba∇dblL2≤1 2√πn1/2 m1/2, d= 3.(A.2) Proof.We first let Ω = Rdand introduce the operators H=V1/2(m2−∆x)−1/2Π,H∗= Π(m2−∆x)−1/2V1/2 acting in [L2(Rd)]d, whereV∈L2(Rd) is a non-negative scalar function which will be specified below and Π is the Helmholtz–Leray projecti on to divergence22 A. ILYIN, A. KOSTIANKO, AND S. ZELIK free vector fields. Then K=H∗His a compact self-adjoint operator acting from [L2(Rd)]dto [L2(Rd)]dand TrK2=Tr/parenleftig Π(m2−∆x)−1/2V(m2−∆x)−1/2Π/parenrightig2 ≤ ≤Tr/parenleftbig Π(m2−∆x)−1V2(m2−∆x)−1Π/parenrightbig = = Tr/parenleftbig V2(m2−∆x)2Π/parenrightbig , where we used the Araki–Lieb–Thirring inequality for trace s [1, 29, 40]: Tr(BA2B)p≤Tr(BpA2pBp), p≥1, and the cyclicity property of the trace together with the fac ts that Π commutes with the Laplacian and that Π is a projection: Π2= Π. We want to show that TrK2≤ 1 4π1 m2/ba∇dblV/ba∇dbl2 L2, d= 2; 1 4π1 m/ba∇dblV/ba∇dbl2 L2, d= 3.(A.3) Indeed, the fundamental solution of ( m2−∆x)2Π inRdis ad×dmatrix Fd ij(x) =Gd(x)δij−∂xi∂xj∆−1Gd(x) withRd-trace atx∈Rd TrRdFd(x) =dGd(x)−d/summationdisplay i=1∂2 xixi∆−1 xGd(x) = (d−1)Gd(x), whereGd(x) is the fundamental solution of the scalar operator ( m2−∆x)2in the whole space Rd: Gd(x) =1 (2π)d/integraldisplay Rdeiξxdξ (m2+|ξ|2)2= 1 8π1 me−|x|m, d = 3; 1 4π1 m2|x|mK1(|x|m), d= 2.(A.4) The first equality here follows from (A.12), while for the sec ond we have (since the function is radial and using formula 13.51(4) in [46]) G2(x)=1 2πF−1/parenleftbig (m2+|ξ|2)2/parenrightbig =1 2π/integraldisplay∞ 0J0(|x|r)rdr (m2+r2)2=1 4π1 m2|x|mK1(|x|m), whereK1is the modified Bessel function of the second kind. Thus, the operator V2(m2−∆x)2Π has the matrix-valued integral kernel V(y)2Fd(x−y)DAMPED 3D EULER–BARDINA EQUATIONS 23 and therefore Tr(V2(m2−∆x)2Π) = =/integraldisplay RdTrRd/parenleftbig V(y)2Fd(0)/parenrightbig dy= (d−1)/ba∇dblV/ba∇dbl2 L2Gd(0) (A.5) which along with (A.4) proves the first inequality in (A.3), a nd also the second one, since ( tK1(t))|t=0= 1. We can now complete the proof as in [27]. Setting ψi:= (m2−∆x)1/2¯θi, we see from (A.1) that {ψj}n j=1is an orthonormal family in L2. We observe that /integraldisplay Rdρ(x)V(x)dx=n/summationdisplay i=1/ba∇dblHψi/ba∇dbl2 L2. By the orthonormality of the ψj’s inL2and the definition of the trace we obtain n/summationdisplay i=1/ba∇dblHψi/ba∇dbl2 L2=n/summationdisplay i=1(Kψi,ψi)≤n/summationdisplay i=1/ba∇dblKψi/ba∇dblL2≤n1/2/parenleftiggn/summationdisplay i=1/ba∇dblKψi/ba∇dbl2 L2/parenrightigg1/2 = =n1/2/parenleftiggn/summationdisplay i=1(K2ψi,ψi)/parenrightigg1/2 ≤n1/2/parenleftbig TrK2/parenrightbig1/2.(A.6) This gives /integraldisplay Rdρ(x)V(x)dx≤n1/2/parenleftbig TrK2/parenrightbig1/2. SettingV(x) :=ρ(x) and using (A.3), we complete the proof of (A.2) for the case of Ω =Rd,d= 2,3. Finally, if Ω is a proper domain in Rd, we extend by zero the vector functions ¯θjoutside Ω and denote the results by/tildewide¯θj, so that/tildewide¯θj∈H1(Rd) and div/tildewide¯θj= 0. We further set /tildewideρ(x) :=/summationtextn j=1|/tildewide¯θj(x)|2. Then setting /tildewideψi:= (m2−∆x)1/2/tildewide¯θi, we see that the system {/tildewideψj}n j=1is orthonormal in L2(Rd) and div/tildewideψj= 0. Since clearly /ba∇dbl/tildewideρ/ba∇dblL2(Rd)=/ba∇dblρ/ba∇dblL2(Ω), the proof of estimate (A.2) reduces to the case of Rdand therefore is complete. /square The proved result can be rewritten in terms of orthogonal fun ctions with re- spect to inner product (4.3) as follows. Corollary A.2. Let the assumptions of Theorem A.1 hold and let {¯θj}n j=1, div¯θj= 0be an orthonormal system with respect to (¯θi,¯θj)L2+α(∇¯θi,∇¯θj)L2=δij. (A.7)24 A. ILYIN, A. KOSTIANKO, AND S. ZELIK Thenρ(x) =/summationtextn j=1|¯θj(x)|2satisfies /ba∇dblρ/ba∇dblL2≤1 2√πn1/2 α1/2, d= 2, /ba∇dblρ/ba∇dblL2≤1 2√πn1/2 α3/4, d= 3.(A.8) Indeed, this statement follows from (A.2) by the proper scal ing. A.2.The case of periodic BC: Estimates for the lattice sums. We now turn to the case Ω = Td. In this case, we naturally have an extra condition that the considered functions have zero mean. Analogously t o the case Ω = Rd, the Laplacian commutes with the Helholtz–Leray projection , so we may define analogously the operator Kand get exactly the same expression (A.5) for its trace. The only difference is that now Gd(x) =Gd,m(x) a fundamental solution of the scalar operator ( m2−∆x)−2on the torus Td(with zero mean condition), so the integral in (A.4) should be replaced by the correspond ing sum over the latticeZd 0=Zd\{0}: Gd(x) =1 (2π)d/summationdisplay k∈Zd 0eik·x (m2+|k|2)2. (A.9) Thus, in order to get estimates (A.2) for the torus T2arguing as in the proof of Theorem A.1, we only need to check that Gd,m(0)< 1 8π1 m, d= 3; 1 4π1 m2, d= 2.(A.10) for allm≥0. Unfortunately, we do not know explicit expressions for th e sum (A.9), so we need to do rather accurate estimates of the assoc iated lattice sum in order get (A.10) based on the Poisson summation formula. I n the cased= 2, (A.10) is proved in [22] and the case d= 3 is considered in Proposition A.4 below. Thus, the following result holds. Theorem A.3. Let a family of divergence free vector functions with zero me an {¯θi}n i=1∈˙H1(T3)be orthonormal with respect to scalar product (A.1). Then estimates (A.2)hold. Analogously, if this family is orthonormal with respe ct to (A.7), thenρsatisfies inequalities (A.8). As explained before, the case d= 2 is verified in [22] and for proving the result ford= 3, it is sufficient to prove the following proposition. Proposition A.4. The following inequality holds for all m≥0: F(m) :=m/summationdisplay k∈Z3 01 (|k|2+m2)2<π2(A.11)DAMPED 3D EULER–BARDINA EQUATIONS 25 Proof.Before we go over to the proof, we first observe that in R3we have the equality/integraldisplay R3dx (|x|2+m2)2=π2 m and, secondly, it is the absence of the term with k= 0 in the sum in (A.11) that makes inequality (A.11) hold at all. We use the Poisson summation formula (see, e.g., [42]) /summationdisplay k∈Znf(k/m) = (2π)n/2mn/summationdisplay k∈Zn/hatwidef(2πkm), where F(f)(ξ) =/hatwidef(ξ) = (2π)−n/2/integraltext Rnf(x)e−iξxdx. For the function f(x) = 1/(1+|x|2)2x∈R3with/hatwidef(ξ) =π2 (2π)3/2e−|ξ|(A.12) (see [42]), and with/integraltext R3f(x)dx=π2this gives F(m) =1 m3/summationdisplay k∈Z3f(k/m)−1 m3= =π2/summationdisplay k∈Z3e−2πm|k|−1 m3=π2+π2/summationdisplay k∈Z3 0e−2πm|k|−1 m3.(A.13) In particular, this formula gives a convenient way to comput eF(m) numerically for the case where mis not very small. We start the proof of inequality (A.11) with the case m≤1. Lemma A.5. Inequality (A.11)holds for all m∈[0,1]. Proof.Since F′(m) =/summationdisplay k∈Z3 0/parenleftbigg1 (|k|2+m2)2−4m2 (|k|2+m2)3/parenrightbigg =/summationdisplay k∈Z3 0|k|2−3m2 (|k|2+m2)3, we see that all terms in the sum for F(m) with|k|2≥3 are monotone increasing (after multiplying by m) with respect to m≤1, so we may write F(m)≤6m (1+m2)2+12m (2+m2)2+/summationdisplay |k|2≥31 (|k|2+1)2≤ ≤max m∈[0,1]/braceleftbigg6m (1+m2)2/bracerightbigg + max m∈[0,1]/braceleftbigg12m (2+m2)2/bracerightbigg +F(1)−6 4−12 9= =9√ 3 8+9√ 6 16−3 2−4 3+π2(1.01306)−1 = 9.4915<π2= 9.8696, where we have used (A.13) in order to compute F(1) =π2(1.01306)−1 (the calculations are reliable since the series has an exponenti al rate of convergence). Thus, the lemma is proved. /square26 A. ILYIN, A. KOSTIANKO, AND S. ZELIK We now turn to the case m≥1. Lemma A.6. Inequality (A.11)holds for all m≥1. Proof.It follows from (A.13) that inequality (A.11) goes over to G(m) :=π2m3/summationdisplay k∈Z3 0e2πm|k|−1<0. We use the inequality |k| ≥1√ 3(|k1|+|k2|+|k3|) for all terms with |k|>1 and leave the first 6 terms with |k|= 1 unchanged. This gives G(m)≤π2m3/summationdisplay k∈Z3 0e2πm(|k1|+|k2|+|k3|)/√ 3−1+6π2m3/parenleftig e−2πm−e−2πm/√ 3/parenrightig and we only need to prove that the right-hand side of this ineq uality is negative. Summing the geometric progression, we get G(m)≤G0(m) :=π2m3/parenleftigg/parenleftbigg 1+2 e2πm/√ 3−1/parenrightbigg3 −1/parenrightigg −1+ +6π2m3/parenleftig e−2πm−e−2πm/√ 3/parenrightig = 6π2m3/parenleftbigg1 e2πm/√ 3−1−e−2πm/√ 3/parenrightbigg + +12π2/parenleftigg m3/2 e2πm/√ 3−1/parenrightigg2 +8π2/parenleftbiggm e2πm/√ 3−1/parenrightbigg3 +6π2m3e−2πm−1 = = 6π2ψ1(m)+12π2ψ2(m)2+8π2ψ3(m)3+ψ4(m). We claim that all functions ψi(m) are monotone decreasing for m≥1. Indeed, the function ψ3(m) is obviously decreasing for all m≥0. The function ψ4(m) is decreasing for m≥3 2π<1. Analogously, as elementary calculations show, the second function is decreasing for m≥m2<1 where m0=√ 3 4π/parenleftig 3+2W/parenleftig −3e−3/2/2/parenrightig/parenrightig ≈0.241, whereWis a Lambert W-function. Finally, let us prove the monotonicity of ψ1(m). Indeed, ψ′ 1(m) =m22mπ√ 3e−2πm√ 3/3−4πm√ 3−9e−2πm√ 3/3+9 3(e2πm√ 3/3−1)2 and we see that 2mπ√ 3e−2πm√ 3/3−4πm√ 3−9e−2πm√ 3/3+9< <2πm√ 3/parenleftig e−2πm√ 3/3−1/parenrightig +9−2π√ 3<0DAMPED 3D EULER–BARDINA EQUATIONS 27 ifm≥1, since 9 −2π√ 3<0. Thus,ψ′ 1(m)<0 form≥1 andψ1(m) is also decreasing. Thus, G0(m) is decreasing for m≥1 and we only need to note that G0(1) =−0.7562<0 and the lemma is proved. /square Finally, we have verified (A.11) for all m≥0 and the proof is complete. /square Remark A.7. Of course, the estimates obtained above hold for families of scalar functions {¯θi}n i=1∈H1that are orthonormal with respect to (A.7). In this case, the factor ( d−1) in formula (A.5) is replaced by 1, and we get a√ 2-times better constant in the 3D case and the same constant in the 2D case. Na mely, the functionρ(x) :=/summationtextn i=1|¯θi(x)|satisfies /ba∇dblρ/ba∇dblL2≤1 2√πn1/2 α1/2, d= 2, /ba∇dblρ/ba∇dblL2≤1√ 8πn1/2 α3/4, d= 3.(A.14) These estimates also hold for all three cases Ω = Td, Ω =Rd, and Ω⊂Rdwith Dirichlet boundary conditions. Appendix B.A pointwise estimate for the nonlinear term In this appendix, we prove a pointwise estimate for the inert ial term which corresponds to the Navier–Stokes nonliearity. Proposition B.1. Let for some x∈Rd,u(x)∈Rdanddivu(x) = 0. Then |((θ,∇x)u,θ)(x)| ≤/radicalbigg d−1 d|θ(x)|2|∇xu(x)|, (B.1) where∇xu(x)is ad×dmatrix with entries ∂iuj, and |∇xu|2=d/summationdisplay i,j=1(∂iuj)2. Proof.Basically, this can be extracted from [28]. For the sake of co mpleteness we reproduce the details. We suppose first that Ais a symmetric real d×dmatrix with entries aijand with Tr A= 0. 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Tao, Finite time blowup for an averaged three-dimensional Navie r-Stokes equa- tion, J. Amer. Math. Soc., vol 29 (2016), 601–674 [44] R.Temam, Infinite Dimensional Dynamical Systems in Mechanics and Phy sics,2nd ed. Springer-Verlag, New York 1997. [45] R. Temam, Navier-Stokes equations and nonlinear functional analysi s, vol.66, Siam, 1995. [46] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, Cambridge, 1995. Email address :ilyin@keldysh.ru Email address :aNNa.kostianko@surrey.ac.uk Email address :s.zelik@surrey.ac.uk 1Keldysh Institute of Applied Mathematics, Moscow, Russia 2University of Surrey, Department of Mathematics, Guildfor d, GU2 7XH, United Kingdom. 3School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China 4Imperial College, London SW7 2AZ, United Kingdom.